MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
MA30089: Stochastic Processes and Finance
Tobias Hartung
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
1 Introduction
2 Background and Revision
3 Fair Games & Martingales
4 Binomial Model
5 Fundamental Theorem of Asset pricing
6 Brownian Motion
7 Stochastic Integration
8 Stochastic Calculus
9 Continuous-time Finance
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Financial Risks
• In the modern world, everyone exposed to lots of (financial)
risks: crashing your car, burglary, illness, unemployment. . .
• E.g. can purchase insurance to cover the financial costs of
unexpected events.
• In addition, need to choose between different financial
products which may have random future payoffs:
• For example, fixed vs. variable interest rates for a
mortgage.
• More complex variants: ‘capped’ floating rate mortgage.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Financial Risks
• A company will also face a number of risks that need to be
managed.
• For example, an airline company will sell tickets today, and
purchase fuel at the time of the flight.
• How to decide the price of the ticket today, without
exposing to the risk that fuel prices will change?
• Idea: enter into a contract with a third party today to
supply fuel at a fixed price at a future date.
• Contract may not even involve physical delivery, may just
be the difference between the contracted price, and the
price on the day.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Financial Risks
• Consider an investment manager for a fund — usually
invests in an index of stocks, e.g. FTSE 100.
• Believes that the market is about to enter a period of high
variability — good chance that the index will increase
substantially over the next month, but an outside chance
that the index will fall substantially.
• What should the fund manager do?
• Imagine that she could enter a contract which gave her the
option, at the end of the month, to buy the index of stocks
at a fixed price £K , if she wished, but was not obligated to.
• Good if index rises, but not bad if it falls.
• To persuade someone to enter into such a contract, have
to offer an incentive, for example by paying a fee upfront
to the other party in the contract. This will typically be
thought of as the price of the contract.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Derivatives
The contracts in the last two examples are special examples of
financial derivatives.
In both cases, the value of the financial exchange made
according to the contract derives its value from that of an
underlying asset.
In this case, the other assets are the price of airline fuel, or the
value of the FTSE 100 index.
In both cases, these correspond to the prices of quantities that
can be bought and sold on an exchange or a financial market,
and whose price today is easily determined (for example, by
looking in a newspaper or online).
A key question in this course will be: given some (probabilistic)
model for the underlying asset, what should the price of a
derivative contract be?
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Probability and Pricing
Example 0.1
A key connection will be to probabilities, and a key notion in
derivatives pricing will be that we often need to use a form of
modified probabilities. To see why, consider the following
example:
A bookmaker is taking bets on a two horse race. Based on a
careful analysis, he correctly identifies that the first horse has a
25% chance of winning, and the second horse has a 75% chance
of winning.
As a result, the bookmaker determines that the odds on the
horses should be 3-1 against, and 3-1 on.
Note that these odds correspond to the bookmaker assigning
‘fair’ probabilities of a 25% chance of the first horse winning,
and a 75% chance of the second horse winning.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Probability and Pricing
Suppose however that the bets taken by the bookmaker are not
very sensitive to the odds the bookmaker sets (within reason),
and the bookmaker expects to have £5,000 bet on the first
horse, and £10,000 bet on the second horse.
Then the bookmaker would make a loss of
£5, 000 = 3× 5, 000− 10, 000 if the first horse wins, and a gain
of £1, 667 = 5, 000− 10, 000/3 if the second horse wins.
Note that the average profit of the bookmaker is
£0 = 0.25×−5, 000 + 0.75× 1, 667.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Probability and Pricing
On the other hand, if he were to set odds of 2-1 against for the
first horse, and 2-1 on for the second horse, then the
bookmaker ends up even which ever horse wins.
For a bookmaker, they may prefer the second outcome (at least
once they build in a bit of profit), since their profit or loss will
not be heavily dependent on the outcome of a few races.
Note that the implied probabilities given by the new odds are
different: with the new odds, the new probabilities are 33% for
the first horse, and 66% for the second horse.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Probability and Pricing
This is an observation that will be crucial to much of what we
discover in this course: in the presence of ‘market information’
(here, the amount bettors will place on each horse), the
‘correct’ probabilities may not be the true probabilities.
Since these are the probabilities that expose the bookmaker to
no risk, we will call these the risk-neutral probabilities.
In what follows, we will mostly think of the market information
as the current price of the underlying asset, and we want to use
the risk-neutral probabilities to help us price derivative
contracts.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
2 Background and Revision
Financial Basics
Probability Background
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
The market
We will begin by modelling the cash-flow of an investor in
discrete time. The basic idea is that an investor will have a set
of K assets (for example, shares in all the companies in the
FTSE 100, commodities, etc.), which, at the end of each day,
they can buy and sell.
There will also be one ‘special’ asset, which we will think of as
(interchangeably), cash, a bank account, or a bond.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 1
• Motivation: Derivatives pricing
• Contract to purchase airline fuel for a fixed price at a fixed
future date
• Option to buy shares in FTSE 100 at a fixed price
• Derivative since: value derived from an underlying asset
• Bookmaking Example: ‘risk-neutral’ probabilities
• Financial setting
• Investor can buy/sell a set of assets (e.g. shares)
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
The market
In particular, we will suppose that at time n, the price of the K
assets are S1n ,S
2
n , . . . ,S
K
n . Typically we will suppose that these
are strictly positive random variables.
In addition, we have our ‘special’ asset (the bank account),
which we think about as the value of £1 invested in a bank
account.
The special feature of a bank account is that we never lose
money invested in the bank, and in fact, we gain an amount of
interest which is known at time n. This means if I have £x
invested at time n in the bank, then at time n + 1 I will have
£x˜ , where x˜ > x .
So we can write x˜ = (1+ rn)x , where rn > 0 is the interest rate.
For the rest of the course, we will think of rn as being a fixed
number, r , known at the outset.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Cashflow Example
Example 1.1
Suppose there are two assets (and the bank account), such
that:
S10 = 100,S
2
0 = 50,S
1
1 = 120,S
2
1 = 40,
S12 = 110,S
2
2 = 60, r = 0.1(= 10%).
An investor begins with £1,000, with which they buy 5 units of
asset 1, and 4 units of asset 2 (the remaining money staying in
the bank account). At time 1, they sell 2 units of the second
asset, and buy one more unit of the first asset. How much is
their portfolio worth at time 2, if they do not spend/earn any
additional money?
Example
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Portfolio Definition
Definition 1.2
A portfolio of assets (and cash) is a sequence of vectors,
φn = (φ
0
n, φ
1
n, φ
2
n, . . . φ
K
n )
where φ0n is the cash held at time n, and φ
i
n is the number of
units of asset i held at time n, for i = 1, . . . ,K .
The value of the portfolio at time n is then:
Vn = φ
0
n +
K∑
i=1
φinS
i
n.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Self-financing definition
Definition 1.3
A portfolio is self-financing if:
φ0n+1 +
K∑
i=1
φin+1S
i
n+1 = φ
0
n(1 + r) +
K∑
i=1
φinS
i
n+1.
That is, the value of the portfolio at time n + 1 is equal to the
value at time n + 1 of the portfolio purchased at time n.
If I buy additional units of an asset at time n + 1, I have to find
this money from somewhere, either by selling other assets, or
using cash from the bank account.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Short-selling
We need one more financial observation: we suppose that we
can short sell the assets — that is, sell units of the asset that
we do not hold, on the basis that we must buy them back at a
later date.
In terms of the portfolio we hold, this corresponds to negative
values of φin.
We can also borrow money from the bank (at the interest rate
r), which corresponds to φ0n being negative.
If a trader has φin > 0, we say that the trader is long the i
th
asset. If φin < 0, we say that the trader is short the i
th asset.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Portfolio Example II
Example 1.4
Suppose there are two assets (and the bank account), such
that:
S10 = 100,S
2
0 = 50,S
1
1 = 120,S
2
1 = 40,
S12 = 110,S
2
2 = 60, r = 0.1(= 10%).
An investor begins with £1,000, with which they buy 20 units of
asset 1, and short sell 5 units of asset 2 (the remaining money
staying in the bank account). At time 1, they sell 40 (or sell 20,
and short sell a further 20) units of the first asset, and buy 20
units of the second asset, to give a total holding of 15 units of
the second asset. How much is their portfolio worth at time 2,
assuming the portfolio is self-financing?
Example
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Perfect & Frictionless
We also make a number of assumptions that will simplify reality
a little bit.
1 Agents are price-takers
• Any trades that we make will not move the market (i.e.
alter the price of the asset)
2 Assets are perfectly divisible
• We can always buy or sell 12 , 13 , 1√2 , . . . units of an asset
3 Short selling is permitted
• We can hold ‘negative’ amounts of an asset
4 There are no transaction costs or taxes:
• We can buy and sell for the same price, and will not accrue
any additional costs for buying or selling.
A market which satisfies these assumptions is said to be ‘perfect
and frictionless’.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 2
• Financial setting
• Investor can buy/sell a set of assets (e.g. shares)
• At time n, the price of the K assets are S1n , S2n , . . . , SKn
• And invest money in their bank account
• £1 at time n in bank worth £(1+ r) at time n + 1. r is
the interest rate.
• Portfolio of assets, self-financing condition
• Examples of trading, computing cash-flow
• Short-selling, borrowing from the bank
• Market assumptions
• Perfect and Frictionless Market
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Forward Contracts
Definition 1.5
A forward contract is a contract which obliges the first party to
purchase an asset from the second party at a future date, at a
fixed price. The price is known as the forward price.
For example, I agree to purchase 100 Barclays shares from you
in 12 months time, at £3 per share. The forward price is then:
£3.
Lemma 1.6
The fair forward price for an asset which is worth £S0 today,
with delivery date N is: (1 + r)NS0.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Fair price of a Forward
Suppose I can enter into either side of a forward contract, with
forward price K , and delivery date N. Consider the following:
• If S0(1 + r)N < K : I ‘sell’ the contract, so I must deliver
the asset at time N. At the outset, I also buy the asset,
borrowing £S0 from the bank to do so.
At time N, I own the asset, which I am contracted to sell
to the other party for K , and owe the bank S0(1 + r)N .
Hence I am left with £(K − S0(1 + r)N), which is greater
than 0, and so I am better off.
The only ‘fair price’ — that is, the only price at which either
the buyer or seller cannot make money, is therefore the forward
price S0(1 + r)N .
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Fair price of a Forward
Suppose I can enter into either side of a forward contract, with
forward price K , and delivery date N. Consider the following:
• If S0(1 + r)N > K : I ‘buy’ the contract, so I must
purchase the asset at time N. At the outset, I also
short-sell the asset, investing the resulting £S0 in the bank.
At time N, I buy the asset from the other party for K , and
hold in the bank S0(1 + r)N .
Hence I am left with £(S0(1 + r)N − K ), which is greater
than 0, and so I am better off.
The only ‘fair price’ — that is, the only price at which either
the buyer or seller cannot make money, is therefore the forward
price S0(1 + r)N .
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
2 Background and Revision
Financial Basics
Probability Background
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Probability background
You should by now be familiar with many of the important
concepts we will need from probability: such as probability
space, events, indicator functions, discrete and continuous
r.v.’s, independence, expectation and conditional probability.
The main ideas are briefly revised in the notes you have — we
will not go over this material again in the lectures, except to
briefly recall conditional expectation, and the properties of
conditional expectation.
The material here is not examinable, in the sense that you will
not be asked (say) to define conditional expectation in an exam,
but you will be expected to be able to use the results/ideas here
in the manner that we use them elsewhere in the notes.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Conditional Expectation (Discrete)
If Y is a discrete r.v., so Y (Ω) = {y1, y2, . . .} = {yi : i ∈ I},
then we define the Conditional Expectation of X given Y to be
the random variable:
E [X |Y ] =
∑
i∈I
1Y=yiE [X |Y = yi ] .
Recall that 1A = 1A(ω) is a r.v. which takes the value 1 if
ω ∈ A, and 0 otherwise.
Note that if we define the function
ψ(y) = E [X |Y = y ]
then E [X |Y ] = ψ(Y ).
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Conditional Expectation (Discrete)
More generally, if Y0,Y1, . . . ,Yn are discrete random variables,
with Yk taking values in a set Ik , we can define the conditional
expectation
E [X |Y0, . . . ,Yn]
=
∑
i0∈I0,...,in∈In
1Y0=yi0 ,...,Yn=yinE [X |Y0 = yi0 , . . . ,Yn = yin ] ,
and it follows that E [X |Y0,Y1, . . . ,Yn] = ψ(Y0,Y1, . . . ,Yn) if
we define:
ψ(y0, y1, . . . , yn) = E [X |Y0 = y0,Y1 = y1, . . . ,Yn = yn] .
A similar definition holds if the r.v.’s are continuous.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Conditional Expectation (Discrete)
Alternatively, we can define (Fn)n≥0 to be the filtration of the
sequence of random variables (Yi )i≥0. That is, Fn is the
collection of events depending only on Y0,Y1, . . . ,Yn. We
usually think of Yk as the value of something observed at time
k , and then Fn is the ‘state of knowledge’ or ‘history’ up to
time n.
Given a filtration, we can then define the conditional
expectation of X ‘given the history up to time n’ as
E [X |Fn] = E [X |Y0, . . . ,Yn]
=
∑
i0∈I0,...,in∈In
1Y0=yi0 ,...,Yn=yinE [X |Y0 = yi0 , . . . ,Yn = yin ] ,
We will also frequently use the shorthand: E [X |Fn] = En [X ].
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Properties of Conditional
Expectation
Lemma 1.7 (Properties of Conditional Expectation)
The following properties hold for all the forms of conditional
expectation above, if X ,Z ,Y0, . . . ,Yn are r.v.’s:
1 (Taking out what is known): for a ‘nice’ function h
E [h(Y0,Y1, . . . ,Yn)X |Y0, . . . ,Yn]
= h(Y0,Y1, . . . ,Yn)E [X |Y0, . . . ,Yn] ;
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Properties of Conditional
Expectation
Lemma 1.7 (Properties of Conditional Expectation)
The following properties hold for all the forms of conditional
expectation above, if X ,Z ,Y0, . . . ,Yn are r.v.’s:
2 (Tower Property): if m ≤ n,
E [E [X |Y0, . . . ,Yn] |Y0, . . . ,Ym] = E [X |Y0, . . . ,Ym] ,
and in particular, if m = 0, then
E [E [X |Y0, . . . ,Yn]] = E [X ] ;
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Properties of Conditional
Expectation
Lemma 1.7 (Properties of Conditional Expectation)
The following properties hold for all the forms of conditional
expectation above, if X ,Z ,Y0, . . . ,Yn are r.v.’s:
3 (Combining (i) and (ii)): for a ‘nice’ function h
E [h(Y0,Y1, . . . ,Yn)X ]
= E [h(Y0,Y1, . . . ,Yn)E [X |Y0, . . . ,Yn]] ;
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Properties of Conditional
Expectation
Lemma 1.7 (Properties of Conditional Expectation)
The following properties hold for all the forms of conditional
expectation above, if X ,Z ,Y0, . . . ,Yn are r.v.’s:
4 (Independence): if X is independent of Y0, . . . ,Yn
E [X |Y0, . . . ,Yn] = E [X ] ;
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Properties of Conditional
Expectation
Lemma 1.7 (Properties of Conditional Expectation)
The following properties hold for all the forms of conditional
expectation above, if X ,Z ,Y0, . . . ,Yn are r.v.’s:
5 (Linearity): for α, β ∈ R
E [αX + βZ |Y0, . . . ,Yn]
= αE [X |Y0, . . . ,Yn] + βE [Z |Y0, . . . ,Yn] .
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Conditional Expectation example
Example 1.8
Consider random variables, on the coin-flipping probability
space ΩC4 = {HHHH,HHHT ,HHTH, . . . ,TTTT}.
Let Yn be the r.v. which is 1 if the nth flip is a head, and −1
otherwise, for n = 1, 2, 3, 4. Let
Xn = #heads on first n flips−#tails on first n flips
=
n∑
i=1
Yi ,
and set also X0 = 0.
If we assume that Yi are independent and identically distributed
(i.i.d.) with P(Yi = 1) = p = 1− P(Yi = −1), then Xn is a
simple random walk.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Conditional Expectation example
We can use conditional expectations to compute certain
probabilities:
1 Find E [X4|Y1,Y2,Y3].
2 Find E [X4|X1,X2,X3] and E [X4|X3].
3 Find E
[
(X4)
2|X1,X2,X3
]
.
4 Suppose that p = 12 . Find E
[
(X4)
2].
Solve
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 3
• Forward Contract: Fair forward price = (1 + r)NS0
• Conditional Expectation: Definition, properties & Example
• Key Properties: Take out what is known, Tower property
• Conditional expectation given a filtration Fn —
information generated by Y0,Y1, . . . ,Yn. Write
En [X ] = E [X |Fn] = E [X |Y0, . . . ,Yn] .
• Coin tossing example
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Financial Basics
Probability
Background
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Definition 1.9
A stochastic process (in discrete time) is a sequence of random
variables indexed by time. E.g. in the previous example, both
X0,X1,X2,X3,X4 and Y1,Y2,Y3,Y4 were stochastic processes.
A stochastic process X0,X1, . . . ,XN is a Markov process if, for
each n = 0, 1, 2, . . . ,N − 1 and for any function f (x), there
exists a function g(x) (depending on n and f ) such that:
E [f (Xn+1)|X0,X1, . . . ,Xn] = g(Xn). (1)
That is, all the information about the future of the process Xn
is contained in its current value.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
3 Fair Games & Martingales
Fair Games
Martingales
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Fair gambling games
Consider a gambler, who can bet on a sequence of fair games.
Specifically, there is a sequence X1,X2,X3, . . . of games such
that if the gambler chooses to bet an amount α ∈ R at time n,
then the gambler receives the random amount αXn+1 (including
the stake, if returned) at time n + 1.
For example, in a game of Roulette, betting on a single number,
then Xn = 36 with probability 1/36, and Xn = 0 (lose the
stake) otherwise, each game being i.i.d..
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Fair gambling games
We call the sequence fair if
Xn ∈ L1, and
the average winnings, given the previous outcomes, is equal to
the amount bet,
i.e. for any α ∈ R:
E [αXn|X1,X2, . . . ,Xn−1] = α =⇒
E [Xn|X1,X2, . . . ,Xn−1] = 1.
In particular, assuming we start with wealth β at time n − 1
and bet α on the game, our new wealth will be β + α(Xn − 1),
since we lose the stake α which we bet.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Gambling Strategies
Definition 2.1
Suppose (Xn)n≥1 is a sequence of fair games. A gambling
strategy (αn)n≥1 is a sequence of functions,
αn = αn(X1,X2, . . . ,Xn−1) depending on the previous games,
which represents the amount wagered on the nth game.
A bounded gambling strategy is a gambling strategy where
there exists K > 0 such that |αn(x1, x2, . . . , xn−1)| ≤ K for all
n and all x1, . . . , xn−1.
In particular,
Gn =
n∑
k=1
αk(X1,X2, . . . ,Xk−1)(Xk − 1).
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Gambling Strategies
Definition 2.1
The gains process (Gn)n≥0 associated with a particular
gambling strategy αn, and the sequence of fair games (Xn)n≥1
is the total winnings after the first n games, so:
G0 = 0,Gn = Gn−1 + αn(X1,X2, . . . ,Xn−1)(Xn − 1).
In particular,
Gn =
n∑
k=1
αk(X1,X2, . . . ,Xk−1)(Xk − 1).
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Gains process
Lemma 2.2
Let Gn be the gains process associated with a bounded
gambling strategy αn and a sequence of fair games Xn. Then
the average gain, Gn, satisfies:
Gn = E [Gn+1|X1,X2, . . . ,Xn] , (2)
and E [Gn] = G0 = 0 for all n.
Note that the condition (2) tells us that my expected winnings
after one more fair game is the same as my current winnings.
Since this is true for any (bounded) gambling strategy, the
result tells us that we cannot find a ‘clever’ gambling strategy
that will allow us to beat a fair game.
Proof of Lemma 2.2
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
3 Fair Games & Martingales
Fair Games
Martingales
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Fair Games and Asset prices
Now think of an asset-price process. We can think of buying φn
units of the asset at time n as making a ‘gamble’ on the change
in the asset’s value between time n and time n + 1.
If we imagine r = 0, then our ‘winnings’ at time n + 1, for
buying φn units of the asset are: φn(Sn+1 − Sn).
In particular, we can think of Xn+1 = 1 + (Sn+1 − Sn) as being
a sequence of fair games, and the condition which we need to
be satisfied in order for the game to be fair is that:
E [Xn+1|X1,X2, . . . ,Xn] = 1.
This is the case if and only if:
E [Sn+1|S1, S2, . . . ,Sn] = Sn.
Show this
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Assumption on ‘information’
Assumption 2.3
From now on, we will make the following assumption: there will
be some fundamental random process, typically the asset price
process, or the fair game process, which generates all the
randomness in our scenario.
Note that, for example, with the gains process, knowing
X1, . . . ,Xn and the (not random) functions, α1, . . . , αn
determines Gn, so all the information that would be relevant at
time n is determined by observing Xn. When we take
conditional expectation given this process, then we will use the
shorthand:
En [Y ] := E [Y |X1,X2, . . . ,Xn]
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Assumption on ‘information’
Assumption 2.3
From now on, we will make the following assumption: there will
be some fundamental random process, typically the asset price
process, or the fair game process, which generates all the
randomness in our scenario.
Note that, for example, with the gains process, knowing
X1, . . . ,Xn and the (not random) functions, α1, . . . , αn
determines Gn, so all the information that would be relevant at
time n is determined by observing Xn.
We will also assume that all our gambling strategies/portfolio
processes are functions of this same underlying process, so we
may take out what is known.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Adapted processes
We will almost always then want to consider processes Yn
which are known at time n.
In particular, we have the following definition:
Definition
We say that a stochastic process Yn is adapted (with respect to
the underlying process Xn) if Yn is a function of X1, . . . ,Xn.
For example, if Xn is a sequence of coin flips (which generates
all our randomness/the underlying information), and we define
Yn to be the number of heads on the first n flips, and Zn to be
the number of heads on the first n + 1 flips, then both Yn and
Zn are stochastic processes, and Yn is adapted, since it is a
function of X1, . . . ,Xn, but Zn is not adapted, since it depends
on X1, . . . ,Xn+1.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Martingales
Definition 2.4
A stochastic process (Mn)n≥0 is a martingale if:
1 E|Mn| <∞ for all n
2 ‘Mn stays the same on average’:
En [Mn+1] = Mn (3)
3 the process Mn is adapted.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Martingales & Fair Games
Theorem 2.5
Suppose Mn is a martingale, and let φn be a bounded gambling
strategy for Mn. Then the gains process defined by:
Gn =
n∑
j=1
φj(Mj −Mj−1),G0 = 0
is a martingale.
Proof
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Fair Games
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 4
• Sequence of Fair Games: Xn winning for £1 staked
E [Xn|X1, . . . ,Xn−1] = 1.
• Gambling strategy: αn = αn(X1,X2, . . . ,Xn−1) is amount
wagered on the nth game
• Gains process Gn:
G0 = 0, Gn = Gn−1 + αn(X1,X2, . . . ,Xn−1)(Xn − 1).
• Gains process for a sequence of fair games ‘stays the same
on average’
• Assumption on ‘information’: write conditional expectation
as: En [Y ] := E [Y |X1,X2, . . . ,Xn].
• Definition: A martingale Mn has E |Mn| <∞ and Mn stays
the same on average: En [Mn+1] = Mn.
• Theorem: If Mn a martingale,
Gn =
n∑
j=1
φj(Mj −Mj−1), G0 = 0 is a martingale.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
1 Introduction
2 Background and Revision
3 Fair Games & Martingales
4 Binomial Model
5 Fundamental Theorem of Asset pricing
6 Brownian Motion
7 Stochastic Integration
8 Stochastic Calculus
9 Continuous-time Finance
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Random Asset Prices
We now want to build a simple model for pricing derivatives.
We will begin in discrete time, and imagine a world in which
there is a single asset, which can only go up or down by a fixed
percentage change, randomly, on which we will want to price a
derivative contract.
Note that, since we are now considering only one asset, we will
drop the superscripts (so we write Sn = S1n ).
We will often refer to this asset as the risky asset to distinguish
it from the bank account.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Coin-flipping model for asset prices
Suppose we have a fixed time-horizon, N ≥ 1, and suppose that
at each time, we flip an independent, (possibly) biased coin. If
we flip heads, then the asset moves up:
Sn 7→ Sn+1 = uSn,
and if we flip tails, the asset will move down:
Sn 7→ Sn+1 = dSn,
where u > d .
E.g. if the asset either goes up or down by 5%, then
u = 1.05, d = 0.95.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Coin-flipping probability space
We can make this probabilistically rigorous by defining
(Sn)0≤n≤N as a sequence of r.v.’s defined on the event space
Ω = {ω = ω1, ω2, . . . , ωN : ωi ∈ {H,T}}.
An outcome in this space is therefore a sequence
ω = HHTHT . . ., which corresponds to the asset going up
twice, then down, then up, then down, . . . .
We assign to elements of this space the usual (biased) coin
probabilities, so if ω is an event with k heads and N − k tails,
then P({ω}) = pk(1− p)N−k . It then follows that flips are
independent.
To specify a particular market, we therefore need to specify the
initial value of the asset, S0, the amount the asset moves up
and down, u, d , the interest rate r , the time-horizon N and the
up probability p.
Graphic Representation
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Binomial Model
Let’s consider the case where N = 1, a one-period Binomial
model, and suppose we have a risky asset Sn, and a derivative,
whose value at time 1 depends on the price of the asset, S1.
For example, a (European) call option with strike price K —
that is, the option (but not the obligation) to buy the asset at
time 1, for the price K .
Then the call option has value
C1 = (S1 − K )+ := max{0, S1 − K} at time 1. We will refer to
the final value of an option as its payoff.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Binomial Example
Example 3.1
Suppose we have an asset in a one period Binomial model with
S0 = 10, u = 32 , d =
4
5 , r =
1
5 , and a call option with strike
price K = 11. Then we construct a (self-financing) portfolio
using only the bank-account and the underlying asset which has
the same payoff as the call option, and initial value 4021 .
Show This
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Arbitrage
This really suggests that the ‘fair’ price for the call option
should be 4021 . Can we make this a bit more rigorous? Consider
the following definition:
Definition 3.2
Suppose (S1n , S
2
n , . . . ,S
K
n )n=0,1,...,N is a sequence of stochastic
processes under some probability measure P, representing the
prices of a set of K assets. Let φn be a self-financing portfolio
where φjn depends only on the assets up to time n, and whose
value at time n is Vn.
If V0 = 0, and
P(VN ≥ 0) = 1 and P(VN > 0) > 0 (4)
then φn is called an arbitrage.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Arbitrage
Note that (4) says that our final portfolio is always worth
something positive (we never lose money), and at least some of
the time, we definitely make money.
An arbitrage means that we can make risk-free profit with
strictly positive probability!
Clearly, any investor would follow an arbitrage strategy, and by
scaling everything up (if φ is an arbitrage, so too is αφ, for any
α > 0) we can make very large profits!
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Arbitrage
Note that (4) says that our final portfolio is always worth
something positive (we never lose money), and at least some of
the time, we definitely make money.
An arbitrage means that we can make risk-free profit with
strictly positive probability!
Clearly, any investor would follow an arbitrage strategy, and by
scaling everything up (if φ is an arbitrage, so too is αφ, for any
α > 0) we can make very large profits!
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 5
• Binomial Model: Asset price goes up by factor u, or down
by d , with probabilities p and 1− p.
• Corresponds to ‘coin-flipping’ probability space.
• Call option with strike K : worth (S1 − K )+ at time 1.
Find a portfolio in asset with same payoff.
• Found ‘fair’ price of call option.
• Arbitrage: Exists self-financing portfolio Vn with V0 = 0
and
P(VN ≥ 0) = 1 and P(VN > 0) > 0
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Arbitrage in practice
In Example 3.1, we can construct an arbitrage if the price of the
derivative is not 4021 :
• if the price of the derivative is more than 4021 , we can sell
the derivative, buy 47 units of the asset, and borrow the
remaining money from the bank; at time 1, we always end
up with a portfolio with a positive value;
• if the price of the derivative is less than 4021 , we can buy the
derivative, (short) sell 47 units of the asset, and invest the
remaining money in the bank; at time 1, we always end up
with a portfolio with a positive value.
Assumption: Arbitrage does not exist!
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Arbitrage-free market
Our first result in this direction is the following:
Theorem 3.3
In a one-period model, consisting only of the asset and the bank
account, with p ∈ (0, 1), there is no arbitrage if and only if
d < (1 + r) < u.
We leave the proof of this result to an example sheet,
but intuitively, the condition d < 1 + r < u says that investing
£1 in the risky asset is never guaranteed to be better, or
guaranteed to be worse than investing £1 in the bank account.
This result is important since it means that we can use a
one-period Binomial model as a (not very realistic) model for
our financial market.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Arbitrage with Derivatives
Theorem 3.4
Suppose we have a one-period Binomial model, with
d < (1 + r) < u, and where we can also purchase a derivative
contract, whose payoff at time 1, C1, depends on the value at
time one of the underlying asset, so:
C1 =
{
cH , if S1 = uS0
cT , if S1 = dS0
.
Then there is no arbitrage if and only if the price of the
derivative at time 0, C0 is:
1
1 + r
[
cT
u − (1 + r)
u − d + cH
(1 + r)− d
u − d
]
. (5)
Proof
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Arbitrage-free price
We call the price (5) an arbitrage-free price of the derivative.
Note that (here) there is a single arbitrage-free price, so we can
talk about the arbitrage-free price.
Example 3.5
Recall Example 3.1. Then cH = (15− 11)+ = 4 and
cT = (8− 11)+ = 0, and so the price at time 0 is:
1
1 + 15
[
0 + 4× 1 +
1
5 − 45
3
2 − 45
]
=
40
21
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Multi-period market
We can use exactly the same idea to work out what happens in
a multiple period market.
If N > 1, we can begin by computing the price of the derivative
at time N − 1, given e.g. we have seen HTHHT . . . up to time
N − 1. This is now a one-period model, and we can compute
the price of the derivative.
Now suppose we are at time N − 2, now we know what the
price of the derivative must be at time N − 1, and so we can
repeat the process to compute an arbitrage-free price at time
N − 2, etc.
Example 3.6
Suppose we have a Binomial model with N = 2, r = 14 ,
u = 43 , d =
5
6 , and S0 = 2025. Find the price of a European call
option with strike K = 1800.
Price
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 6
• Example: ‘fair’ price of a one-period option by construction
of ‘equivalent’ portfolio
• Arbitrage: Exists self-financing portfolio Vn with V0 = 0
and
P(VN ≥ 0) = 1 and P(VN > 0) > 0
• Binomial model is free of arbitrage iff d < (1 + r) < u.
• Theorem 3.3 : No arbitrage if and only if the price of the
derivative at time 0 is:
1
1 + r
[
cT
u − (1 + r)
u − d + cH
(1 + r)− d
u − d
]
.
• Proof of Theorem:
• Idea of proof: look at all self-financing portfolios with
V0 = 0 involving the asset and the derivative.
• Multiple time-periods: consider N-period model as many
one-period models.
• Example of computing option price in 2-period model.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Path-independent options
The European Call option in Example 3.6 is path-independent:
the payoff of the option only depends on the final value of the
asset, and not on the path that the asset takes.
An alternative option, which has a path-dependent payoff, could
be something like a knock-in call option:
this is an option with a call payoff, but which will only pay out
if the asset value exceeds some barrier level some time before
the maturity date.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Path-dependent option pricing
Example 3.7
Suppose we have a Binomial model with
N = 3, r = 16 , u =
3
2 , d =
5
6 ,S0 = 1080, and suppose we want
to price a knock in call option with barrier B = 2400 and strike
K = 1700, so the payoff of the option is:
(S3 − K )+1{Sn>B, for some n=0,1,2,3}.
What is the arbitrage-free price of the derivative?
Price
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Hedging
Finally, we recall that the arbitrage-free price was determined by
the fact that we could ‘replicate’ the derivative by trading in the
underlying asset.
We can do a similar thing with multiple time-periods, and we
can interpret this as hedging. Imagine a bank who sells a client
an option. The bank is then exposed to the risk associated to
the final value of the product that they have sold, and may
want to ‘hedge’ this risk.
They can do this by trading in the underlying to ‘replicate’ the
payoff of the derivative that they have sold to the client.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Hedging
Given some derivative, we will call the portfolio which has the
same value at maturity as the derivative the hedging portfolio,
since it is how a bank who has sold a derivative would hedge
their exposure to the risk associated with the contract.
For a financial institution selling many derivatives, it is often as
important to know the hedging portfolio as it is to know the
price of the option.
The number of units of the asset that we need to hold at a
given time to hedge a derivative is often called the delta of the
derivative.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Hedging Example
Example 3.8
Consider a bank who has sold the option in Example 3.7, and
wishes to hedge their exposure. Find the portfolio that they
should hold at time 1 if S1 = 1620.
Find Hedge
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
1 Introduction
2 Background and Revision
3 Fair Games & Martingales
4 Binomial Model
5 Fundamental Theorem of Asset pricing
6 Brownian Motion
7 Stochastic Integration
8 Stochastic Calculus
9 Continuous-time Finance
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Pricing as Expectation
The method described above allows us to price derivatives in a
multi-period Binomial model, but it is a bit time-consuming,
and if we let N get big, it can be lengthy to compute prices.
Can we find a quicker way?
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Pricing as Expectation
The answer lies in the expression:
1
1 + r
[
cT
u − (1 + r)
u − d + cH
(1 + r)− d
u − d
]
.
which we can rewrite as:
1
1 + r
[
cT
(
1− (1 + r)− d
u − d
)
+ cH
(1 + r)− d
u − d
]
.
or:
1
1 + r
[cT (1− q) + cHq] .
where q := (1+r)−du−d ∈ (0, 1) (by Theorem 3.3). This looks like
the average payoff, where we do not use the real probability p
of a move up, but a new probability q.
Observe that the actual value of p plays no role in computing
the price of the derivative. We will call q the risk-neutral
probability.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Example 3.1 revisited
Example 4.1
Recall Example 3.1, we had u = 32 , d =
4
5 , r =
1
5 , so
q =
(1 + r)− d
u − d =
6
5 − 45
3
2 − 45
=
4
7
,
and we can write
1
1 + r
E˜ [C1] =
1
1 + r
[(1− q)C1(T ) + qC1(H)]
=
1
6
5
[
3
7
× 0 + 4
7
× 4
]
=
5
6
× 16
7
=
40
21
as expected.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
‘New’ probabilities
From now on, we will use the notation E˜ [X ] and P˜(A) to
denote expectation and probability taken using the risk-neutral
probabilities.
We can in fact make this a bit more rigorous on the
coin-flipping probability space.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Change of Measure
Definition 4.2
Let Hn(ω) be the number of heads we flip on the first n coin
flips on the outcome ω ∈ Ω. Then we can define
Zn(ω) =
(
q
p
)Hn(ω)(1− q
1− p
)n−Hn(ω)
.
We say that E˜ [·] is the change of measure of E [·] induced by
the Radon-Nikodym derivative ZN by:
E˜ [Y ] = E [ZNY ]
for any random variable.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Change of Measure on Coin-flipping
space
Observe that we can express the probability of a given sequence
of heads and tails in terms of the number of heads, so if ω ∈ Ω
is a sequence of N heads and tails, with k = HN(ω) heads, then
P({ω}) = pk(1− p)N−k = pHN(ω)(1− p)N−HN(ω).
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Path probabilities
Observe also that we can deduce statements about probability
from statements about expectation using:
P˜(A) = E˜ [1A] = E [ZN1A] .
In particular, the probability of the path ω = ω1ω2ω3 · · ·ωN ,
where ωi ∈ {H,T} is given by:
P˜({ω}) = E˜ [1{ω}] = E [ZN1{ω}]
= E
[(
q
p
)HN(ω)(1− q
1− p
)N−HN(ω)
1ω(ω)
]
=
(
q
p
)HN(ω)(1− q
1− p
)N−HN(ω)
P({ω})
=
(
q
p
)HN(ω)(1− q
1− p
)N−HN(ω)
pHN(ω)(1− p)N−HN(ω)
= qHN(ω)(1− q)N−HN(ω)
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 7
• Pricing in a multi-period model:
• Path-independent option
• Path-dependent option
• Multi-period pricing and hedging
• Price of an option: if q = (1+r)−du−d ,
Price =
1
1 + r
[cT (1− q) + cHq] = 11 + r E˜ [C1] .
• E˜ [·] and P˜(·) are the risk-neutral probability measures. Use
q instead of p!
• Change of measure: Can define E˜ [·] by
E˜ [Y ] = E [ZNY ]
for any random variable Y , where ZN is called the
Radon-Nikodym derivative.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
‘New’ probability model
Since the probability of any event on a coin-flipping probability
space can be built up from the probabilities of individual
outcomes, it follows from this computation that P˜(·) really does
correspond to a Binomial model with up-probability q instead of
p.
Observe also that for all ω ∈ Ω, 0 < Zn(ω) <∞, so for A ⊆ Ω:
P(A) > 0 ⇐⇒ E [1A] > 0
⇐⇒ E [ZN · Z−1N 1A] > 0
⇐⇒ E˜ [Z−1N 1A] > 0
⇐⇒ P˜(A) > 0.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Discounting Asset Prices
Why is this probability space important? Let’s consider the
discounted asset price — that is, the price of the asset
discounted into today’s money by dividing by the value of £1
invested in the bank:
S∗n =
Sn
(1 + r)n
.
This has some nice consequences for the value of a
self-financing portfolio.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Asset prices as martingales
Lemma 4.3
Let Vn be the value of a self-financing portfolio in the assets
S1n , S
2
n , . . . ,S
K
n and the bank account. If we write
V ∗n =
Vn
(1 + r)n
,
for the discounted portfolio value, and
S i ,∗n =
S in
(1 + r)n
then
V ∗n+1 = V
∗
n +
K∑
i=1
φin(S
i ,∗
n+1 − S i ,∗n ). (6)
Only specify risky assets — the bank account ‘takes care of
itself’.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Proof of Lemma 4.3
We have
Vn+1 = φ
0
n(1 + r) +
K∑
i=1
φinS
i
n+1
= (1 + r)
[
φ0n +
K∑
i=1
φinS
i
n
]
+
K∑
i=1
φin(S
i
n+1 − (1 + r)S in)
= (1 + r)Vn +
K∑
i=1
φin((1 + r)
n+1S i ,∗n+1
− (1 + r) · (1 + r)nS i ,∗n ),
and so dividing through by (1 + r)n+1 we get (6).
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Asset prices are martingales
The discounted asset prices also have another nice property:
Theorem 4.4
Under the probability measure P˜, the discounted asset price S∗n
is a martingale.
Proof
Because of this property, we will often call P˜ the risk-neutral
probability measure.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Self-financing portfolios are
martingales
We can use both these results to see the following:
Lemma 4.5
Suppose that the discounted asset prices are martingales. Then
the discounted value of any bounded self-financing portfolio φ
is a martingale.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Self-financing portfolios are
martingales
Proof.
If S i ,∗n is a martingale for each i , then so too is
Mn =
n∑
k=1
φik−1(S
i ,∗
k − S i ,∗k−1)
by Theorem 2.5.
The sum of martingales is a martingale (check the definition!),
so we also see that
V ∗n = V
∗
0 +
n∑
k=1
K∑
i=1
φik−1(S
i ,∗
k − S i ,∗k−1)
is a martingale.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Martingale Representation Theorem
These results say that self-financing portfolios (discounted) are
martingales — the next result is a sort of converse. As we shall
see, it says essentially that martingales (under P˜) are
self-financing portfolios!
Theorem 4.6 (Martingale Representation Theorem)
Let Mn be a martingale under P˜. Then there exists a process φn
which depends only on S1, . . . ,Sn such that
Mn = E˜ [MN ] +
n∑
k=1
φk−1(S∗k − S∗k−1).
Proof
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Fundamental Theorem of Asset
Pricing
Theorem 4.7 (Fundamental Theorem of Asset Pricing)
Consider a derivative contract on the N-period Binomial model
with payoff CN at time N. Then there is no arbitrage if and
only if the price of the contract at time n is:
Cn =
1
(1 + r)N−n
E˜n [CN ] , for n = 0, 1, . . . ,N. (7)
In particular, C0 = 1(1+r)N E˜ [CN ].
Proof
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 8
• Discounted asset prices:
S∗n =
Sn
(1 + r)n
.
• Under P˜, discounted asset price a martingale.
• If discounted asset prices are martingales, so too is the
value of a discounted portfolio process.
• Martingale Representation Theorem: If Mn is a martingale
under P˜, then there exists a portfolio φn such that:
Mn = E˜ [MN ] +
n∑
k=1
φk−1(S∗k − S∗k−1).
• Fundamental Theorem of Asset pricing: no arbitrage if and
only if the price of the contract at time n is:
Cn =
1
(1 + r)N−n
E˜n [CN ] , for n = 0, 1, . . . ,N.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Consequences of the FTAP
This allows us to make some quick observations about how to
price options. For example, suppose that we have a derivative
whose payoff depends only on the value of the asset at time N,
say CN(ω) = f (SN(ω)) for some function f .
We know (under P˜) that the probability of a single ‘up’ move is
q, and that each step is independent of the previous steps, so
P˜(SN = S0ukdN−k) =
(
N
k
)
qk(1− q)N−k , for k = 0, . . . ,N.
(8)
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Consequences of the FTAP
Therefore
E˜ [f (SN)] =
N∑
k=0
f (S0u
kdN−k)P˜(SN = S0ukdN−k)
=
N∑
k=0
f (S0u
kdN−k)
(
N
k
)
qk(1− q)N−k .
If follows from Theorem 4.7 that the arbitrage-free price of a
derivative with path-independent payoff CN = f (SN) is:
E˜ [CN ]
(1 + r)N
=
1
(1 + r)N
N∑
k=0
f (S0u
kdN−k)
(
N
k
)
qk(1−q)N−k (9)
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Example: Asian Option
Example 4.8
An Asian option is a derivative which pays the holder
(AN − K )+
where AN is the average of the asset price between time 0 and
time N:
AN =
1
N + 1
N∑
n=0
Sn.
Suppose N = 3, u = 2, d = 1/2, r = 1/4,S0 = 1,K = 1. What
is the price of the Asian option?
Solution
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 9
• Fundamental Theorem of Asset pricing: no arbitrage if and
only if the price of the contract at time n is:
Cn =
1
(1 + r)N−n
E˜n [CN ] , for n = 0, 1, . . . ,N.
• Proof: uses fact that self-financing portfolios are
martingales, and Martingale Representation Theorem
• Use FTAP to price options as expectations. If
CN = f (SN), the option price is:
1
(1 + r)N
N∑
k=0
f (S0u
kdN−k)
(
N
k
)
qk(1− q)N−k .
• Using FTAP to price options. (Asian option example)
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
1 Introduction
2 Background and Revision
3 Fair Games & Martingales
4 Binomial Model
5 Fundamental Theorem of Asset pricing
6 Brownian Motion
7 Stochastic Integration
8 Stochastic Calculus
9 Continuous-time Finance
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Limit of Binomial model
Suppose we fix a terminal time T > 0, and we want to think
about the behaviour of a Binomial model with a large number
of intermediate steps.
For example, we could consider T = 1 year, and think of each
step as being a day, an hour, a minute, a second,. . . . Can we
say anything about the limit?
Suppose we divide [0,T ] into n steps, of length T/n. We can
first think of what we should do with the bank account.
Perhaps we initially get paid an interest rate r over 1 year, so at
time 1 we have £(1 + r).
An alternative would be to receive the interest rate r
compounded biannually, which means that we receive the
interest r2 paid every half-a-year. So at the end of a year, $1 is
worth £(1 + r2)
2.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Limit of Binomial model
This extends to compounding n times a year, so after one year,
we receive £(1 + rn )
n, and if we let n→∞, we get continuous
compounding:
if we have £1 at time 0, at time 1 we have
£ lim
n→∞
(
1 +
r
n
)n → £er .
More generally, if we keep £1 in the bank for t units of time,
then at the end of the period we have £ert .
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Limit of Binomial model
So the question now is, what happens to the asset under the
risk-neutral measure as we let n→∞?
Recall that, under P˜, we know the value of the asset at time 1
is: Sn = S0dn
(
u
d
)Hn where Hn is the number of heads we get
on the first n flips, so Hn ∼ Bin(n, q).
Let’s fix q = 12 , and think about how to choose u and d . Since
we want a sensible limit as n→∞, we should choose the
parameters of the model, un, rn and dn to depend on n, and the
argument above suggests we should take rn = rn .
In addition, if q = 12 , then there must exist sn such that
un = 1 + rn + sn and dn = 1 + rn − sn.
We now want to choose sn so that in the limit as n→∞, we
get something that is random, but not ‘too random’ that is, in
the limit, we should have a well-defined random variable.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Limit of Binomial model
It turns out to be easier to consider log(Sn), which is:
log(Sn) = log
(
S0d
n
n
(
un
dn
)Hn)
= log(S0) + n log(dn)
+ Hn(log(un)− log(dn)). (10)
We note that log(1 + x) ≈ x − x22 for small x .
Suppose that sn = σ√n , so that
un = 1 + rn +
σ√
n
, and
dn = 1 + rn − σ√n .
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Limit of Binomial model
Then we have
log(un) = log
(
1 +
r
n
+
σ√
n
)
≈ r
n
+
σ√
n
− 1
2
(
r
n
+
σ√
n
)2
≈ r
n
+
σ√
n
− σ
2
2n
where we have ignored terms of order n−
3
2 and smaller.
Similarly, we get:
log(dn) ≈ r
n
− σ√
n
− σ
2
2n
.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Limit of Binomial model
If we put these into (10) we get:
log(Sn) ≈ log(S0) +
(
r
n
− σ√
n
− σ
2
2n
)
n + 2
σ√
n
Hn
≈ log(S0) + 2σ
Hn − 12n√
n
+ r − σ
2
2
≈ log(S0) + σ
Hn − 12n√
1
4n
+ r − σ
2
2
.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Limit of Binomial model
Now, recall that Hn is the number of heads, so we can write
Hn = Y1 + Y2 + · · ·+ Yn, where Yi are i.i.d. with mean 12 and
variance 12 ×
(
1− 12
)
= 14 .
The Central Limit Theorem tells us that we get the convergence
in distribution:
Hn − 12n√
1
4n
D−−→ Z ∼ N(0, 1)
that is, for large n this looks like a Normal random variable with
mean 0 and variance 1.
If we call this B , we see that:
S = S0eσB+(r−
1
2σ
2).
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Limit of Binomial model
Now, this was all assuming that we are looking for the price of
the asset at time 1. We can also do a similar analysis for the
asset at a general time t (we do a similar analysis, but look at
the asset after m steps, where m ≈ tn), and we get an
expression for the value of the asset at time t, St as:
St = S0eσBt+(r−
1
2σ
2)t ,
where Bt is now N(0, t) for each t.
What can we say about Bt?
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Brownian motion
Definition 5.1
A process (Bt)t≥0 is a (standard) Brownian motion if B0 = 0
and:
1 t 7→ Bt is continuous
2 Bt has independent increments: i.e.
(Bt1 − B0), (Bt2 − Bt1), . . . , (Btn − Btn−1) are independent
if 0 ≤ t1 ≤ t2 ≤ · · · ≤ tn−1 ≤ tn
3 Bt − Bs ∼ N(0, (t − s)) for 0 ≤ s ≤ t.
Note that the third condition implies Bt ∼ N(0, t), by taking
s = 0.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Brownian Motion II
‘Standard’ here refers to the fact that EB1 = 0 = B0, and
Var(B1) = 1. More generally, for ν ∈ R and ξ > 0 we can have:
B˜t = νt + ξBt
where Bt is a standard Brownian motion. Then we say that B˜t
is a Brownian motion with drift ν and diffusion coefficient ξ.
In fact, a rather surprising (and hard) theorem says that the
standard Brownian motion actually exists!
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 10
• Limit of Binomial model. Fix T > 0, suppose we have n
steps Binomial model, step size T/n, and let n→∞.
• Price of an asset at time t is (via Binomial limit):
St = S0eσBt+(r−
1
2σ
2)t ,
where Bt ∼ N(0, t).
• Bt is a (standard) Brownian motion (BM): B0 = 0 and
• t 7→ Bt is continuous;
• Bt has independent increments;
• Bt − Bs ∼ N(0, t − s).
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Geometric Brownian Motion
Definition 5.2
Suppose that Bt is a standard Brownian motion, and x0, ξ > 0.
Then the process
Xt = x0 exp
{(
ν − 1
2
ξ2
)
t + ξBt
}
is a geometric Brownian motion (GBM) with infinitesimal drift
ν and diffusion coefficient ξ.
• x0 is the initial value, so X0 = x0
• exp(·) > 0, so Xt > 0 for all t ≥ 0
• From the connection to the binomial model, we see that ν
is the average proportionate change (per unit of time) in
Xt over a small time-step, and ξ2 is the variance of this
change.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Transformations of Brownian
motion
Since Brownian motion will form a key part of what follows, we
will look at a few properties of Brownian motion.
Theorem 5.3
If Bt is a standard Brownian motion, then:
1 X 1t = −Bt ,
2 X 2t = Bt+s − Bs , for s > 0,
3 X 3t = aBa−2t , for a 6= 0
are all (standard) Brownian motion.
Proof
The final property corresponds to ‘zooming’ into the origin. If
we rescale correctly, the process when we zoom in is still a
Brownian motion.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
‘Uniqueness’ of Brownian motion
Think of B,X 1,X 2,X 3 etc. as being “statistically” identical. It
is not true that Bt = X 1t always, even though both are
Brownian motions.
However, it is still true that Brownian motion is unique as a
random variable, in the sense that the probability that B does
something is the same as the probability that X 1 or X 2 do the
same thing.
This is a common thing in probability: the Normal distribution
with mean 0 and variance 1 is unique in the sense that e.g. the
distribution function is unique, however if X ∼ N(0, 1), then
−X ∼ N(0, 1) as well, but they are not equal, in fact
P(X = −X ) = 0.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 11
• Geometric Brownian motion:
Xt = x0 exp
{(
ν − 1
2
ξ2
)
t + ξBt
}
is a geometric Brownian motion (GBM) with infinitesimal
drift ν and diffusion coefficient ξ.
• Transformation of BM: −Bt ,Bt+s − Bs and aBa−2t are all
BM.
• Proof: check satisfy definition.
• ‘Uniqueness’ of Brownian motion
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Brownian motion as a martingale
Theorem 5.4
Brownian Motion is a martingale.
Note that we will informally assume that many of the properties
we had in discrete time, such as being able to take conditional
expectations, also hold for continuous time. To prove these
results rigorously is rather harder, but for the rest of this course,
we will be a bit more informal regarding such details.
Proof
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Paths of Brownian motion
One of the features of Brownian motion is that the paths of the
process are not very smooth at all.
It can be shown (we won’t!) that the path Bt is not
differentiable anywhere.
We will be interested in a related property: suppose f (t) is a
differentiable function, with a bounded derivative. Then there
exists some constant K such that |f (t)− f (s)| ≤ K |t − s|.
Suppose now we consider the function evaluated at smaller and
smaller time steps. Specifically, suppose that we set tni =
i
n , so
tni+1 − tni = 1n , and Tn = btnc, so tnTn ≈ t.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Paths of Brownian motion
Then we get:
Tn−1∑
i=0
(f (tni+1)− f (tni ))2 ≤ K 2
Tn−1∑
i=0
|tni+1 − tni |2
= K 2
Tn−1∑
i=0
1
n2
=
K 2
n2
× btnc → 0,
as n→∞.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Paths of Brownian motion
However, if we do the same for Brownian motion:
Tn−1∑
i=0
(Btni+1 − Btni )2
is a sum of Tn ≈ tn i.i.d. r.v.’s with mean (tni+1 − tni ) = 1/n,
and finite variance.
This has a non-zero average, so perhaps the limit is not zero.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Paths of Brownian motion
We make this precise by considering the limit in the following
sense:
Lemma 5.5
We have:
lim
n→∞E
∣∣∣∣∣
Tn−1∑
i=0
(Btni+1 − Btni )2 − t
∣∣∣∣∣
2 = 0.
That is,
∑Tn−1
i=0 (Btni+1 − Btni )2 → t in mean-square, or
(equivalently) in L2.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Proof of Lemma 5.5
To simplify the mathematics, we will assume that
Tn = btnc = tn. The result remains true in general, but we
need to be a bit more careful in the proof.
Note that if we write ∆ni = (Btni+1 − Btni ), then we know that
∆ni ∼ N(0, 1n ) and so, using standard properties of the Normal
distribution:
E
[
(∆ni )
2
]
=
1
n
, E
[
(∆ni )
4
]
=
3
n2
.
In addition, if i 6= j , ∆ni and ∆nj are independent, since they are
increments of the Brownian motion.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Proof of Lemma 5.5
So:
E
∣∣∣∣∣
Tn−1∑
i=0
(Btni+1 − Btni )2 − t
∣∣∣∣∣
2
= E
(Tn−1∑
i=0
(
(∆ni )
2 − 1
n
))2
= E
Tn−1∑
i ,j=0
(
(∆ni )
2 − 1
n
)(
(∆nj )
2 − 1
n
)
=
Tn−1∑
i ,j=0
E
[(
(∆ni )
2 − 1
n
)(
(∆nj )
2 − 1
n
)]
But by the independence of Brownian increments, and since
E
[
(∆ni )
2 − 1n
]
= 0, the terms where i 6= j are all zero.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Proof of Lemma 5.5
We therefore have:
E
∣∣∣∣∣
Tn−1∑
i=0
(Btni+1 − Btni )2 − t
∣∣∣∣∣
2 = Tn−1∑
i=0
E
[(
(∆ni )
2 − 1
n
)2]
=
Tn−1∑
i=0
E
[
(∆ni )
4 − 2
n
(∆ni )
2 +
1
n2
]
=
Tn−1∑
i=0
(
3
n2
− 2
n2
+
1
n2
)
= t n
2
n2
=
2t
n
→ 0,
where we have used the values for E
[
(∆ni )
4] and E [(∆ni )2] we
gave earlier and Tn = tn.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
1 Introduction
2 Background and Revision
3 Fair Games & Martingales
4 Binomial Model
5 Fundamental Theorem of Asset pricing
6 Brownian Motion
7 Stochastic Integration
8 Stochastic Calculus
9 Continuous-time Finance
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Integration
Why is the roughness of Brownian motion a problem? Suppose
we consider the path of a financial asset, and suppose we
choose to invest in the asset in a way that depends on the
behaviour of the asset, so perhaps we always choose to hold a
number of units of the asset which is equal to the current value
of the asset. How much profit/loss will we make?
If the asset price at time t is St , and we approximate by
changing the number of units of the asset we hold at times tni ,
then our profit/loss from this strategy V nt is:
V nt =
Tn−1∑
i=0
Stni (St
n
i+1
− Stni ).
If we now let n→∞, then we expect to get the value of
following this strategy ‘in continuous time’.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Integration
First we observe that, for a, b ∈ R:
a(b − a) = 1
2
(b2 − a2)− 1
2
(b − a)2.
Then if we take a = Stni and b = Stni+1
V nt =
Tn−1∑
i=0
Stni (St
n
i+1
− Stni )
=
Tn−1∑
i=0
[
1
2
(
S2tni+1 − S
2
tni
)
− 1
2
(
Stni+1 − Stni
)2]
=
1
2
(
S2btnc/n − S20
)
− 1
2
Tn−1∑
i=0
(
Stni+1 − Stni
)2
where we have used the fact that tnTn = btnc/n, which is
approximately t.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 12
• Brownian motion is a martingale
• Paths of Brownian motion are ‘rough’:
Tn−1∑
i=0
(Btni+1 − Btni )2 → t in L2.
• This compares with the fact that if f is a differentiable
function with bounded derivative, then∑Tn−1
i=0
(
f (tni+1)− f (tni )
)2 → 0.
• Intuition: Bt+ε − Bt ≈
√
ε, but some +, some −.
• Gains from ‘trading’, holding φt = St :
V nt =
Tn−1∑
i=0
Stni (St
n
i+1
− Stni )
=
1
2
(
S2btnc/n − S20
)
− 1
2
Tn−1∑
i=0
(
Stni+1 − Stni
)2
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Integration
But we have just observed above: if St = f (t), a continuous,
differentiable function, then the second term on the right
disappears as n→∞.
However, if St = Bt is Brownian motion, then this term will not
disappear. In fact, it will converge to t in the limit.
Since we are thinking about small time steps, we will write
(informally) dt for the time-step (in this case, 1n ),
dSt = St+dt − St for the change in St , and similarly,
df (t) = f (t + dt)− f (t), and dBt = Bt+dt − Bt .
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
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Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Integration
If we are thinking about letting the time step go to 0, then the
sum will become an integral, and we can rephrase these results
as:
Tn−1∑
i=0
f (tni )(f (t
n
i+1)−f (tni ))→
∫ t
0
f (s) df (s) =
1
2
(f (t)2−f (0)2),
but for Brownian motion:
Tn−1∑
i=0
Btni (Bt
n
i+1
− Btni )→
∫ t
0
Bs dBs =
1
2
(B2t − B20 )−
1
2
t. (11)
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Integration
For differentiable f , this is nothing more than the product rule:
since df (t) = f (t + dt)− f (t) ≈ f ′(t)dt, we have:∫ t
0
f (s)df (s) =
∫ t
0
f (s)f ′(s) ds =
∫ t
0
(
1
2
f (s)2
)′
ds
=
[
1
2
f (s)2
]t
0
=
1
2
(f (t)2 − f (0)2).
But for Brownian motion, which is not differentiable, the
roughness of the paths gives us the additional terms involving t.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Integration
Recall that when we considered processes in discrete time, the
processes of the form
∑n−1
i=0 φi (Si+1 − Si ) played an important
role to understanding our possible trading portfolios.
The same will be true in continuous time, but we need to make
sense of the limit as the gap between times where we trade get
smaller.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Integration
In particular, (using the dSt notation above), we are interested
in the limit
Tn∑
i=0
φtni (St
n
i+1
− Stni )→
∫ t
0
φsdSs
and we will first consider the case where St = Bt is Brownian
motion so we want to make sense of:
Tn∑
i=0
φtni (Bt
n
i+1
− Btni )→
∫ t
0
φsdBs ,
and later see how this relates to the financial interpretation.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Simple Processes
So we begin by considering what class of processes (φs)s≥0
which we are allowed to take as the integrand in the integral.
In general, we will want φs , the value of the integrand at time
s, to only depend on the information available up to time s.
An easy case is when we allow φ only to change a finite number
of times.
These integrands will play a special role, so we define:
Definition 6.1
A process (φt)t≥0 is called a simple process if it can be written
as
φt =
n−1∑
i=0
Xi1{t ∈ [ti , ti+1)}, (12)
where 0 = t0 < t1 < · · · < tn, and each Xi is a random
variable, known at time ti , such that E
[
X 2i
]
<∞.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Stochastic Integral for Simple
Processes
This definition is nice, since we are essentially in the discrete
case!
We hold φti units of the asset between time ti and time ti+1,
and we can compute the ‘gains’ from this as the sum of the
terms φti (Bti+1 − Bti ).
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Stochastic Integral for Simple
Processes
This leads us to the definition:
Definition 6.2
We define the stochastic integral (It)t≥0 (with respect to a
Brownian motion Bt) of a simple process (φt)t≥0 given by (12)
as: for t ∈ [tk , tk+1)
It =
k−1∑
i=0
Xi (Bti+1 − Bti ) + Xk(Bt − Btk ).
Graphical Example
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 13
• The non-differentiability of Brownian motion causes
difficulties, because, for a nice function f (differentiable):
Tn−1∑
i=0
f (tni )(f (t
n
i+1)−f (tni ))→
∫ t
0
f (s) df (s) =
f (t)2 − f (0)2
2
but for Brownian motion:
Tn−1∑
i=0
Btni (Bt
n
i+1
− Btni )→
∫ t
0
Bs dBs =
1
2
(B2t − B20 )−
1
2
t.
• Simple process: φt =
∑n−1
i=0 Xi1{t ∈ [ti , ti+1)}
• Stochastic Integral of a Simple process:
It =
k−1∑
i=0
Xi (Bti+1 − Bti ) + Xk(Bt − Btk ).
• Stochastic Integral (of a simple process) is a martingale
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Properties of Stochastic Integral
Then we have some important properties of the stochastic
integral for a simple process:
Theorem 6.3
The stochastic integral of a simple process is a martingale.
Sketch Proof
And the Stochastic integral has another important property:
Theorem 6.4 (Itô isometry)
The stochastic integral of a simple process satisfies:
E
[I2t ] = E [∫ t
0
φ2s ds
]
Proof
MA30089:
Stochastic
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and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
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Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
General integrands
Finally, we define our class of potential integrands.
Let V be the set of stochastic processes, (φt)t≥0 such that φt is
known at time t, and:
E
[∫ t
0
φ2s ds
]
<∞.
The key idea is that we can now define the stochastic integral
for a general integrand in V by considering the limit of integrals
of simple processes.
Approximation of φ by simple functions
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Stochastic Integral
Definition 6.5
Suppose φt ∈ V, and suppose φnt is a sequence of simple
processes such that:
E
∫ t
0
|φs − φns |2 ds → 0 as n→∞.
Then we define the stochastic integral of φt with respect to a
Brownian motion Bt by:
It =
∫ t
0
φs dBs = lim
n→∞
∫ t
0
φns dBs = limn→∞ I
n
t , (13)
where the limit is in the sense of E
[
(Int − It)2
]→ 0 as n→∞.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Stochastic Integral
Definition 6.5
Suppose φt ∈ V, and suppose φnt is a sequence of simple
processes such that:
E
∫ t
0
|φs − φns |2 ds → 0 as n→∞.
Then we define the stochastic integral of φt with respect to a
Brownian motion Bt by:
It =
∫ t
0
φs dBs = lim
n→∞
∫ t
0
φns dBs = limn→∞ I
n
t , (13)
where the limit is in the sense of E
[
(Int − It)2
]→ 0 as n→∞.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Stochastic Integral
Of course, there are a lot of mathematical questions which, if
we were being rigorous, we would try to answer —
for example does there always exist an approximating sequence
of simple functions, and
does the limit in (13) exist, and
is it unique?
The answer to these questions are yes but we will not go into
the details. (See the lecture notes for a non-examinable
discussion.)
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 14
• Itô isometry:
E
[I2t ] = E [∫ t
0
φ2s ds
]
• Class of general integrands, V: φ ∈ V if φ adapted (φt
known at time t), and E
[∫ t
0 φ
2
s ds
]
<∞.
• Definition of Stochastic integral for φ ∈ V by
approximation with simple integrands
• Matlab example — visualisation of approximating sequence
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Example
Example 6.6
This is all very well, but can we actually find the stochastic
integral for any processes?
Well, recall the calculations at the beginning of this chapter.
There, we showed (replacing S with B) that if we take
Tn = btnc and tni = i/n, we can write:
Tn−1∑
i=0
Bti n(Btni+1−Bti n) =
1
2
(
B2btnc/n − B20
)
−1
2
Tn−1∑
i=0
(
Btni+1 − Btni
)2
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Example
Now, the left-hand side is really just the stochastic integral of
the simple process:
φnt =
Tn−1∑
i=0
Btni 1{t ∈ [tni , tni+1)}
and as n→∞, this looks more and more like Brownian motion,
and in particular, it can be shown that
∫ t
0 |φns − Bs |2 ds → 0.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Example
So we have convergence as required in the definition of the
stochastic integral. This means that the right-hand side should
converge to the stochastic integral as n→∞.
But by the continuity of Brownian motion, Bbtnc/n → Bt since
btnc/n→ t, and the sum of the squared Brownian increments
converges to t by Lemma 5.5.
Hence ∫ t
0
Bs dBs =
1
2
(
B2t − B20
)− 1
2
t. (14)
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Properties of Stochastic Integral
Theorem 6.7 (Properties of the Stochastic Integral)
Suppose (φt)t≥0, (ψt)t≥0 ∈ V, and (Bt)t≥0 is a Brownian
motion. Then the stochastic integrals:
It =
∫ t
0 φs dBs ,Jt =
∫ t
0 ψs dBs satisfy:
1 Continuity: t 7→ It is continuous;
2 Known: It is known at time t;
3 Linearity: if α, β ∈ R, then αφt + βψt ∈ V and:∫ t
0
(αφs + βψs) dBs = αIt + βJt ;
4 Martingale: It is a martingale;
5 Itô Isometry: E
[I2t ] = E [∫ t0 φ2s ds].
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
1 Introduction
2 Background and Revision
3 Fair Games & Martingales
4 Binomial Model
5 Fundamental Theorem of Asset pricing
6 Brownian Motion
7 Stochastic Integration
8 Stochastic Calculus
9 Continuous-time Finance
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Stochastic Integral equations
Recall our ‘intuitive’ notation, where dXt meant the change in
the process Xt over an interval from t to t + dt. Then∫ t
0
dXs ≈
∑
i :ti≤t
(Xtni+1 − Xtni ) ≈ Xt − X0 (15)
since all the other terms cancel out.
If we know that Xt is a stochastic integral, perhaps
Xt = x0 +
∫ t
0
αsdBs
we might write this instead as the ‘differential equation’:
dXt = αtdBt , X0 = x0.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Stochastic Integral equations
More generally, we may consider stochastic processes which
have a non-stochastic integral term as well:
Xt = x0 +
∫ t
0
αsdBs +
∫ t
0
βsds
which would be equivalent to:
dXt = αtdBt + βtdt, X0 = x0.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Stochastic Integral equations
For example, remember that we know (from (14)):
B2t = B
2
0 +
∫ t
0
2Bs dBs + t. (16)
However, using (15) with Xt = B2t ,∫ t
0
d(B2s ) = B
2
t − B20
and
t =
∫ t
0
1 ds
we can rewrite (16) in the form:
d(B2t ) = 2Bt dBt + 1 dt = 2Bt dBt + dt.
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
‘Rules’ of Stochastic Calculus
Theorem 7.1 (Rules of Stochastic Calculus)
1 We have (dBt)2 = dt and dBtdt = (dt)2 = (dBt)3 = 0;
2 If dYt = αt dBt + βt dt then:
Xt dYt = αtXt dBt + βtXt dt
or equivalently:∫ t
0
Xs dYs =
∫ t
0
αsXs dBs +
∫ t
0
βsXs ds;
3
d(XtYt) = Xt dYt + YtdXt + dXtdYt
where we use the rules in 1 to work out what the last term
is.
MA30089:
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Hartung
Introduction
Background
and Revision
Fair Games
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Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
‘Rules’ of Stochastic Calculus
Here, the first half of 7.1.1 is essentially Lemma 5.5.
To see why 7.1.3 might be true, observe that:
(x2y2 − x1y1) = x1(y2 − y1) + y1(x2 − x1) + (x2 − x1)(y2 − y1).
We can write this expression in ‘integrated’ form as:
XtYt − X0Y0 =
∫ t
0
d(XsYs)
=
∫ t
0
Xs dYs +
∫ t
0
Ys dXs +
∫ t
0
dXs dYs .
MA30089:
Stochastic
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and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Using the rules of stochastic
calculus
We can also compute our favourite stochastic integral using
7.1.1 and 7.1.3. If we take Xt = Yt = Bt :
d(BtBt) = d(B2t ) = Bt dBt + Bt dBt + dBt dBt ;
and using (dBt)2 = dt we get:
d(B2t ) = 2Bt dBt + dt
or equivalently:
B2t −B20 =
∫ t
0
d(B2t ) = 2
∫ t
0
Bs dBs +
∫ t
0
ds = 2
∫ t
0
Bs dBs +t.
We can repeat these kinds of calculations to find, for example,
d(B3t ).
Compute d(B3t )
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 15
• Example: ∫ t
0
Bs dBs =
1
2
(
B2t − B20
)− 1
2
t.
• Properties of Stochastic integral
• Stochastic Calculus notation:
Xt = x0 +
∫ t
0
αsdBs +
∫ t
0
βsds
is equivalent to:
dXt = αtdBt + βtdt, X0 = x0.
• ‘Rules’ of Stochastic Calculus
• (dBt)2 = dt, dBt dt = (dt2) = (dBt)3 = 0
• dYt = αt dBt + βt dt =⇒ Xt dYt =
αtXt dBt + βtXt dt
• d(XtYt) = Xt dYt + YtdXt + dXtdYt
• Example: using rules of stochastic calculus to compute
d(B3t ).
MA30089:
Stochastic
Processes
and Finance
Tobias
Hartung
Introduction
Background
and Revision
Fair Games
&
Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Itô’s Lemma
Lemma 7.2 (Itô’s Lemma)
Let f (x) be a twice-differentiable function. Then
d(f (Bt)) = f ′(Bt) dBt +
1
2
f ′′(Bt) dt, (17)
where f ′(x), f ′′(x) are the first and second derivatives.
More generally, if f (x , t) is a ‘nice’ function, then:
d(f (Bt , t)) =
∂f
∂x
(Bt , t) dBt +
(
1
2
∂2f
∂x2
(Bt , t) +
∂f
∂t
(Bt , t)
)
dt.
(18)
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Itô’s Lemma
Both these results also hold if we replace Bt by a process Xt
satisfying dXt = αt dBt + βt dt, provided we also replace dBt
by dXt and dt by (dXt)2 = α2t dt except in the term
accompanying ∂f∂t (Xt , t), so:
d(f (Xt)) = f ′(Xt) dXt +
1
2
f ′′(Xt) (dXt)2, (19)
and
d(f (Xt , t)) =
∂f
∂x
(Xt , t) dXt+
1
2
∂2f
∂x2
(Xt , t) (dXt)2+
∂f
∂t
(Xt , t) dt.
(20)
Sketch proof of (17)
Note that to remember Itô’s formula, you just need to
remember Taylor’s Theorem!
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Examples using Itô’s Lemma
Example 7.3
Use Itô’s Lemma to find dXt , when:
1 Xt = B
n
t , n ∈ N,
2 Xt = tBt ,
3 Xt = exp {αBt + βt}.
Example
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Identifying Martingales
Recall (Theorem 6.7.4) that stochastic integrals against a
Brownian motion are also martingales (at least, provided the
integrand is ‘nice’ in the sense of Theorem 6.7).
As a result, if we can write dXt as an expression which does not
involve any dt term, just a dBt term, then the process Xt is a
martingale.
This means, for example, from Example 7.3.3, that if
Xt = exp
{
αBt − 12α2t
}
, then:
dXt = αXt dBt +
(
1
2
α2 − 1
2
α2
)
Xt dt = αXt dBt .
So Xt is a martingale.
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Stochastic Differential Equations
Observe that in Example 7.3.3, we were able to write an
equation for Xt of the form:
dXt = σ(Xt) dBt + µ(Xt) dt, (21)
where σ(x) = αx and µ(x) =
(1
2α
2 + β
)
x .
Equations of this form are called Stochastic Differential
Equations (SDE), and we can define processes which are the
solution to an equation of the form (21), together with an
initial condition (such as X0 = x0) as solutions to a Stochastic
Differential Equation.
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Stochastic
Calculus
Continuous-
time
Finance
GBM as solution to an SDE
Recall that we conjectured our asset prices should be Geometric
Brownian Motion (GBM), where a GBM was defined
(Definition 5.2) to be the process Xt where
Xt = x0 exp
{(
ν − 1
2
ξ2
)
t + ξBt
}
and we call ν the infinitesimal drift and ξ the diffusion
coefficient.
From Example 7.3.3 we can see that GBM Xt with infinitesimal
drift ν and diffusion coefficient ξ is the solution to the SDE:
dXt = ξXt dBt + νXt dt, X0 = x0.
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Examples of SDEs
Example 7.4
Solve the following SDEs:
1 dXt = 3Xt2/3 dBt + 3X
1/3
t dt, X0 = 1;
2 dXt = αXt dBt , X0 = x0 ;
3 dXt = e−rt dBt − rXt dt, X0 = x0.
Example
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Last Lecture: Lecture 16
• Itô’s Lemma
• Taylor’s Theorem & Rules of Stochastic Calculus!
• Identifying martingales: only a dBt term, no dt term
• Stochastic Differential Equations: process Xt solves:
dXt = σ(Xt) dBt + µ(Xt) dt,
for some functions σ(x) and µ(x).
• GBM Xt with infinitesimal drift ν and diffusion coefficient
ξ is the solution to the SDE:
dXt = ξXt dBt + νXt dt, X0 = x0.
• Example of solving SDEs.
MA30089:
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Hartung
Introduction
Background
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Fair Games
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Binomial
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Fundamental
Theorem of
Asset pricing
Brownian
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Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Outline
1 Introduction
2 Background and Revision
3 Fair Games & Martingales
4 Binomial Model
5 Fundamental Theorem of Asset pricing
6 Brownian Motion
7 Stochastic Integration
8 Stochastic Calculus
9 Continuous-time Finance
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Fundamental
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Brownian
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Stochastic
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Stochastic
Calculus
Continuous-
time
Finance
Black-Scholes Model
We begin by choosing a model for our asset price:
Definition 8.1
We say an asset price St obeys the Black-Scholes model if:
St = s0 exp
{
σBt +
(
µ− 1
2
σ2
)
t
}
(22)
where σ is the volatility parameter, µ is the mean rate of return
of the asset, and s0 the value of the asset at time 0.
This means that St is the solution to the Stochastic Differential
Equation:
dSt = StσdBt + Stµdt, S0 = s0.
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Discounted Asset Price
In discrete time, we found it helpful to consider the discounted
asset price. In continuous time, we discount from time t to time
0 by multiplying by e−rt where r was the (continuously
compounded) interest rate.
So the discounted asset price will be:
S∗t = e
−rtSt = s0 exp
{
σBt +
(
µ− r − 1
2
σ2
)
t
}
and so also
dS∗t = S
∗
t σ dBt + S
∗
t (µ− r) dt (23)
= e−rt(Stσ dBt + St(µ− r) dt)
= e−rt(dSt − rSt dt) (24)
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Portfolio Value
What about the value of our portfolio? Suppose we have a
portfolio worth Vt at time t, which consists of φt units of the
asset (worth φtSt), and the remaining cash (Vt − φtSt)
invested in the bank account. Then at time t + dt, the portfolio
has value:
Vt+dt = φtSt+dt + erdt(Vt − φtSt)
since £1 in the bank at time t is worth £er(t+dt)−rt = £erdt at
time t + dt. Therefore:
dVt = Vt+dt − Vt
=
(
φtSt+dt + erdt(Vt − φtSt)
)
− (φtSt + Vt − φtSt)
= φt(St+dt − St) + (erdt − 1)(Vt − φtSt)
= φt dSt + r(Vt − φtSt) dt
since erdt ≈ 1 + rdt.
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Fundamental
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Brownian
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Stochastic
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Stochastic
Calculus
Continuous-
time
Finance
Discounted Portfolio Value
By itself, this isn’t very useful, but what happens if we consider
the discounted portfolio value, V ∗t = e−rtVt?
For example, by applying Itô’s Lemma to the function
f (x , t) = e−rtx , we get:
d(V ∗t ) = d(e
−rtVt)
= e−rtdVt − re−rtVtdt
= e−rt(φt dSt + r(Vt − φtSt) dt)− re−rtVtdt
= e−rtφt dSt − re−rtφtSt dt
= φt dS∗t (25)
where we have used (24) in the last line.
This result should not be unexpected!
Compare this last result with Lemma 4.3
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Brownian
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Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Girsanov’s Theorem
Theorem 8.2 (Girsanov’s Theorem)
Fix T > 0 and let (Bt)t∈[0,T ] be a standard Brownian motion.
Define
Zt = exp
{
αBt − 12α
2t
}
, t ∈ [0,T ],
for α ∈ R.
Let E˜ [·] be the change of measure of E [·] induced by the
Radon-Nikodym derivative ZT , so:
E˜ [Y ] = E [ZT · Y ]
for any r.v. Y .
Then B˜t = Bt − αt is a standard Brownian motion under E˜.
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Fundamental
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Brownian
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Stochastic
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Stochastic
Calculus
Continuous-
time
Finance
Girsanov’s Theorem: interpretation
Girsanov’s Theorem says that we can change the drift of the
Brownian motion through a suitable change of measure! How
might we use this?
Suppose we can ‘change’ the drift of the Brownian motion so
that µ = r by switching from Bt to B˜t , then (23) simplifies to:
dS∗t = e
−rtStσ dB˜t
so that S∗t is a martingale, which is exactly what we wanted.
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Background
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Fair Games
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Fundamental
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Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Girsanov’s Theorem
By Girsanov’s Theorem, under the probability measure P˜
(corresponding to E˜), B˜t = Bt − αt is a standard Brownian
motion.
If we substitute Bt = B˜t + αt into (22), we get:
St = s0 exp
{
σ(B˜t + αt) +
(
µ− 1
2
σ2
)
t
}
= s0 exp
{
σB˜t +
(
µ+ σα− 1
2
σ2
)
t
}
.
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Fundamental
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Stochastic
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Stochastic
Calculus
Continuous-
time
Finance
Asset price under P˜
We want the discounted asset price to be a martingale under P˜,
and we said this would happen exactly when we had r instead
of µ in the exponential, so we want:
µ+ σα = r =⇒ α = r − µ
σ
.
By analogy with the discrete case, we call P˜ the risk-neutral
probability measure, where P˜ is the change of measure of P
given by the Radon-Nikodym derivative:
ZT = exp
{
r − µ
σ
BT − 12
(
r − µ
σ
)2
T
}
.
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Hartung
Introduction
Background
and Revision
Fair Games
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Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Asset price as a martingale
Then we have the continuous-time analogue of Theorem 4.4:
Lemma 8.3
In the Black-Scholes model, under the probability measure P˜,
the discounted asset price S∗t is a martingale.
Proof.
Since Bt = B˜t + αt, then dBt = dB˜t + α dt, so (23) can be
rewritten as:
dS∗t = S
∗
t σ dBt + S
∗
t (µ− r) dt
= S∗t σ dB˜t + S
∗
t σ
(
r − µ
σ
)
dt + S∗t (µ− r) dt
= S∗t σ dB˜t .
And hence S∗t is a martingale under P˜ (since then B˜t is a
Brownian motion).
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Hartung
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Fundamental
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Brownian
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Stochastic
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Stochastic
Calculus
Continuous-
time
Finance
Martingale Representation Theorem
We now want to prove the Fundamental Theorem of Asset
Pricing in continuous time (c.f. Theorem 4.7). This result had
two main ingredients:
we needed our portfolios of traded assets to be martingales, and
we needed the Martingale Representation Theorem
(Theorem 4.6).
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Martingale Representation Theorem
Theorem 8.4 (Martingale Representation Theorem)
Let P˜ be a probability space on which (B˜t)t∈[0,T ] is a Brownian
motion. Let Y be a random variable with E˜
[
Y 2
]
<∞ which
depends only on (B˜t)t∈[0,T ]. Then there exists a process φt ,
which depends only on the Brownian motion up to time t such
that
Y = E˜ [Y ] +
∫ T
0
φs dB˜s .
We will not prove this result, but note simply that this is the
continuous-time equivalent of Theorem 4.6.
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Continuous-
time
Finance
Fundamental Theorem of Asset
Pricing
Theorem 8.5 (Fundamental Theorem of Asset Pricing in
Continuous Time)
Suppose the asset (St)t∈[0,T ] behaves according to the
Black-Scholes model, and suppose P˜ is the corresponding
risk-neutral measure. If CT is the payoff at time T of a
derivative, whose value depends only on the path of (St)t∈[0,T ],
then
there is no arbitrage if and only if the price of the derivative at
time t is:
Ct = e−r(T−t)E˜t [CT ] . (26)
We will not prove this (a non-examinable sketch proof is in the
notes), but note that the structure of the proof is very similar
to the proof of the FTAP in discrete time!
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Brownian
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Stochastic
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Stochastic
Calculus
Continuous-
time
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Last Lecture: Lecture 17
• Black-Scholes model: asset price St is GBM
• Discounted asset and portfolio value:
dS∗t = S
∗
t σ dBt + S
∗
t (µ− r) dt, dV ∗t = φtdS∗t
• Girsanov’s Theorem: If we define a change of measure to
E˜ [·] by the Radon-Nikodym derivative:
ZT = exp
{
αBT − 12α2T
}
. Then B˜t = Bt − αt is a
Brownian motion under E˜ [·]
• Girsanov =⇒ S∗t ,V ∗t are martingales under P˜
• Using Girsanov to find risk-neutral measure
• Martingale Representation Theorem
• FTAP: Absence of Arbitrage if and only if price of
derivative with payoff CT at time T is
Ct = e−r(T−t)E˜t [CT ] .
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Black-Scholes Formula
Theorem 8.6 (Black-Scholes Formula)
In the Black-Scholes model, the arbitrage-free price of a
European Call option with strike K and maturity T is:
C (K ,T ) = s0Φ(d1)− Ke−rTΦ(d2),
where
d1 =
log
(
s0
K
)
+
(
r + 12σ
2)T
σ
√
T
d2 =
log
(
s0
K
)
+
(
r − 12σ2
)
T
σ
√
T
= d1 − σ
√
T
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Black-Scholes Prices
• In the theorem, Φ(·) is the cumulative (standard) normal
distribution function,
Φ(x) =
∫ x
−∞
e−y2/2√
2pi
dy .
• Note that there is no µ appearing in the formula! This
should not be too surprising: under the risk-neutral
measure, µ is ‘replaced’ by r , so µ should not appear in the
option price.
• To understand how the Black-Scholes formula behaves, we
look at the prices for different parameter values.
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Black-Scholes Prices: Volatility
K
Price
10050 150 200
100
50
Volatility σ: upper curve is σ = 0.94, lower curve is σ = 0.56.
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Black-Scholes Prices: Asset Price
K
Price
10050 150 200
100
50
Asset price s0: upper curve is s0 = 125, lower curve is s0 = 75.
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Black-Scholes Prices: Maturity
K
Price
10050 150 200
100
50
Maturity T : upper curve is T = 0.63, lower curve is T = 0.38.
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Continuous-
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Black-Scholes Prices: Interest Rate
K
Price
10050 150 200
100
50
Interest Rate r : upper curve is r = 0.063, lower curve is
r = 0.038.
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Proof of Theorem 8.6
We use the fact that the price of the call option is the
expectation of the payoff under the risk-neutral probability.
So the price is given by:
C (K ,T ) = E˜
[
e−rT (ST − K )+
]
= E˜
[
e−rT
(
s0 exp
{
σB˜T + (r − 12σ
2)T
}
− K
)
+
]
Since the (. . .)+ means we only need count when this is
positive, we can compute the expectation over the set where
s0 exp
{
σB˜T + (r − 12σ
2)T
}
≥ K
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Proof of Theorem 8.6
In particular, the payoff is positive if:
B˜T ≥ 1
σ
[
log
(
K
s0
)
− (r − 1
2
σ2)T
]
=: λ
where we write λ for the term on the right.
If we write E˜ [X ;A] to mean the expectation of X on the set A,
we get
C (K ,T ) = e−rT E˜ [(ST − K )+]
= e−rT E˜ [ST − K ; ST ≥ K ]
= e−rT E˜ [ST ;ST ≥ K ]− e−rT E˜ [K ; ST ≥ K ] .
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Stochastic
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Stochastic
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Continuous-
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Proof of Theorem 8.6
Since K is just a constant, the second term here,
E˜ [K ; ST ≥ K ], must be K times the probability (under the
risk-neutral probability) that the process ends above K .
For the first term, we get:
e−rT E˜ [ST ; ST ≥ K ] =
Complete
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Brownian
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Stochastic
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Stochastic
Calculus
Continuous-
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Finance
Proof of Theorem 8.6
Since B˜T is a Brownian motion (under P˜), B˜T ∼ N(0,T ), and
we can calculate:
E˜
[
eσB˜T ; B˜T ≥ λ
]
=
Complete
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Proof of Theorem 8.6
This final integrand is the density of an N(0,T ), and this is
integrated between λ− Tσ and ∞, so that this is just:
1− Φ
(
λ− Tσ√
T
)
= Φ
(
Tσ − λ√
T
)
but note also that
Tσ − λ√
T
=
Tσ − 1σ
[
log
(
K
s0
)
− (r − 12σ2)T
]
√
T
=
Tσ2 + log
(
s0
K
)
+ (r − 12σ2)T
σ
√
T
=
log
(
s0
K
)
+ (r + 12σ
2)T
σ
√
T
= d1
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Proof of Theorem 8.6
This gives us:
e−rT E˜ [ST ;ST ≥ K ] = s0e−
1
2σ
2T E˜
[
exp
{
σB˜T
}
; B˜T ≥ λ
]
= s0e−
1
2σ
2T eTσ
2/2Φ(d1)
= s0Φ(d1)
We now consider the term:
e−rT E˜ [K ; ST ≥ K ]
But since K is fixed, and {ST ≥ K} = {B˜T ≥ λ}, this term is
just Ke−rT times the probability a Brownian motion at time T
is above λ.
MA30089:
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Hartung
Introduction
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Fair Games
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Binomial
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Fundamental
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Brownian
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Stochastic
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Stochastic
Calculus
Continuous-
time
Finance
Proof of Theorem 8.6
Which is the probability that a N(0,T ) is greater than λ:
1− Φ
(
λ√
T
)
= Φ
(
− λ√
T
)
= Φ
− 1σ
[
log
(
K
s0
)
− (r − 12σ2)T
]
√
T

= Φ
([
log
(
s0
K
)
+ (r − 12σ2)T
]
σ
√
T
)
= Φ (d2)
Putting this all together, we get the price of the option to be:
C (K ,T ) = s0Φ(d1)− e−rTKΦ(d2).
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Black-Scholes formula at general
time
We can also use the Black-Scholes formula to give us the price
of a Call option with maturity time T at any time t < T .
If the price of the asset a time t is St , then this is equivalent to
having a call option with maturity date T − t and with initial
asset price St .
So we can find the price of the call option at time t by
replacing T by T − t and s0 by St .
If we write the price as f (St , t) then f is given by:
f (x , t) = xΦ
(
log
(
x
K
)
+
(
r + 12σ
2) (T − t)
σ
√
T − t
)
− e−r(T−t)KΦ
(
log
(
x
K
)
+
(
r − 12σ2
)
(T − t)
σ
√
T − t
)
.
(27)
MA30089:
Stochastic
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Tobias
Hartung
Introduction
Background
and Revision
Fair Games
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Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 18
• Black-Scholes Formula: arbitrage-free price of a European
Call option with strike K and maturity T is:
C (K ,T ) = s0Φ(d1)− Ke−rTΦ(d2)
• Examined behaviour of call price as parameters change
• Proof of Black-Scholes formula:
C (K ,T ) = e−rT E˜ [(ST − K )+]
= e−rT E˜ [ST ;ST ≥ K ]− e−rT E˜ [K ; ST ≥ K ]
• Key step:
e−rT E˜ [ST ;ST ≥ K ] = s0e−
1
2σ
2T E˜
[
exp
{
σB˜T
}
; B˜T ≥ λ
]
and
E˜
[
exp
{
σB˜T
}
; B˜T ≥ λ
]
= Φ(d1)
• Black-Scholes Formula at a General Time
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Stochastic
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Stochastic
Calculus
Continuous-
time
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Put-Call Parity
Once we have a formula for the Call price, we can derive the
price of a Put option with the same strike and maturity using
Put-Call Parity.
The Payoff of a Call option is: (ST − K )+, and the payoff of a
Put option is:
(K − ST )+ = max{0,K − ST}
= max{0,−(ST − K )}
= −min{0, ST − K}
So:
(ST−K )+−(K−ST )+ = max{0, ST − K}+ min{0,ST − K}
= ST − K .
Show Graphically
MA30089:
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Hartung
Introduction
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Binomial
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Fundamental
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Brownian
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Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Put-Call Parity
In particular,
(K − ST )+ = (ST − K )+ − ST + K .
If we write P(K ,T ) for the price of the Put option with strike
K and maturity T , and C (K ,T ) for the price of the
corresponding call option, from the FTAP we get:
P(K ,T ) = e−rT E˜ [(K − ST )+]
= e−rT E˜ [(ST − K )+ − ST + K ]
= e−rT E˜ [(ST − K )+]− E˜
[
e−rTST
]
+ e−rT E˜ [K ]
= C (K ,T )− s0 + e−rTK ,
where we have used the fact that S∗t is a martingale (under P˜),
so E˜
[
e−rTST
]
= S∗0 = s0 and K is a constant, so we can ‘take
out what is known’. This formula is known as the Put-Call
Parity formula.
MA30089:
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Hartung
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Fundamental
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Brownian
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Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Put-Call Parity
Finally, recall that the Black-Scholes formula for the Call price
was:
C (K ,T ) = s0Φ(d1)− Ke−rTΦ(d2),
We can rearrange to get a formula for the price of the Put
option:
P(K ,T ) = s0Φ(d1)− Ke−rTΦ(d2)− s0 + e−rTK
= s0(Φ(d1)− 1) + Ke−rT (1− Φ(d2))
= Ke−rTΦ(−d2)− s0Φ(−d1).
(Recall that 1− Φ(x) = Φ(−x) by properties of the standard
Normal distribution).
MA30089:
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Binomial
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Fundamental
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Brownian
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Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Hedging in Continuous time
Finally, we consider the problem of hedging an option in
continuous time. Can we find the hedging portfolio?
Suppose I sell an option with payoff g(ST ) at time T , and I
wish to trade in the underlying stock in a way that I have
g(ST ) at time T .
Suppose the option is worth f (St , t) at time t, and we are short
the option, long φt units of the asset, and hold f (St , t)− φtSt
in the bank,
so our portfolio has value
Vt = (f (St , t)− φtSt) + φtSt − f (St , t) (the value of the cash,
asset and option terms respectively).
MA30089:
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Hartung
Introduction
Background
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Binomial
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Fundamental
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Brownian
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Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Hedging in Continuous time
Now consider the change in the value of our portfolio from time
t to time t + dt. For each pound in the bank, the bank account
grows by r dt,
while the change in our holding in the asset is φt times the
change in St , and the change in the value of the option is
d(f (St , t)).
Then:
dVt = (f (St , t)− φtSt)r dt + φt dSt − d(f (St , t)).
MA30089:
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Binomial
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Fundamental
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Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Hedging in Continuous time
By Itô’s Lemma:
d(f (St , t)) =
∂f
∂x
(St , t) dSt +
1
2
∂2f
∂x2
(St , t) (dSt)2 +
∂f
∂t
(St , t) dt.
Also
(dSt)2 = (Stσ dB˜t + rSt dt)2
= S2t σ
2 (dB˜t)2 + 2S2t σr dB˜t dt + r
2S2t (dt)
2
= S2t σ
2 dt
MA30089:
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Hartung
Introduction
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and Revision
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Binomial
Model
Fundamental
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Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Hedging in Continuous time
So
dVt = (f (St , t)− φtSt)r dt +
(
φt − ∂f
∂x
(St , t)
)
dSt
−
(
1
2
S2t σ
2 ∂
2f
∂x2
(St , t) +
∂f
∂t
(St , t)
)
dt.
Now suppose we choose φt = ∂f∂x (St , t).
Then the dSt term disappears and we have:
dVt =
(
rf (St , t)− rSt ∂f
∂x
(St , t)
− 1
2
S2t σ
2 ∂
2f
∂x2
(St , t)− ∂f
∂t
(St , t)
)
dt.
MA30089:
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Tobias
Hartung
Introduction
Background
and Revision
Fair Games
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Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Hedging in Continuous time
Remember that we chose our portfolio such that Vt = 0. If the
dt term is greater than 0, we can create a portfolio worth 0 at
time t which will have a strictly positive value at time t + dt.
This means that an arbitrage exists!
A similar argument holds if the dt term is less than zero if we
buy the option, go short ∂f∂x (St , t) units of the asset, and
borrow/invest the rest in the bank.
MA30089:
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Tobias
Hartung
Introduction
Background
and Revision
Fair Games
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Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Hedging in Continuous time
1 If the price of the option at time t is f (St , t) and we wish
to hedge a short position in the option, we should hold
∂f
∂x (St , t) units of the asset. The amount
∂f
∂x (St , t) is often
called the delta of the option.
2 If the price of the option at time t can be written as a
function f (St , t), then the function f (x , t) should satisfy
the Black-Scholes PDE:
rx
∂f
∂x
(x , t) +
1
2
x2σ2
∂2f
∂x2
(x , t) +
∂f
∂t
(x , t) = rf (x , t).
The Black-Scholes PDE gives us another way to try and
compute option prices: try to solve the PDE together with
a boundary condition f (x ,T ) = g(x), where g(ST ) is the
payoff of the option at the maturity date.
MA30089:
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Tobias
Hartung
Introduction
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Fair Games
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Binomial
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Fundamental
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Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Black-Scholes Delta
Example 8.7
Compute the Delta of a Call option under the Black-Scholes
model.
Recall the value of a Call option at time t with asset price St ,
strike K and maturity T was given by (27). To find the delta of
the option, we differentiate f (x , t) by x (that is, the St
variable).
Find ∂f∂x
MA30089:
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Hartung
Introduction
Background
and Revision
Fair Games
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Binomial
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Fundamental
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Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Last Lecture: Lecture 18
• Put-Call Parity: P = C − s0 + Ke−rT
• Hedging: if f (x , t) is arbitrage-free price of option at time
t when St = x , to hold a portfolio with no risk (i.e. hedge)
should have:
φt =
∂f
∂x
(St , t)
units of asset at time t. This is the delta of the option.
• In addition f (x , t) satisfies the Black-Scholes PDE:
rx
∂f
∂x
(x , t) +
1
2
x2σ2
∂2f
∂x2
(x , t) +
∂f
∂t
(x , t) = rf (x , t).
• Computed delta of a Call option: Φ(d1).
MA30089:
Stochastic
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Tobias
Hartung
Introduction
Background
and Revision
Fair Games
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Martingales
Binomial
Model
Fundamental
Theorem of
Asset pricing
Brownian
Motion
Stochastic
Integration
Stochastic
Calculus
Continuous-
time
Finance
Beyond Black-Scholes
• The Black-Scholes theory relies on a number of
assumptions about the way the market behaves
• To get more accurate prices, we need to find ways of
relaxing the assumptions, and constructing more
complicated models — for example: stochastic volatility
models. σ is not a constant, but (for example) σ2t is the
solution to:
d(σ2t ) = θ(σ
2 − σ2t ) dt + ξσt dBt .
• Other important issues: stochastic interest rates,
American-style options, market incompleteness, market
imperfections (transaction costs, illiquid markets, taxes),
energy & commodity markets, . . .
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