MAS362/462/6053/61017 Financial Mathematics

Lecture Notes1

Sam Marsh

Semester 1, 2023

1basedonnotesbyMotyKatzmanandDimitriosRoxanas

Contents

What is this course about? 5

1. What does “correct price” mean? 5

2. Portfolio Theory 6

Chapter 1. Interest, present value and bonds 9

1. Interest, periodic and continuous compounding 9

2. Present value 11

3. Bonds 13

4. Discount and yield curves 14

5. Forward rates 16

6. Exercises 17

Chapter 2. Forward and Futures Contracts 19

1. Short-selling 19

2. The forward price 20

3. Foreign exchange forward contracts 22

4. Futures contracts 22

5. Exercises 23

Chapter 3. Options 25

1. The payoffs of options 25

2. Elementary inequalities satisfied by option prices 26

3. Optimal exercise for American style options 27

4. Parameters affecting the prices of options 28

5. Exercises 28

Chapter 4. Binomial trees and risk neutral valuation 31

1. A motivating example 31

2. Derivatives in a simple, up-down world 32

3. An example 33

4. A two step process 33

5. An example 34

6. n-step trees 35

7. Example: an American option 36

8. Using binomial trees for approximating values of derivatives - Not examinable 37

9. Exercises 38

Chapter 5. The stochastic process followed by stock prices 39

1. Brownian motion 39

3

4 CONTENTS

2. The Ito integral 40

3. Modelling stock prices 41

4. Ito’s Lemma. 43

5. Exercises 45

Chapter 6. The Black-Scholes pricing formulas 47

1. The Black-Scholes differential equation 47

2. Boundary conditions 48

3. The Black-Scholes pricing formulas 49

4. More exotic derivatives 50

5. The Black-Scholes pricing formulas: the risk neutral valuation approach. 51

6. Volatility 52

7. Exercises 52

Chapter 7. Portfolio Theory 55

1. Axioms satisfied by preferences 55

2. Indifference curves 56

3. Portfolios consisting entirely of risky investments 57

4. The feasible set 58

5. Efficient portfolios 58

6. Different choices of portfolios for different appetites for risk 59

7. Portfolios containing risk-free investments 60

8. Exercises 61

Chapter 8. The Capital Asset Pricing Model 63

1. The market portfolio. 63

2. The market price of risk 64

3. Exercises 65

Solutions 67

Chapter 1 67

Chapter 2 69

Index 71

What is this course about?

In this course you will learn some basic facts about an important component of modern life:

finance. This is a mathematics course hence we will be interested in mathematical ideas used in the

context of finance. Our task is to build a mathematical model of financial securities. A crucial first

stageisconcernedwiththepropertiesofthemathematicalobjectsinvolved. Thisisdonebyspecify-

ing a number of assumptions, the purpose of which is to find a compromise between the complexity

of the real world and the limitations and simplifications of a mathematical model imposed to make

ittractable.1 Theassumptionswearegoingtomakereflectourcurrentpositiononthiscompromise

and will be modified in the future. You may find that some situations look simple based on these

assumptions; however I hope you will see they can be sufficiently interesting to convey the flavour

of the theory to be developed later on.

Specifically, in this course we aim to address two questions:

(1) What is the “correct price” of financial assets?

(2) What are optimal investment strategies?

ThefirstquestionwillleadustotheBlack-Scholespricingformula,whichearnedRobertC.Mer-

ton and Myron S. Scholes the 1997 Nobel prize in Economics. William F. Sharpe and Harry M.

Markowitz received the 1990 Nobel prize in Economics for answering the second question.

1. What does “correct price” mean?

Lets first see an example of an incorrect price. Suppose that a barrel of oil trades at £100 (i.e.,

one can buy it for £100 and one can sell it for £100). I want to start trading in oil, too. Would

£120 be a correct price? No! Other traders will buy oil in large quantities for £100, sell it to me for

£120. Everyone except me will become rich, I end up with a pile of barrels of oil I can’t sell. Would

£90 be a correct price? Think about it. (The answer is “no”).

Amoresubtleexample: considerthreemerchantswhoarewillingtobuyandsellbagscontaining

apples and oranges (all of identical size and quality) as follows:

Bag content Price

Merchant A 3 apples, 2 oranges £5

Merchant B 2 apples, 3 oranges £6

Merchant C 4 apples, 3 oranges £8

To see that these prices are incorrect, let me show you how to get rich:

(a) borrow £36 for a short while (with negligible interest);

(b) buy 6 bags from merchant A and buy 1 bag from merchant B for a total cost of £36;

1Thereisawell-knownrelevantquoteforthis: “Allmodelsarewrong,somemodelsareuseful”.

5

6 WHAT IS THIS COURSE ABOUT?

(c) rearrange the fruit in five bags of 4 apples and 3 oranges each;

(d) sell the five bags to merchant C for £40;

(e) return the £36 loan and pocket a £4 profit.

Repeat this process until you are very rich.

Lots and lots of people would be buying from merchants A and B and selling to merchant C.

WhenthishappensthepricesofbagsAandBwillriseandthoseofbagCwillfall, andthisprocess

will continue until there are no more easy profits to be made.

Notice:

(a) we didn’t assume an intrinsic or objective price of apples and oranges,

(b) correct pricing is an equilibrium price,

(c) we expose incorrect pricing by exhibiting profit-making trading strategies which change

prices.

The making of a certain profit with no investment is called arbitrage. Correct pricing means

setting the unique price which does not introduce arbitrage opportunities. The correct price of bag

Cwouldbe£36/5asthatistheuniquepriceforwhichthestrategyabove(oritsopposite)doesnot

produce a profit. We call this method of pricing no-arbitrage pricing, which is also known colloqui-

ally as “There is no such thing as a free lunch!”

Thegeneralideaforsucharguments, usedtofindthe“correct”priceofsomething, istousethe

assumptionofnoarbitragetoshowthatthecorrectpricecanbeneitherhighernorlowerthanwhat

weclaimtobetherightone. The guiding principle for such arguments is that if something

has a price that is perceived to be “too low”, we buy it in the hopes that we can sell

it for a higher price and make a profit; similarly, if we think that something is priced

“too high”, we seek to sell it.

No-arbitrage pricing is the most fundamental (and mathematically fruitful) assumption we will

make about the market. It is an assumption that’s close enough to reality. As we will witness our-

selves in this course, arguments based on the no-arbitrage-principle are the main tools of financial

mathematics. Arbitrageopportunitiesrarelyexistinpractice. Thisisreasonable: ifarbitrageexists,

arbitragetradingwilloccurandpriceswillquicklychangetoeliminatethesearbitrageopportunities.

Ifandwhentheydo,thegainsaretypicallyextremelysmallcomparedtothevolumeoftransactions,

makingthembeyondthereachofsmallinvestors. Theycanhoweverarisewhenmarketparticipants

make a “mistake”, but they are usually very subtle, short-lived and difficult to spot.

2. Portfolio Theory

Arethereinvestmentstrategieswhicharebetterthanothers? Andwhatcould“better”meanin

thiscontext? Wearegreedy,i.e.,wewanthighreturns. Butaveragehighreturnsarerisky! Suppose

you are offered to make a bet on the outcome of tossing a fair coin: heads – you lose everything

you own, including your clothes, and are forced to live on the street; tails – you double your wealth.

Would you bet? This is a fair game, i.e., the expected gain is zero, but most people would not

bet. People are not only greedy – they also have an aversion towards risk. Even if a win triples or

quadruples their wealth, most people would not place a bet. If we used a coin which produces tails

with 99% probability some people might choose to place a bet. Now the expected gain may tempt

some, but not others.

2. PORTFOLIO THEORY 7

There is an interplay between expected returns and risk investors are willing to take. So a

“good”waytoinvestisastrategythatcompensatesforitsriskwithanappropriatelyhighexpected

return.

The last part of the course will make this statement precise: we will see that these optimal

investments exist and consist of portfolios 2 of risk-free deposits/bonds and market-index trackers.

2Portfolio: acollectionofassets

CHAPTER 1

Interest, present value and bonds

From our daily experience we understand that £100 to be received after one year is worth less

than the same amount today. The main reason is that money due in the future (or locked in a

fixed term account) cannot be spent right away. One would therefore expect to be compensated for

postponed consumption. In addition, prices may rise in the meantime, and in that scenario, this

amount will not have the same purchasing power as it would have in the present. Finally, there is

always a risk, even if a negligible one, that the money will never be received. Whenever a future

paymentisuncertaintosomedegree,itsvaluetodaywillbereducedtocompensatefortherisk. For

now, we shall restrict ourselves to risk-free assets, such as a bank account or a bond (more about

bonds later in this chapter.)

Thewayinwhichmoneychangesitsvalueintimeisacomplexissueoffundamentalimportance

in finance. This topic is often referred to as the time value of money. We will be mainly addressing

the following two questions:

• What is the future value of an amount invested (or borrowed) today ?

• What is the present value of an amount to be paid or received at a certain time in the

future ?

1. Interest, periodic and continuous compounding

Money can be lent and thus earn interest; think of interest as getting paid “rent” for your

money. For what follows, suppose that an amount P is paid into a bank account (you are “lending

thebank”), whereitistoearn(or“accrue”)interest. Thefuture valueofthisinvestmentconsistsof

the original deposit, called the principal, plus all the interest earned since the money was deposited

in the account. We will assume that the principal A is attracting annual interest at a constant rate

r > 0. Interest rates apply for a given period, and interest is often compounded a fixed number

of times in that period. In other words, the interest earned will now be added to the principal

periodically, for example, annually, semi-annually, quarterly, monthly, daily, etc. Subsequently,

interest will be attracted not just by the original deposit, but also by all the interest earned so far.

In such cases, we talk of discrete or periodic compounding.

Example. Let’s consider the case of monthly compounding. The first interest payment of r A

12

willbedueafteronemonth(thisishowmuchinterestouroriginaldepositwillattractinonemonth).

Therefore, now our principal (how much we currently have in the bank) is A+ r A = (1+ r )A.

12 12

This whole amount will now attrack interest in the future. The next interest payment, due after

two months (2 since we started, 1 month since the last payment), will be r (1+ r )A, increasing

12 12

our total capital (what we have in the bank) to (1+ r )2A (check this!). After one year, our capital

12

will have become (1+ r )12A, after m months it will be (1+ r )mA, and after n years it will be

12 12

(1+ r )mnA. The last formula admits n equal to a whole number of months, that is, a multiple of

12

1 .

12

9

10 1. INTEREST, PRESENT VALUE AND BONDS

In general, if m interest payments are made per annum, the time between two consecutive

paymentsmeasuredinyearswillbe 1,thefirstpaymentbeingdueattime 1.Eachinterestpayment

m m

will increase the principal by a factor of 1+ r. Given that the interest rate remains unchanged,

m

after n years the future value of an initial principal A, will become

(cid:16) r (cid:17)mn

(1) V(n)= 1+ A,

m

because there will be nm interest payments during this period. In this formula n nust be a whole

multiple of the period 1. The number (cid:0) 1+ r(cid:1)mn is referred to as the growth factor.

m m

To summarize, an amount A invested for n years at a yearly interest rate r com-

pounded m times a year, yields at the end of this period

A(cid:0)

1+

r(cid:1)mn

.

m

The exact value of the investment may sometimes need to be known at time instants between

interest payments. In particular, this may be so if the account is closed on a day when no inter-

est payment is due. For example, what is the value after 10 days of a deposit of £100 subject

to monthly compounding at 10%? One possible answer is £100, since the first interest payment

wouldbedueonlyafteronewholemonth. Itishoweverpossibletoextend(1)foralln≥0,notjust

integermultiplesofthecompoundingfrequency,andfromnowthiswillbeastandingassumption.

(Under this assumption, that we allow for any t≥0) formula (1) for the future value at time t

of a principal A attracting interest at a rate r >0 compounded m times a year can be written as

(cid:20)(cid:16)

r

(cid:17)m(cid:21)tr

V(t)= 1+ r A,

m

Now think of a case of periodic compounding with the frequency m is very large (e.g., 1 second or

even milliseconds). In the limit as m→∞, we obtain1

(2) V(t)=etrA,

This is known as continuous compounding (with corresponding growth factor etr). Formula (2) is a

good approximation of the case of periodic compounding when the frequency m is very large. Since

it is a simpler formula, which is easier to manipulate and transform when necessary (as we will see

many times in the course), in this course we want to work with continuously compounded interest

rates (which is also the one most commonly used in practice).

To summarize, an amount A invested for t years accruing a yearly interest of r com-

pounded continuously, yields at the end of this period £Aert.

As a closing remark for this section, note that the same payment of interest can be described

usingdifferentinterest-compoundingconventions. Forexample,adepositof£100yieldingabalance

of £110 in one year pays

• 10% annual interest compounded yearly,

√

• 2( 1.1−1)≈9.76% annual interest compounded every six months,

• ln1.1≈9.53% annual interest compounded continuously.

(cid:18) 1(cid:19)n

1Recallthat lim 1+ =e.

n→∞ n

2. PRESENT VALUE 11

(Quick Exercise: verify the claims in the above example.)

Unless stated otherwise, we always assume that interest is compounded continu-

ously.

2. Present value

The notion of interest is linked to another important notion: present value.

Definition. The present value of apayment occurring in the future, isthe present price of the

entitlement to that payment.

Suppose that a deposit of £A yields £B when we withdraw it in t years (with A < B). We

can now ask how much would we pay now to receive £B in t years. The answer is, of course £A!

Depositing £A now is equivalent to paying that amount now to receive £B in t years. We say that

the present value of £B paid in t years is £A.

To understand better the notion of present value, let’s look at a numerical example:

Example. How much are people willing to pay now in order to receive £1 in 10 years? We

make the assumption that 10-year deposits pay 3% per year compounded yearly.

Call the amount we are willing to pay now, X. If you deposit £X for 10 years you end up with

£X ×1.0310 ≈ 1.34×X. To obtain £1 in 10 years, you need to deposit £1/1.34 ≈ 0.74 now. I

claim that this is the correct present value of £1 paid in 10 years. Let’s see why. In what follows, A

denotes the amount of money people are willing to pay now to receive £1 in 10 years. We will see

that if A is either higher or lower than 1.03−10, there is a strategy that allows us to take advantage

and get a free lunch!

To make the argument that follows easier to understand, imagine that everybody in the market

can issue certificates that declare “the holder of this certificate is to be paid £1, 10 years from this

date”,andhasthesignatureoftheperson/institutionissuingit,thereforemakingitalegalcontract

that people have to uphold. Our original question can be thus rephrased in terms of this certificate:

“what is the correct price for this certificate? How much are people willing to pay now to get £1 in

10 years? (by showing this certificate to the issuer)”.

Case 1: Assume that people are willing to pay A>1.03−10 for this certificate. We believe that

thispriceis“toohigh”,soaccordingtoourguidingprincipleforsomethingthatispriced“toohigh”,

we issue such a certificate, sign it, and sell it to them for £A. Then we deposit A in the bank, and

since the interest rate is 3%, we will collect A×1.0310 > 12 after 10 years (that’s the capital for a

principal deposit of A). At the same time (this is after 10 years), they will come to us and show us

the certificate we issued. We are under the obligation to pay them £1, and so we do. But notice:

our capital in the bank after 10 years is A×1.0310, and we have to pay the holder of the certificate

£1, so we pocket the profit A×1.0310−1>0.

2usingtheassumptionA>1.03−10 andrearranging

12 1. INTEREST, PRESENT VALUE AND BONDS

Case 2: Now assume that people are willing to pay A < 1.03−10. We think this price to be

“too low” and we saw that in such cases we will buy the asset (here, the certificate). So, we will

borrow £A from the bank and spend them on buying the certificate. This means that we can now

present the certificate (in 10 years time) and collect £1. This is exactly what we do, wait 10 years

and “exchange” the certificate for £1. We have to pay the bank back (with interest), that will be

A×1.0310 <13. But then we make a profit of 1−A×1.0310 >0 (collect 1, pay A×1.0310 >0 and

pocket the positive difference.)

In both cases, a guaranteed profit was achieved with no investment whatsoever; a free lunch!

This contradicts our assumption that there are no arbitrage opportunities in the market, which

implies that A=1.03−10 ≈0.74 pounds. Consequently, the present value of £1 paid 10 years from

now is 1.03−10.

Example. An annuity is a sequence of finitely many payments of a fixed amount due at equal

time intervals. An example of this can be a mortgage, which is a loan given with some asset held as

a “security” for the loan. If you want to buy a house, you can ask to get a mortgage, which is the

loan to buy the house; if you don’t repay the loan you will lose your house.

Consider a £100,000 mortgage paying a fixed annual interest rate of 6% which is compounded

monthly. What are the monthly repayments if the mortgage is to be repaid in 20 years?

The loan will be repaid as a sequence of finitely many payment of a fixed amount every month.

Call this monthly repayment P, which is this number we seek to calculate. We can find the value

of P in two steps.

First, let’s ask ourselves how much is the bank willing to pay you now in order to receive £P

in m months? (e.g. £P in 3 months; it’s always the same amount P that the bank will receive).

Basically, we are looking for the present value of £P in m months. Under monthly periodic com-

pounding at a fixed annual rate of 6%, £1 now will be worth £(cid:0) 1+ 0.06(cid:1)m in m months4 therefore

12

the present value of £P in m months is clearly P ×(cid:0) 1+ 0.06(cid:1)−m .

12

This means in particular, that an amount of £P in 1 month has a present value of P ×

(cid:0) 1+ 0.06(cid:1)−1 , an amount of £P in 6 months has a present value of P ×(cid:0) 1+ 0.06(cid:1)−6 , etc. In other

12 12

words, the present value is exactly how much the bank is willing to pay now in order to receive £P

in m months.

The second step, is to now find the present value of all such monthly payments of £P for 20

2 (cid:88)0×12 (cid:18) 0.06(cid:19)−m

years. This is P 1+ and we must have

12

m=1

(cid:88)240 (cid:18) 0.06(cid:19)−m (cid:88)240 (cid:18) 200(cid:19)m

100,000=P 1+ =P

12 201

m=1 m=1

3exactlyasbeforeusetheassumptiononAandrearrange

4thinkofdepositing£1inthebankandcollectinginterestattheprescribedrate

3. BONDS 13

The latter is a geometric sum that we can explicitly calculate; then solving for P :

100,000 100,000 100,000

P = = = ≈716.43,

(cid:80)240 (cid:0)200(cid:1)m 2001−(200/201)240 200(cid:16) 1−(cid:0)200(cid:1)240(cid:17)

m=1 201 201 1/201 201

which is how much you will be paying the bank every month for the next 20 years.

3. Bonds

The money market consists of risk-free securities. An example is a bond, which is a financial

security promising the holder a sequence of guaranteed future payments. Risk-free here means that

these payments will be delivered with certainty.

Definition. A bond is a contract in which the issuer commits to pay the holder of the bond

payments of certain amounts on certain dates.

Typically, a bond would specify

• a maturity date, which is the date when final payments are made,

• a face value, which is an amount paid on the maturity date, and

• coupons: a sequence of payments by the issuer, their amounts and payment dates.

In effect, the person or institution who buys the bond is lending money to the bond writer.

Bonds are issued mainly by governments and corporations often in auctions, and can be traded

in exchanges (sometimes referred to as secondary markets). There are many kinds of bonds, like

treasurybillsandnotes,mortgageanddebenturebonds,commercialpapers,andotherswithvarious

arrangements concerning the issuing institution, duration, number of payments, embedded rights

and guarantees. 5

Example. Suppose that we have a bond with face value of £100, paying 6% annual interest,

with 2 years to maturity and with semi-annual coupons.

This means that the bond pays the holder 6% of the face value (0.06×100=6) in two install-

mentseveryyear(becauseithassemi-annualcoupons),whichmeansthatitpays£3every6months.

Consequently, this bond entitles its owner to the following payments:

£3 in 6 months (first coupon payment),

£3 in 12 months (second coupon payment),

£3 in 18 months (third coupon payment), and

£3+100 in 24 months (fourth -and last- coupon payment plus face value since we have reached the

maturity date).

The simplest case of a bond is a zero-coupon bond, which involves just a single payment. (In

practice, this is not very common, but it is useful for theoretical purposes.)

Definition. A zero coupon bond has no coupons, i.e., the issuer makes only one payment: the

face value is paid at the maturity of the bond.

ThepriceP ofa£1facevaluezerocouponbondwithmaturityintyearsispreciselytheamount

of money you are willing to pay now in order to be receive £1 in t years. So P is the present value

of £1 paid t years into the future.

5YoucanfinddescriptionsofbondsissuedbytheUKgovernmenthere.

14 1. INTEREST, PRESENT VALUE AND BONDS

Proposition 1. Let P be the price of a zero-coupon bond with face value £1 and maturing in

t years. Let r be the (continuously compounded) interest rate for t-year deposits and loans. Then

P =e−rt.

In proving this statement we will make our usual assumption about the world: there are no

arbitrage opportunities. We also assume that everyone can issue bonds, lend and borrow money

under the same terms (these assumptions certainly don’t apply to you and me, but they do apply

approximately to stable governments and large corporations).

Proof. We will consider two cases (higher/lower and argue by contradiction)

If P >e−rt 6,

• issue a zero-coupon bond with face value of £1 and maturing in t years, sell it, and collect

its price £P,

• deposit £P for t years earning a (continuously compounded) interest rate of r,

• wait t years,

• collect the balance of your deposit amounting to Pert,

• pay the owner the face value of the bond you issued amounting to £1,

• pocket Pert−1>0.

This strategy, which required no initial investment and produced a certain profit, is an arbitrage

strategy, which we assume cannot exist.

If P

• borrow £P for t years; this will earn the lender a (continuously compounded) interest rate

of r,

• use these £P to buy a zero-coupon bond with face value of £1 and maturing in t years,

• wait t years,

• receive the face value of your bond amounting to £1,

• repay the balance of your loan amounting to Pert,

• pocket 1−Pert >0.

This is another arbitrage strategy, which we assume not to exist.

Since both inequalities cannot occur, we are forced to deduce that P =e−rt. □

4. Discount and yield curves

Definition (Discount curves and yield curves). A discount curve is a function of time P(t)

whose value at any t ≥ 0 is the present value of one unit of currency paid in t years. Equivalently,

P(t) is the price of a zero coupon bond with face value 1 maturing in t years.

A yield curve is a function Y(t) whose value at any t > 0 is the interest rate paid on t-year

deposits.

The values of Y(t) are also called spot (interest) rates or yields. 8

6wethinkthisishighsowe“sell”tomakeprofit

7wethinkthisislowsowe“buy”hopingtosellhighertomakeprofit

8YoucancheckwhatisthecurrentUKyieldcurvehere.

4. DISCOUNT AND YIELD CURVES 15

With these definitions, we can rephrase Proposition 1 in the previous section as,

P(t)=e−Y(t)t.

Yield curves are not directly observable – these are constructed by finding interest rates which

fit observed prices of assets whose prices depend on interest rates (e.g., bonds) and obtaining Y(t)

from these interest rates by linear interpolation9. This process is often called the bootstrap method.

So far, we valued bonds under the assumption that their issuers will make the payments they

contracted to make. In real life this is not always the case: e.g., banks go bankrupt 10 and govern-

ments default on their debt. 11 The price of bonds and the rates of deposit reflect this: risky bonds

arecheaper anddepositsinshakybankspayhigher interest. Thesameistrueofsecuredloans,e.g.,

mortgages, compared to non-secured loans.

Henceforth when working with interest rates we will disregard risk. We always assume that for

everycurrencythereexistsarisk-freeinstitutionissuingbondsinthatcurrencyandouryieldcurves

and discount factors will refer to these bonds. We also assume that everyone can both deposit and

borrow any amount of cash for any maturity date at these risk-free rates. (Don’t try to tell that to

your bank manager.)

4.1. Constructing yield curves: the bootstrap method. We now describe a method of

producing yield curves based on the prices of bonds illustrated in the following example.

Consider the following bonds with semi-annual coupons:

Face value (£) Maturity (years) Annual interest Price

100 0.25 0 98.3

100 0.5 0 96.5

100 1 0 93.7

100 1.5 4 95.5

100 2 6 97.2

The bond in the first line will have no coupon payments (since 0×100/2=0), we can therefore

see it as a zero coupon bond, and so P(t), which is the price of a zero coupon with face value £1

maturing in 0.25 years will be P(0.25)=0.983. Since P(t)=e−Y(t)t, we can immediately calculate

the spot rate Y(0.25)=−ln0.983 ≈6.86%.

0.25

Arguing in exactly the same way:

P(0.5)=0.965, Y(0.5)≈7.13%

P(1)=0.937, Y(1)≈6.51%.

After that, it becomes more interesting. Annual interest of 4% and face value £100 means that

the owner of the bond is due 0.04 × 100 = 4 every year, and since they are getting 2 coupon

payments per year, every coupon payment produces £2, and the owner will get these coupon

payments in 6,12, and 18 months (semi-annual coupons). But a £2 in 6 months is worth now

£2e−r0.5 = 2e−Y(0.5)0.5(present value). This is however nothing but 2P(0.5)12. Similarly, the £2-

coupon payment we will get in one year is worth 2P(1) now. At the 18 month mark, which is

9thismeansthatgivenpoints(t1,Y(t1)),(t2,Y(t2))wewillfindthelinethatgoesthroughbothofthemanduse

theequationofthislineastheformulaforY(t)fort∈[t1,t2].

10E.g.,the Icelandbankcrisis.

11E.g., Russiandebtdefault

12formulaconnectingPandY

16 1. INTEREST, PRESENT VALUE AND BONDS

the maturity time for this bond, the owner will get the last coupon payment and the price of

the face value, i.e., £102, whose present value is 102e−Y(1.5)1.5 = 102P(1.5). This means that the

price we are willing to pay now in order to get this face value along with the coupon payments is

2P(0.5)+2P(1)+102P(1.5), therefore,

2P(0.5)+2P(1)+102P(1.5)=95.5

and since we have already calculated P(0.5),P(1), solving for P(1.5) we obtain P(1.5) ≈ 0.89898

and Y(1.5)≈7.10%.

We argue in exactly the same way for the bond on the 5th line (now the time to maturity is 2

years and we will get 4 coupon payments):

3P(0.5)+3P(1)+3P(1.5)+103P(2)=97.2 and solving for P(2) we obtain

P(2)≈0.8621073672, Y(2)≈7.42%.

Throughout the previous example we thought of P(t) as both the present value of one unit of

currency paid in t years, and as the price of a zero coupon bond with face value 1 maturing in t

years.

5. Forward rates

Definition. Aforward rate agreement isacontractinwhichonepartyagreestopaytheother

party a pre-specified interest rate on a deposit occurring during a specified period of time in the

future.

The point is that we want to agree now for the rate of a future deposit/loan.

Let’s say that you plan to deposit £1 t years from now and to withdraw the balance t years

1 2

from now (t > t ), but you would like to agree now on the interest rate r for this deposit. Of

2 1 12

course,themostnaturalquestionwouldbetoaskwhat this rate r should be. Letthespotinterest

12

rates for t and t year deposits be r and r respectively. One option you have is to deposit your

1 2 1 2

£1 for t

1

years at r 1; after t

1

years you will have 1×er1t1. You could then deposit your balance at

the pre-agreed rate r

12

for t 2−t

1

years. At the end of t

2

years you will have er1t1er12(t2−t1).

A second option available to you is to just deposit £1 for t

2

years; balance =1×er2t2.

Intuitively, we would expect that either way should result in the same balance, i.e.,

So we should expect to receive

r t −r t

er1t1er12(t2−t1) =er2t2 ⇒r = 2 2 1 1.

12 t −t

2 1

Definition. Let 0≤t

1 2 1 2 1 2

r t −r t

The forward rate for the period from t to t , is r = 2 2 1 1.

1 2 12 t −t

2 1

This number was found by comparing two different scenaria and by intuitively expecting the

resulttobethesame. So,atthispointyoucanthinkofr astheratethatweexpecttobecorrect.

12

Nowthatwehaveanideaaboutwhattheanswer“ought”tobe,wewilluseno-arbitragearguments

to reach a rigorous proof.

Proposition 2. In a market with no arbitrage opportunities the interest rates of forward rate

agreements are equal to the corresponding forward rates.

6. EXERCISES 17

Proof. Let the forward rate agreement time period start in t years and end in t years, let r

1 2

be the pre-specified interest rate and let

r t −r t

r = 2 2 1 1

12 t −t

2 1

be the corresponding forward rate for the period from t to t .

1 2

If r >r ,

12

• enter the agreement as a depositor,

• borrow e−r1t1 for t

2

years and deposit this amount for t

1

years.

• Attimet=t

1

receivethebalanceofyourdepositwhichwillbee−r1t1er1t1 =1,anddeposit

it until t=t earning an interest rate of r.

2

• At time t=t

2

obtain the balance of your deposit which will equal er(t2−t1) and repay the

balance of your loan, which will be e−r1t1er2t2 =er2t2−r1t1.

• Since

r t −r t

r > 2 2 1 1 ⇒er(t2−t1) >er2t2−r1t1

t −t

2 1

you can now pocket the difference er(t2−t1)−er2t2−r1t1 >0.

If r

12

• enter the agreement as a borrower,

• borrow e−r1t1 for t

1

years and deposit this amount for t

2

years.

• At time t=t borrow £1 until t=t paying an interest rate of r, and use this £1 to repay

1 2

your loan.

• At time t = t

2

receive the balance of your deposit which will be e−r1t1er2t2 = er2t2−r1t1,

and use it to repay the balance of your second loan which will be er(t2−t1).

• Now

r t −r t

r < 2 2 1 1 ⇒er(t2−t1)

t −t

2 1

and you can pocket the difference er2t2−r1t1 −er(t2−t1) >0.

□

6. Exercises

(1) Find the price of a bond with face value £100 and £5 annual coupons that matures in 4

years, given that the continuous compounding rate is i) 8% or ii)5%. Hint: in this case

they tell us in advance what the coupon payment is.

(2) How long will it take to earn £1 in interest if £1,000,000 is deposited at 10% compounded

continously?

(3) What is the interest rate if a deposit subject to annual compounding is doubled after 10

years?

(4) Supposethatwehaveadepositof£100attractinginterestat10%compounded(i)annually,

and (ii) semi-annually. Find the future value of the deposit after 2 years in both cases.

(5) Whatinitialinvestmentsubjecttoannualcompoundingat12%isneededtoproduce£1,000

after 2 years?

(6) You have £1,000 and you are given the choice between depositing your money attracting

interest (i)at 15% compounded daily, or (ii) at 15.5% compounded semi-annually. Which

option will you choose?

(7) Howmuchcanyouborrowiftheinterestrateis18%, youcanaffordtopay£10,000atthe

end of each year, and you want to clear the loan in 10 years?

18 1. INTEREST, PRESENT VALUE AND BONDS

(8) Consider the following five bonds with face value of £100:

Time to maturity Annual interest Bond price

(in years) (paid every 6 months) (in £)

0.25 0 99.0

0.5 0 97.8

1.0 0 95.5

1.5 8% 104.5

2.0 12% 113.0

(a) Find the 0.25, 0.5 and 1-year spot interest rates.

(b) Use the bootstrap method to find the 1.5 and 2-year spot interest rates.

(c) Now, suppose that you are offered by a risk-free institution the opportunity to deposit

or borrow £1000 in six months for a period of six months earning an interest rate of 5%.

Describe in detail an arbitrage opportunity available to you.

(9) Consider any traded asset whose value v in T years is known with certainty. Prove that

T

the current value of the asset must be v e−rT where r is the continuously-compounded

T

T-years interest rate.

CHAPTER 2

Forward and Futures Contracts

Oneofthemainaimsofthiscourseistofindthecorrect(i.e.,no-arbitrage)pricesofderivatives.

We now define these formally as follows.

Definition. A derivative is an asset whose price depends on the value of other assets.

In this chapter we study two derivatives, namely, forward contracts and futures contracts.

Definition. Aforward contract isanagreementbetweentwopartiesinwhichonepartyagrees

to sell a particular asset at an agreed price (the forward price), on a certain date (maturity date or

delivery date), and the other party agrees to buy that asset at that price and on that date.

There is an asymmetry in the parties to a forward contract: The party receiving the asset has

a long position and the party delivering the asset has a short position. 1

Forward contracts are entered freely by both parties and have no cost. The question we now

raise is what should the forward price which makes a forward contract valueable be. To answer this

questionwewilldistinguishbetweenforwardcontractsonassetsthatprovideincome(e.g., dividend

yielding shares in a company) and those that do not (e.g., bars of gold). To do this we’ll need to

assume short-selling which we describe next

1. Short-selling

Definition. Short-selling is selling something which one does not own.

Thus one may own a positive or a negative amount of assets.

In this course we will assume that any tradable asset can be bought, sold and

short-sold.

Short-selling sounds vaguely criminal, after all, it involves selling something owned by someone

else without their consent– however this is no different from the situation in which a bank takes

your deposit and lends it to someone else. As long as the bank can repay your deposit on request,

all is well and proper.

How does short-selling work? You short sell an asset through a broker, the broker borrows the

asset from another client and sells it, and gives you the proceeds. You can use most of this money

to invest in assets through the broker, but you have to keep some of the proceeds in cash held in

a margin account managed by the broker. The balance on the margin account should be a certain

percentage of the spot value of the short-sold asset, and it is balanced daily: if the spot price of

the short-sold asset goes up, the balance in the margin account needs to be increased and you will

receive a margin call, i.e., a demand to add cash to your margin account. The broker has the right

to sell your assets and to use your balance in the margin account to buy back the short-sold asset

and return it to the other client, if the other client demands it or if you do not respond to a margin

call.

1Infinance,“short”oftenmeans“sell”and“long”oftenstandsfor“buy”.

19

20 2. FORWARD AND FUTURES CONTRACTS

To “close” a short position, you buy back asset and return it to its owner. A profit is made if

the sale price was higher than the purchase price, i.e., if the asset’s price decreased.

We will use repeatedly the following consequence of our ability to short-sell assets:

Theorem 3. (Present-Future Price Comparison) Let p and p be the current prices of two

1 2

assets, and suppose that we know with certainty that their prices q and q at some point in the

1 2

future satisfy q ≤ q . Then p ≤ p . In particular, if we know that two assets will have the same

1 2 1 2

price at some point in the future, then the two assets must have the same price at any prior time.

Proof. Assume, for a contradiction, that p > p . Short sell item 1, and buy item 2 (we sell

1 2

the “expensive” and buy the “cheap”). The balance after this: p −p > 0 (by our assumption).

1 2

Then wait until we reach the time when the prices are q ,q , respectively. We buy back item 1 (to

1 2

return to the rightful owner) and sell item 2. The balance of these transactions is q −q ≥ 0 (by

2 1

assumption). To sum it up, our final balance is (q −q )+(p −p )>0. This is a contradiction to

2 1 1 2

the no arbitrage assumption since we exhibited how to make a profit with no investment. □

2. The forward price

In what follows, we will make the usual assumption of no arbitrage. We will also assume that

therearenotaxesongoodsbought/sold,thattherearenotransactioncosts(e.g.,thebrokerdoesn’t

ask for a commission) – these assumptions are often referred to as “no friction” assumptions. Last,

wewillalsoassumethattraderscanborrowandlendmoneyatthesame(risk-free)rate, aswehave

done so far. We are now ready to deduce the forward price in a forward contract:

Theorem 4. Consider a forward contract on an asset which provides no income and whose

price at the present (its spot price) is S. The forward price is F = SerT where T is the time to

maturity and r =Y(T), the T-year spot interest rate.

Proof. If F >SerT there is the following arbitrage strategy:

• take a short position in the forward contract,

• borrow £S for T years at spot interest rate of r,

• buy the asset for £S [remember that we have agreed to sell it at time T],

• wait T years,

• deliver (sell) the asset and collect £F,

• use £SerT to repay the loan,

• pocket the difference F −SerT >0.

If F

• take a long position in the forward contract (commit to buy the asset in T years for £F),

• short sell the asset for £S,

• deposit £S for T years at spot interest rate of r,

• wait T years,

• withdraw £SerT from your deposit,

• have the asset delivered for £F,

• use it to close the short position on the asset (I have to return what I short-sold),

• pocket the difference SerT −F >0.

Both cases contradict the no-arbitrage assumption.

□

2. THE FORWARD PRICE 21

We now consider a forward contract on an asset that provides an income during the duration

of the forward contract. For example, you can consider a bond with coupon payments. To be more

succinct, we will be referring to payments X (generated by the asset, e.g., bond coupons) received

k

attimest k,k =1,...,n,whosepresentvaluesareI

k

=X ke−Y(tk)tk.So,I =I 1+I 2+...+I k+...+I

n

is the present value of this collection of payments, i.e., of the income generated by the asset until

the maturity of the forward contract.

Theorem 5. Consider a forward contract on an asset that provides income, with maturity

date in T years. Let S be the spot price of the asset and let I be the present value of the income

generated by the asset until the maturity of the forward contract.

Then F =(S−I)erT where r =Y(T), the T-year spot interest rate.

Proof. Suppose that the asset produces payments X at times 0 < t < ··· < t < T whose

k 1 n

present values are I ,...,I and write r =Y(t ),...,r =Y(t ).

1 n 1 1 n n

If F >(S−I)erT there is the following arbitrage strategy:

• Take a short position in the forward contract. (You commit to sell, so make sure you will

have the asset at maturity. If you don’t already have it, you need to buy it; its current

price is S =(S−I)+I.)

• Borrow £(S−I) for T years at spot interest rate of r.

• for every 1≤k ≤n, borrow I for t years at a spot rate of r .

k k k

Borrowed I +I +...+I +...+I =I in total.

1 2 k n

• Buy the asset for £S (=(S−I)+I).

• WaitT years: meanwhileusethepaymentsfromtheassettorepaythecorrespondingloan.

Note: a loan of I

k

for t

k

years at r

k

grows to I

k

·erktk = X ke−rktk ·erktk = X k, which is

exactly the payment you will get from the asset at time t .

k

• At the end of this period the outstanding debt is £(S−I)erT.

• Deliver the asset and collect £F.

• Use £(S−I)erT to repay the loan.

• Pocket the difference F −(S−I)erT >0.

If F <(S−I)erT there is the following arbitrage strategy:

• Take a long position in the forward contract.

• Short sell the asset for £S (with the idea that I will return it to the owner after having

bought it for myself with the forward contract).

• Deposit£(S−I)forT yearsatspotinterestrateofr. Alsodeposittheremaining£I from

the sale as follows: for every 1≤k ≤n deposit I for t years at a spot rate of r .

k k k

• Wait T years: meanwhile make the payments due from the asset (to the rightful owner)

while withdrawing the corresponding deposits I kerktk =X k.

• At the end of this period the deposit balance is £(S−I)erT.

• Withdraw £(S−I)erT from your deposit.

• Have the asset delivered for £F.

• Use it to close the short position on the asset, i.e., return to owner.

• Pocket the difference (S−I)erT −F >0.

□

22 2. FORWARD AND FUTURES CONTRACTS

3. Foreign exchange forward contracts

An interesting example of a forward contract is one in which the underlying asset is foreign

currency– this is an example of a forward contract in which the underlying asset provides income

continuously (in the form of continuously compounded interest on a deposit of foreign currency.)

We will deal with these by introducing a new variant of no-arbitrage arguments which will be used

repeatedly in this course: we will construct two portfolios whose values at a certain time in the

future are known to be identical. We will then apply Theorem 3 to deduce that the spot values of

these two portfolios are identical and this will allow us to find the correct no-arbitrage price.

Theorem 6. Consider now a forward contract on one unit of foreign currency whose spot rate

is S i.e., it costs S units of domestic currency to purchase one unit of foreign currency. Let the

maturity date be T years in the future, and let the yield curve in the foreign currency be Y (t). Let

f

F be the forward rate, i.e., the party with the short position in the forward contract will deliver in

T years one unit of foreign currency and receive a payment of F units of domestic currency. Write

r =Y(T) and r

f

=Y f(T). We have F =Se(r−rf)T.

Proof. Consider the following two portfolios:

Portfolio A:alongpositionintheforwardcontractandFe−rT unitsofdomesticcurrencyearning

interest rate of r for T years.

Portfolio B: e−rfT units of foreign currency earning interest rate r f.

At time T portfolio A will generate F units of domestic currency which can be used to pay

for the one unit of foreign currency. At time T portfolio B will also be worth one unit of foreign

currency. Theorem 3 implies that both portfolios must have the same value at any time t with

0≤t≤T. If we consider t=0, this gives us Fe−rT =Se−rfT. □

4. Futures contracts

A futures contract is similar to a forward contract: it is also an agreement to deliver an asset at

an agreed price F, the futures price, and at an agreed date, the maturity date or delivery date

Again, one can have a short position (a commitment to deliver the asset) or a long position (a

commitment to buy the asset.)

Unlike forward contracts, which are private agreements and not traded, futures contracts are

traded through brokers in specialised exchanges such as London’s LIFFE and Chicago Board of

Trade (CBOT). 2 These exchanges regulate both trading and delivery of assets. Underlying assets

of futures contracts are standardized. 3

One closes out a position in a futures contract by entering an identical contract but with the

opposite position, e.g., to close out a long position in a March 2022 Cocoa futures, you take a short

position on a March 2022 Cocoa futures.

Futures are settled daily: e.g., assume you took a short position in a May 2022 Brent crude

futures, with futures price $24.10, i.e., you agreed to provide 1,000 US barrels of petroleum of a

certain quality for $24.10 per barrel.

IfattheendofthedaythefuturespriceforMay2022Brentcrudeis$24.20,thenyoulose$0.10

per barrel, because now your barrel of oil can be delivered for $0.10 more than what you will get.

You pay your broker $1,000·0.10 = $100 and your futures price is changed to $24.20. If the next

2Check,forexampleOilfuturespricesfromtheNYmercantileexchange.

3Check,forexampleCME’swheatfuturescontractspecification.

5. EXERCISES 23

dayseesadropof$0.20oftheMay2022Brentcrudeyourbrokerwillcredityou$200andyournew

futures price is changed to $24. The daily settlement of futures is called marking-to-market.

The marking-to-marketof futures contractsinvalidates theno-arbitrage argumentsusedto pro-

duceforwardprices. Thereasonforthisisthattheinterestpaidorobtainedfromthedailycash-flows

generated by the futures contracts are stochastic, i.e., they are not known in advance.

However, one has the following.

Theorem 7 (The Futures-Forwards Equivalence Principle)). If interest rates are deterministic

then forward price and futures price coincide.

For a proof, consult Chapter 3 of John Hull’s “Options, Futures and other derivatives”.

5. Exercises

(1) Suppose that the 1-year risk free -interest rates for British Pounds (£) and US Dollars ($)

are 4% and 2%, respectively. Suppose further that the spot £/$ exchange rate is now 0.54

(i.e., $1 costs £0.54).

(a) What is the 1-year £/$ forward exchange rate?

(b) Someone is willing to buy/sell in one year British pounds at a £/$ exchange rate of

0.54. Describe in detail an arbitrage opportunity available to you.

(2) Let A and B be two portfolios of financial assets and assume that it is possible to have

both long and short positions on both portfolios. Suppose that we know with certainty

that at some time in the future the value of portfolio A will be at least as high as that of

portfolio B. Prove that the current value of portfolio A must be at least as high as that of

portfolio B.

CHAPTER 3

Options

Definition. Options are contracts conferring certain rights regarding the buying or selling of

assets. We call the asset referred by the option as the underlying asset of the option.

A European call option gives the owner the right to buy its underlying asset at a certain price

on a certain date.

A European put option gives the owner the right to sell its underlying asset at a certain price

on a certain date.

ThedateonwhichtheownerofEuropeanoptionscanexercisetheirrightiscalledtheexpiration

date and the price is called the strike price.

An American call option gives the owner the right to buy its underlying asset at a certain price

by a certain date.

An American put option gives the owner the right to sell its underlying asset at a certain price

by a certain date.

ThedatebywhichtheownerofAmericanoptionscanexercisetheirrightiscalledtheexpiration

date and the price is called the strike price.

1. The payoffs of options

Consider a European call option with strike price X which has just expired at time t = T and

let S be the spot price of the underlying asset at the expiration of the option.

T

If S > X the holder of the option will buy the asset for X and sell it for S , thus getting a

T T

payoff of S −X from the option.

T

If S ≤X the option holder will not exercise it; in this case the option generates no payoff.

T

Payoff

S

T

X

Thus at expiration, the owner of the option will receive a payoff which is a function of the

underlying asset price S at expiration, and that function is max{S −X,0}.

T T

Similarly, the payoff of a corresponding European put option is given by max{X −S ,0} and

T

is described by the following graph:

25

26 3. OPTIONS

Payoff

S

T

X

2. Elementary inequalities satisfied by option prices

We will now consider different option types on the same stock with spot price S, expiring in T

years and with strike price X. We denote with c and p the current prices of European call and put

options, and with C and P the current prices of American call and put options.

Proposition 8.

(a) c,C,p,P ≥0.

(b) c≤C ≤S and p≤P ≤X.

Proof. If any of the options has negative value −v, buy it for −v, i.e., receive the option plus

an amount of v in cash, and forget about the option. If c>C, sell a European call option, buy an

Americancalloption, pocketc−C >0andwaitforexpiration. IftheEuropeanoptionisexercised,

exercise your American option and deliver the stock. A similar argument shows that p≤P.

If C > S, sell the call option, buy the stock, pocket C −S > 0 and wait. If the option is

exercised, deliver your stock, otherwise keep it. If P >X, sell the option and wait. If the option is

exercised, buy the stock for X and pocket P −X > 0 plus the stock; if the option is not exercised

pocket P. □ □

Proposition 9. Assume that the stock does not pay dividends and let r be the T-year spot

interest rate.

(a) c>S−Xe−rT.

(b) p>Xe−rT −S.

Proof. To prove the first inequality consider:

Portfolio A: one European call option plus an amount of cash equal to Xe−rT deposited for T

years at an interest rate of r.

Portfolio B: one share.

After T years portfolio A will yield an amount of cash equal to X.

If, after T years, the stock price S is above X, the call option in portfolio A will be exercised,

T

thesharesoldandtheportfoliowillbeworthS . Otherwise,afterT years,S ≤X,theoptionisnot

T T

exercisedandtheportfoliowillbeworthX. SoafterT yearsportfolioAisworthmax(S ,X)≥S ,

T T

and since portfolio B is always worth S after T years,

T

the initial value of portfolio A must be no less that the initial value of portfolio B, which is just

S. But since sometimes portfolio A is worth more than portfolio B we have a strict inequality

c+Xe−rT >.S

To prove p>Xe−rT −S consider:

Portfolio C: one European put option plus one share.

3. OPTIMAL EXERCISE FOR AMERICAN STYLE OPTIONS 27

Portfolio D: an amount of cash equal to Xe−rT deposited for T years at an interest rate of r.

After T years portfolio D will be worth X.

If, after T years, S < X, then the put option in portfolio C will be exercised; the share is

T

sold for X and the portfolio will be worth X. Otherwise, if , after T years, S ≥ X, the option is

T

not exercised and the portfolio will be worth S . So after T years portfolio C is worth max(S ,X)

T T

and the initial value of portfolio C must be no less that the initial value of portfolio D which is just

Xe−rT. But since sometimes portfolio C is worth more than portfolio D we have a strict inequality

p+S >Xe−rT.

□

Proposition 10 (Put-call parity). Assume that the stock does not pay dividends and let r be

the T-year spot interest rate. Then c+Xe−rT =p+S.

Proof. Recall portfolios A and C above: after T years they are both worth max(S ,X) where

T

S is the stock price after T years. These two portfolios must have identical initial values, i.e.,

T

c+Xe−rT =p+S. □

3. Optimal exercise for American style options

When should a rational investor exercise an American option?

Proposition 11. Assume that the stock does not pay dividends. The optimal exercise time for

the American call option occurs at the expiration of the option and hence c=C.

Proof. For any time 0 < τ < T write S ,c ,C for the values at time τ years of the stock,

τ τ τ

EuropeanoptionandAmericanoption,respectively,andletr betheT−τ spotinterestrateattime

τ. We have

C ≥c >S −Xe−r(T−τ) >S −X,

τ τ τ τ

where the second inequality follows from Proposition 9. but S −X is the payoff, at time τ, from

τ

theexerciseoftheAmericanoption, andsincethevalueoftheAmericanoptionexceedsthatofthis

payoff, it should not be exercised (e.g., you can hold it or try to sell it at an options market for C ).

τ

The optimal exercise of the American option will produce the same payoff as the European option,

hence c=C. □

American put options may have an early optimal exercise date: e.g., suppose that on June

25th, 2002 you held American put options on Worldcom 1 stock with strike price of $65 expiring in

September 2002. Since you bought the stock the company disclosed that it inflated profits for over

a year by improperly accounting for more than $3.9 billion and the stock now sells for $0.20. The

payofffromexercisingyouroptionnowwouldbe$64.8,almostitstheoreticalmaximum. Thingscan

only get worse as time progresses and you should exercise your options now.

Proposition 12.

S−X

Proof. Since P ≥p always, and since some of the time the payoff of the American option will

be greater than that of the corresponding European option, P >p.

The second inequality is a consequence of Put-Call Parity, P >p and c=C.

To prove S−X

1AparticularlycrookedAmericancorporationwhichwentbankruptinthe00’s.

28 3. OPTIONS

Portfolio E: one European call option plus an amount of cash equal to X deposited for T years at an

interest rate of r.

Portfolio F: one American put option plus one share.

At the time of expiration, portfolio E will be worth

max(S −X,0)+XerT = max(S −X,0)+XerT +X−X

T T

= 2max(S ,X)−X+XerT

T

= max(S ,X)+X(erT −1)

T

> max(S ,X)

T

and, Case 1, if the American option has not been exercised before, portfolio F will be worth

max(X−S ,0)+S =max(X,S ).3 So portfolio E expires with higher value than portfolio F.

T T T

Case 2, if the American option was (rationally) exercised at an earlier time 0≤τ

that time portfolio F was worth (X−S )+S =X. At that time, the value of the cash in portfolio

τ τ

E is at least X.

SinceineithercasethereisalwaysatimeatwhichportfolioEismorevaluablethanportfolioF,

the current value of portfolio E is greater than the current value of portfolio F, i.e., C+X >P +S.

□

□

4. Parameters affecting the prices of options

Suppose you hold a call option on a stock whose present price is £10 expiring in T years. What

should happen to the price c of the call option if the stock price goes up to £15? The payoff of the

option at expiration is max(S −X,0) where S is the price of the stock at expiration. The rise in

T T

the stock price suggests that the market expects S to be higher as well, so c and C are increasing

T

functions of S. For similar reasons, p and P are decreasing functions of S. Obviously, c and C are

decreasing functions of X while p and P are increasing functions of X.

Suppose that the strike price of your call option is £20. If you are told that the variability of

the stock price is very small, your option is not worth much because when the option expires the

stock price is likely to be very close to £10, far below the strike price.

In general, share price movements can result in both higher and lower payoffs from an option,

but the downside is limited (to losing the price paid for the option) while the upside is unlimited

(in the case of a call option) or bounded by the strike price which is usually much higher than the

option price. So we expect that c, C are increasing functions of the volatility of the stock price.

It is reasonable to assume that for longer expiration times T, the value of the stock at time T

will have more variability. But on the other hand, for bigger T, the payoff of the option has to be

discounted by a smaller discount factor. Obviously, C and P are increasing functions of T.

5. Exercises

(1) A European call option, expiring in one year and with strike price £14, on a non-dividend

paying stock, currently priced at £12, is traded at £0.94. Use Put-Call Parity to find the

price of a European put on the same stock with the same strike price and expiry time.

3weusedtheabovepropertyagain

5. EXERCISES 29

(2) Describe the payoff function of an option which entitles its holder to buy a given stock for

£X, but only if the price is at least £Y, with Y >X.

(3) Showthat,ifc andc arethepricesoftwoEuropeancalloptionsonthesamenon-dividend

1 2

paying stock with price S at time t=0 , with the same strike price X and expiry times T

1

and T , respectively, with T > T > 0, then c ≥ c . (Hint: consider the corresponding

2 1 2 1 2

American options.)

CHAPTER 4

Binomial trees and risk neutral valuation

A derivative is an asset whose value depends on the value of another asset, e.g., Call/Put

European/American options. In this chapter we find prices of derivatives providing a single payoff

at a future date when the underlying asset price evolves in a particularly simple way.

1. A motivating example

Considera1-yearEuropeancalloptiononastockwithstrikeprice£10. Assumethatthecurrent

price of the stock is S =£10 and that at the end of the one year period the price of the stock will

0

be either S =£11 or S =£9. Assume further that the 1-year interest rate is 5%.

u d

We picture this world as follows:

S =£11

u

(cid:55)(cid:55) • Option payoff=£1

S =£10 •

(cid:40)(cid:40)

• S =£9

d

Option payoff=£0

What should the price c of the option be? Consider a portfolio with δ shares of this stock, and

short in one option.

S =£11

u

(cid:55)(cid:55) • Option payoff=£1

S =£10 •

(cid:40)(cid:40)

• S =£9

d

Option payoff=£0

Ifthestockpricegoesuptheportfoliowillbeworth11δ−1andifthestockpricegoesdownitwill

be worth 9δ. What if we choose our δ so that

1

11δ−1=9δ, i.e., δ = ?

2

The value of this portfolio is the same in all possible states of the world! The portfolio must

have a present value equal to its value in one year discounted to the present, i.e., 9e−0.05×1 but the

2

current price of stock in the portfolio is £10/2, so

9 9

e−0.05 =5−c⇒c=5− e−0.05 ≈0.72.

2 2

The probability of up or down movements in the stock price plays no role whatsoever!

31

32 4. BINOMIAL TREES AND RISK NEUTRAL VALUATION

2. Derivatives in a simple, up-down world

We generalise: consider a financial asset which provides no income and a financial derivative on

that asset providing a single payoff t years in the future. The current price of the asset is S in t

years theprice of thestock will be either Su (u>1), resultingin a payoff of P from thederivative,

u

or Sd (0≤d<1) resulting in a payoff of P from the derivative. Let r be the t-year interest rate.

d

asset price=Su

(cid:55)(cid:55) • Option payoff=P

u

S •

(cid:39)(cid:39)

• asset price=Sd

Option payoff=P

d

Construct a portfolio consisting of δ units of the asset and -1 units of the derivative and choose

δ so that the value of the portfolio after t years is certain: δ must satisfy

P −P

δSu−P =δSd−P ⇒δ = u d .

u d S(u−d)

The value of the portfolio in t years will be

P −P P −P

u d Su−P = u du−P

S(u−d) u u−d u

and its present value is

(cid:18) (cid:19)

P −P

e−rt u du−P .

u−d u

Let x be the price of the derivative. We must have the following equality of present values

(cid:18) (cid:19)

P −P P −P P −P

e−rt u du−P =δS−x= u d S−x= u d −x

u−d u S(u−d) u−d

Solving for x we obtain

x= P u−P d −e−rt(cid:18) P u−P du−P (cid:19) = e−rt (cid:0) (ert−d)P +(u−ert)P (cid:1)

u−d u−d u u−d u d

ert−d

andifwelet q = we canrewrite x as x=e−rt(qP +(1−q)P ). Notice: 0≤q = ert−d ≤1

u−d u d u−d

and we can interpret q as a probability 1. Now In a world where the probability of the up movement

in the asset price is q, the equation x=e−rt(qP +(1−q)P ) says that the price of the derivative is

u d

the expected present value of its payoff. Using these probabilities, stock price at time t has expected

value

ert−d

E =qSu+(1−q)Sd=qS(u−d)+Sd= S(u−d)+Sd=(ert−d)S+Sd=ertS,

u−d

i.e., the world where the probability of the up movement in the asset price is q is one in which the

stock price grows on average at the risk-free interest rate.

So in this world investors are indifferent to risk (unlike real-life investors).

We refer to the probabilities q and 1−q as risk neutral probabilities and to equation above as a

risk neutral valuation.

1Wedidthisinthelecture;foreachcaseassumetheinequalityisfalseandshowsomethingabsurd.

4. A TWO STEP PROCESS 33

3. An example

Consider a stock whose current price is £20 and whose price in 3 months will be either £22 or

£18. Let c be the price of a European call option on this stock with strike price £20 and expiring

in three months. Assume that the 3-month interest rate is 5%.

Let p be the probability of an upward movement in the stock price in a risk neutral world.

As we are not told the values for u and d we can’t directly calculate p, the “up” probability in

a risk-neutral world. However, we know that in such a world the value of an asset grows on average

at the risk-free rate.

Assume that the price of the asset at time 0 is S. Let’s examine the above paragraph in detail

and make sense of the statement there. If I have £S, I can either put them in the bank to accrue

interest or invest them in the asset. In the first case, after time T I have £SerT. In the latter case,

on average (i.e., in expectation), my asset is worth pSu+(1−p)Sd = 22p+18(1−p) as we know

what the possible “up” and “down” values will be. In other words, my portfolio (which is in this

case is just this unit of the asset) is worth 22p+18(1−p). When we say that “the value of an asset

grows on average at the risk-free rate” we actually mean that SerT = pSu+(1−p)Sd. Now the

numbers:

In such a world the expected price of the stock must be 20e0.05/4 = 20e1/80, so p satisfies

9

22p+18(1−p)=20e1/80 ⇒p=5e1/80− ≈0.5629.

2

Having found p, we can now easily find the price of the option, recalling that the price will be the

present value of the expected payoff. I.e., the expected payoff times the discount factor.

So,astheexpectedpayoffoftheoptionisnow2p+0(1−p)=2p,itspresentvalueis2pe−0.05/4 ≈

1.112, which is exactly the price of the option that we wanted to find.

4. A two step process

Consider now a world where the price of the underlying changes twice, each time by either a

factor of u > 1 or d < 1. After two periods the stock price will be Su2, Sud = Sdu or Sd2. The

derivative expires after the two periods producing payoffs of P , P = P and P respectively.

uu ud du dd

Assume also each period is ∆t years long and that interest rates for all periods is r.

Su (cid:54)(cid:54) • Su2,payoff=P

uu

(cid:53)(cid:53) •

(cid:41)(cid:41)

S• (cid:53)(cid:53) • Sud,payoff=P

ud

(cid:41)(cid:41)

•

(cid:40)(cid:40)

Sd • Sd2,payoff=P

dd

34 4. BINOMIAL TREES AND RISK NEUTRAL VALUATION

Tofind x, thevalueofthe derivative, we now workourwaybackwards, from theend ofthetree

(i.e., the end of the second period) to the root (i.e., the present.)

B (cid:53)(cid:53) •D

(cid:53)(cid:53) •

(cid:41)(cid:41)

A• (cid:53)(cid:53) •E

(cid:41)(cid:41)

•

(cid:41)(cid:41)

C •F

The value of the derivative is known at vertices D,E and F; these are the payoffs P , P and P .

uu ud dd

How about nodes B and C?

We can find the value of the derivative at node B by considering the following one period tree:

Su2

(cid:53)(cid:53) • P

uu

v •

B

(cid:41)(cid:41)

• Sud

P

ud

er∆t−d

The risk neutral probability q of an upward movement is given by q = and so the value

u−d

v of the derivative at node B is v =e−r∆t(qP +(1−q)P ).

B B uu ud

Similarly, the value of the derivative at node C is obtained from

Sud

(cid:53)(cid:53) • P

ud

v •

C

(cid:41)(cid:41)

• Sd2

P

dd

v =e−r∆t(qP +(1−q)P ).

C ud dd

Now we can work our way back one more step to node A;

(cid:52)(cid:52) •v

B

v •

A

(cid:42)(cid:42)

•v

C

and the value at node A is

v =e−r∆t(qv +(1−q)v ).

A B C

5. An example

Consider a European put option on stock currently traded at £10, with strike price £11 and

expiring in one year. Interest rates for all periods are 4%.

Weuseatwo6-month-periodtreewithu=5/4andd=3/4toestimatethepriceoftheoption.

6. n-STEP TREES 35

S =15.625

S =12.5 (cid:52)(cid:52) • Payoff=0

(cid:51)(cid:51)

•

(cid:42)(cid:42)

S =10• (cid:52)(cid:52) • S =9.375

(cid:43)(cid:43)

• Payoff=1.625

(cid:42)(cid:42)

S =7.5 •

S =5.625

Payoff=5.375

Risk neutral probability of an upward movement of stock price

er∆t−d e0.04/2−3/4

q = = ≈0.5404.

u−d 5/4−3/4

Value v of the derivative at node B

B

v =e−r∆t(qP +(1−q)P )≈0.7321,

B uu ud

value of the derivative at node C

v =e−r∆t(qP +(1−q)P )≈3.282,

C ud dd

value at node A

v =e−r∆t(qv +(1−q)v )≈1.866

A B C

6. n-step trees

Consider a 18-month European put option with strike £12 on a stock whose current price is

£10. Assume that interest rates for all periods are 5%. Use u = 6/5 and d = 4/5 to construct the

following three step binomial tree.

(cid:50)(cid:50) 17.28 0

(cid:50)(cid:50) 14.4 0.2044

(cid:44)(cid:44)

(cid:51)(cid:51) 12 1.008 (cid:50)(cid:50) 11.52 0.48

(cid:44)(cid:44)

10 2.008 (cid:50)(cid:50) 9.6 2.104

(cid:43)(cid:43) (cid:44)(cid:44)

8 3.415 (cid:50)(cid:50) 7.68 4.32

(cid:44)(cid:44)

6.4 5.304

(cid:44)(cid:44)

5.12 6.88

er∆t−d e0.05/2−4/5

q = = ≈0.5633.

u−d 6/5−4/5

e−0.05/2(0.5633×0+0.4367×0.48)≈0.2044

e−0.05/2(0.5633×0.48+0.4367×4.32)≈2.104

e−0.05/2(0.5633×4.32+0.4367×6.88)≈5.304

e−0.05/2(0.5633×0.2044+0.4367×2.104)≈1.008

e−0.05/2(0.5633×2.104+0.4367×5.304)≈3.415

e−0.05/2(0.5633×1.008+0.4367×3.415)≈2.008

36 4. BINOMIAL TREES AND RISK NEUTRAL VALUATION

7. Example: an American option

Consider a 18-month American put option with strike £12 on a stock whose current price is

£10. Assume that interest rates for all periods are 5%. Use u = 6/5 and d = 4/5 to construct a

three step binomial tree. Consider the “dd” node in the previous figure.

Immediate exercise gives payoff of 12−6.4=5.6>5.304 and that is the value of the option at this

node.

(cid:51)(cid:51) 7.68 4.32

6.4 5.304

(cid:43)(cid:43)

5.12 6.88

When we have an American (instead of European) option to find the price of with a binomial

tree, at each node there are two steps. (1) First we apply the formula: x=e−r∆t(qP +(1−q)P )

u d

andwefindthatnumber. ForaEuropeanoption,wefinishhere,nothingelsetodo,thatistheprice

of our option at that point. However, (2) for an American one, before we move on to the next node,

weshouldalwaysaskourselves: “whatifIexercisemyoptionrightnow?”(whichwasnotsomething

available to us with a European option). So, we look at the spot price at that node (that would be

the red numbers) and we compare with the strike price. Of course, at this point, it matters whether

it’s a put or a call (btw, if it is a call, we have already seen that the prices are exactly the same but

if it is a put that is not true anymore), but we can easily find the payoff in either case. This will

give us a second number (either 0 or positive).

Between the two numbers that we find, we will select the largest. Why? Think of a third

observer watching our pricing and trading. From their point of view, we can either (Case 1) treat

our American option as a European one (so we can either keep it and wait for expiration and then

decideifweexercise, orwecansellthecontractitselftoacontractsmarket-forthepricexthatwe

found above - and after that is not our problem anymore if the holder decides to exercise the option

or not, it’s their contract now, not ours). At the same time, (Case 2), this being American, gives

us the right to exercise early.

In essence, if we decide that it is beneficial to exercise, we can think of handing the contract to

someone and they give us the payoff we calculated we should be getting, i.e., we are “selling” the

contract at the price of the payoff. This is the second number. So, any observer, at any point, will

say that if we are making more money by exercising at that node, that would be the proper price

forthederivative(wewouldn’tsellacontractfor£2.104ifexercisingitmakesus£2.4. Inthatcase,

the price of the contract will be £2.4).

This discussion (and the calculations listed below) leads to the modified tree for the American

option

8. USING BINOMIAL TREES FOR APPROXIMATING VALUES OF DERIVATIVES - NOT EXAMINABLE 37

(cid:50)(cid:50) 17.28 0

(cid:51)(cid:51) 14.4 0.2044

(cid:44)(cid:44)

(cid:51)(cid:51) 12 1.1345 (cid:51)(cid:51) 11.52 0.48

(cid:43)(cid:43)

10 2.327 (cid:51)(cid:51) 9.6 2.4

(cid:43)(cid:43) (cid:43)(cid:43)

8 4 (cid:51)(cid:51) 7.68 4.32

(cid:43)(cid:43)

6.4 5.6

(cid:44)(cid:44)

5.12 6.88

Underlined values differ from the European style case. However, this does not mean that in these

caseswedecidedtoexercisetheoption, e.g., lookatnode“u”: itisdifferentfromthecorresponding

European node “u” because the value at node “ud” is contributing to the formula, and this is the

first value that is different for an American option.

er∆t−d e0.05/2−4/5

q = = ≈0.5633.

u−d 6/5−4/5

e−0.05/2(0.5633×0+0.4367×0.48)≈0.2044

e−0.05/2(0.5633×0.48+0.4367×4.32)≈2.104,12−9.6=2.4>2.104

e−0.05/2(0.5633×4.32+0.4367×6.88)≈5.304,12−6.4=5.6>5.304

e−0.05/2(0.5633×0.2044+0.4367×2.4)≈1.1345

e−0.05/2(0.5633×2.4+0.4367×5.6)≈3.7037,12−8=4>3.7037

e−0.05/2(0.5633×1.1345+0.4367×4)≈2.327

8. Using binomial trees for approximating values of derivatives - Not examinable

The assumption that the price of the asset underlying a derivative changes at a finite number

ofmoments canapproximaterealityonlyifweallowthepriceto changeatalargenumber ofpoints

in time, many more than two or three. This can lead to n-step trees for large values of n. These

will contain 1+2+3+···+n = n(n+1)/2 nodes and for even a modest value of n, say n = 10,

these computations are best left to computers. In the case of certain exotic derivatives their value

depends not only on the final price of an asset but on its history as well. These derivatives require

even larger binomial trees.

Whichvaluesofuanddshallweuseforthesenumericalapproximations? Differentchoiceswould

√

result in different prices for derivatives! The common practice is to take d = 1/u and u = eσ ∆t

where σ is the yearly standard deviation of the logarithm of the stock price and ∆t is the length in

years of every step in the tree. 2

2Onecanshowthatthisimpliesthatthelogarithmofstockprices∆tyearsinthefuturearenormallydistributed

√

random variables with standard deviation σ ∆t. This will conform to our model of the evolution of share prices

describedinthenextchapter

38 4. BINOMIAL TREES AND RISK NEUTRAL VALUATION

9. Exercises

(1) In a binomial tree with u = 1.2 and d = 0.8, calculate the risk-neutral probability as a

function of r, the interest rate per annum. Comment on the case where the interest rate

is 25%.

(2) Consider a one-step binomial tree with initial stock price £3, u = 1.2 and d = 0.8, and

interest rate 5% (per step). Price the following options (expiring after the single step)

assuming that the stock does not pay dividends:

(a) A European call option with strike price £3.20,

(b) A European put option with strike price £2.90,

(c) An option which pays £1 if and only if the price of the stock after the single step is

at most £2.90.

(3) Again consider a one-step binomial tree with initial stock price £3, u = 1.2 and d = 0.8,

and interest rate 5% (per step). A European call option on the stock (which does not pay

dividends), expiring after the single step with strike price £3.20, is traded at 30 pence.

Describe an arbitrage opportunity available to you.

CHAPTER 5

The stochastic process followed by stock prices

Prices S of a an asset at a future time t is uncertain and we model it as a random variable. In

t

this chapter we ask the question “what sort of random variable are S for t ≥ 0?” Our answer to

t

this question will consist of model involving Brownian motion as a major ingredient.

1. Brownian motion

Definition. A Brownian motion is a family of random variables

{B |t≥0}

t

on some probability space (Ω,F,P) such that:

(1) B =0,

0

(2) for 0≤s

t s

(3) for any 0≤t

1 2 n

B −B ,B −B ,...,B −B

t1 0 t2 t1 tn tn−1

are independent random variables, and

(4) For any ω ∈Ω the function t(cid:55)→B (ω) is continuous.

t

Figure 1. Three instances of Brownian Motion corresponding to ω ,ω ,ω ∈Ω

1 2 3

Brownian motion exists (this statement is Kolmogorov’s Existence Theorem) and has some

surprising properties. For example,

(a) The function t(cid:55)→B (ω) is nowhere differentiable with probability 1,

t

(b) if B =x for some t then for any ϵ>0 the set {τ :|τ −t|<ϵ and B =x} is infinite with

t τ

probability one.

Brownian motion is useful for describing the jiggling of prices: buying and selling jiggle prices.

39

40 5. THE STOCHASTIC PROCESS FOLLOWED BY STOCK PRICES

2. The Ito integral

Brownianmotionisanexampleofastochastic process i.e.,afamilyofrandomvariablesindexed

by time t ≥ 0. We now construct a more general kind of stochastic processes whose definition is

based on Brownian motion as follows.

We want to construct a stochastic process {X |t≥0} on the same probability space (Ω,F,P)

t

on which Brownian motion {B |t≥0} is defined with the property that the change of X over an

t

infinitesimal period of time dt is given by

dX =a(ω,t)dt+b(ω,t)dB,

whereaandbarethemselvesstochasticprocesseson(Ω,F,P)withcontinuouspaths,andwheredB

is the change in the Brownian motion over the infinitesimal period of time dt.

To construct such a process, we first need to understand how to interpret the term b(ω,t)dB.

Fix any ω ∈ Ω; for any partition P of [0,t] into small intervals [s ,s ],[s ,s ],...,[s ,s ]

0 1 1 2 n−1 n

where s =0 and s =t, we compute the sum

0 n

n−1

(cid:88) (cid:0) (cid:1)

ΣP = b(ω,s i) B(ω)

si+1

−B(ω)

si

.

i=0

Define the norm of the partition P to be

∥P∥=max{s −s ,s −s ,...,s −s };

1 0 2 1 n n−1

If b satisfies some technical conditions, the limit as ∥P∥→0 of ΣP exists.

This limit is known as an Ito integral and we denote it with

(cid:90) t

b(ω,s)dB(ω) .

s

0

We now define the process

(cid:90) t (cid:90) t

X (ω)=X (ω)+ a(ω,s)ds+ b(ω,s)dB(ω)

t 0 s

0 0

for all t≥0.

Example. We calculate (cid:82)t sdB from first principles.

0 s

(1) Take any partition P of [0,t] into small intervals [s ,s ],[s ,s ],...,[s ,s ] where s =0

0 1 1 2 n−1 n 0

and s =t.

n

(2) The sum

n−1

(cid:88) (cid:0) (cid:1)

ΣP = s

i

B

si+1

−B

si

i=0

is a sum of independent normally distributed random variables with mean 0 and variance s2(s −

0 1

s ),s2(s −s ),...,s2 (s −s ).

0 1 2 1 n−1 n n−1

(3)ΣP isnormallydistributedwithmean0andvariances2 0(s 1−s 0)+s2 1(s 2−s 1)+···+s2 n−1(s n−

s ).

n−1

(4) As ∥P∥→0 this variance converges to (cid:82)t s2ds=t3/3.

0

(cid:82)t

(5)Weconcludethat sdB isanormallydistributedrandomvariablewithmean0andvariance

0 s

t3/3.

3. MODELLING STOCK PRICES 41

Henceforth we write

dX =a(X,t)dt+b(X,t)dB

to denote that fact that X is a stochastic process defined by

(cid:90) t (cid:90) t

X (ω)=X (ω)+ a(X (ω),s)ds+ b(X (ω),s)dB(ω) .

t 0 s s s

0 0

We shall refer to stochastic processes of this form as Ito processes. .

Sometimes we are given an equation as above and asked to find the process X. We then regard

thisequationasakindofdifferentialequation,astochastic differential equation (SDEforshort)and

refer to a process X satisfying it as a solution to this equation.

It is usually very hard to solve these SDEs.

3. Modelling stock prices

As a first approximation we model the proportional1 increase in stock prices as a Brownian

motion. We could then derive the following discrete time version

dS

=σdB,

S

where dS is the change in the stock price over a short time from t to t+dt, dB =B −B , and

t+dt dt

B is a Brownian motion. (In particular proportional increases in S are independent, e.g., today’s

increase in a stock price is independent of tomorrow’s increase.)

This model in unsatisfactory: it implies that the values of stocks vary without any long term

trend, i.e., E(dS)=0.

S

A quick glance at historical data shows that this is not very plausible: Despite the randomness

Figure 2. The Dow Jones Industrial Index

of the value of the DJIA, its long term exponential growth is quite visible.

To take into account this upward trend in stock prices we introduce a drift term

dS

=σdB+µdt.

S

We refer to such a process S as a geometric Brownian motion.

1If we were to model the price variation of a stock simply as a Brownian motion, it would be possible for the

pricetogobelowzero,althoughstockpricesarealwayspositive. Moreover,changesinpriceoveratimeperiodwould

notbeafunctionoftheinitialprice. Itwouldsuggestthatstockswithdifferentinitialpricescanhavesimilargains

or losses in the same time interval. This is not the case in reality. For example, the probability of observing a £1

changeinpriceoveradayislessforastockpricedat£10thanitisforastockthatisworth£1,000

42 5. THE STOCHASTIC PROCESS FOLLOWED BY STOCK PRICES

Now

dS

E( )=µdt.

S

We shall refer to µ as the expected return of the stock and to σ as the volatility of the stock.

Figure 3. Instances of the process dS =0.1Sdt+σSdB

One can raise several objections to this model. Consider the following list of all trades in

Vodafone shares between 16:22:08 and 16:23:08 on August 16th, 2002:

Trade time Trade price Bid Ask Volume Block price Buy/Sell

16:23:08 100.75p 100.75p 101p 180,614 £181,969 SELL

16:23:08 100.75p 100.75p 101p 1,185 £1,194 SELL

16:23:08 100.75p 100.5p 100.75p 250,000 £251,875 BUY

16:23:01 100.5p 100.25p 100.75p 2,500 £2,512

16:22:55 100.75p 100.5p 100.75p 28,383 £28,596 BUY

16:22:55 100.75p 100.5p 100.75p 3,000 £3,022 BUY

16:22:55 100.75p 100.5p 100.75p 248,815 £250,681 BUY

16:22:55 100.75p 100.5p 100.75p 179,802 £181,151 BUY

16:22:55 100.75p 100.5p 100.75p 40,000 £40,300 BUY

16:22:42 100.5p 100.5p 100.75p 1,500 £1,508 SELL

16:22:41 100.75p 100.5p 100.75p 23,117 £23,290 BUY

16:22:39 100.527p 100.5p 100.75p 10,000 £10,053 SELL

16:22:38 100.75p 100.5p 100.75p 48,500 £48,864 BUY

16:22:38 100.75p 100.5p 100.75p 150,000 £151,125 BUY

16:22:38 100.75p 100.5p 100.75p 25,000 £25,188 BUY

16:22:38 100.75p 100.5p 100.75p 26,500 £26,699 BUY

16:22:29 100.688p 100.5p 100.75p 25,000 £25,172 BUY

16:22:18 100.75p 100.5p 100.75p 25,000 £25,188 BUY

16:22:08 100.75p 100.5p 100.75p 11,000 £11,082 BUY

(1) The model allows any real number to be a value for S, but in real life there is a smallest

unit.

(2) We assume that prices are changing continuously but trades occur at discrete times.

(3) Shares are often traded through market makers. They buy at the bid price and sell at the

ask price. So there are two prices! Sometimes it is crucial to model both.

(4) Sometimes, e.g., during market crashes, changes seem to have a “memory”.

4. ITO’S LEMMA. 43

We should regard our model as an approximation to real life prices and trades, not as an accurate

description.

4. Ito’s Lemma.

Our main goal is to understand the prices of derivatives, which are themselves (by definition)

dependent on the prices of their underlying assets. In particular, given a model for the prices of the

underlying asset, X(t), we would like to get a model for the prices of the derivatives, seeing them

as “functions” of the stock price and time, i.e., Y =G(X,t). To this end, we need to develop some

technical tools.

Consider a stochastic process X whose change over a small interval of time from t to t+dt is

t

given by

dX =a(X,t)dt+b(X,t)dB

where a(x,t) and b(x,t) are functions of x and t. Consider a new stochastic process Y = G(X,t)

where G(x,t) is a function of x and t. What sort of process is Y?

If B where just a variable rather than Brownian motion, the chain rule would give us

(cid:18) (cid:19)

∂G∂X ∂G ∂G∂X

dY = + dt+ dB

∂x ∂t ∂t ∂x ∂B

(cid:18) (cid:19)

∂G ∂G ∂G

= a+ dt+ b dB.

∂x ∂t ∂x

However, this is false: B is not a variable, and it turns out that we have to add a term:

Theorem 13 (Ito’s Lemma). Assume that G(x,t) is twice continuously differentiable with re-

spect to x and continuously differentiable with respect to t. The process Y = G(X,t) is also an Ito

process. In fact,

(cid:18) ∂G ∂G 1∂2G (cid:19) ∂G

dY = a+ + b2 dt+ b dB.

∂x ∂t 2 ∂x2 ∂x

ThereasonwhythistheoremholdsisthatdB2 isnotnegligiblecomparedtodtanddB, infact,

roughly speaking ”dB2 =dt”.

√

To see this write dB =B −B as ∆tZ for some standard normal Z.

t+∆t t

Note that 1=Var(Z)=E(Z2)−E(Z)2 =E(Z2), hence E(dB2)=E(∆tZ2)=∆tE(Z2)=∆t, and

Var(dB2)=∆t2Var(Z2) is of order ∆t2.

As ∆t→0, Var(dB2)≪∆t=E(dB2), so dB2 “converges to” ∆t.

Here is a sketch of a (very) informal proof of Ito’s Lemma:

Using the Taylor series for G we can write

∂G ∂G 1∂2G ∂2G 1∂2G

dY = ∆X+ ∆t+ ∆X2+ ∆X∆t+ ∆t2+ higher order terms.

∂x ∂t 2 ∂x2 ∂x∂t 2 ∂t2

Now

(∆X)2 =(a∆t+bdB)2 =b2∆t+ higher terms in ∆t

So for small ∆t we approximate

∂G ∂G 1∂2G

dY ≈ ∆X+ ∆t+ b2∆t+ higher terms in ∆X,∆t=

∂x ∂t 2 ∂x2

44 5. THE STOCHASTIC PROCESS FOLLOWED BY STOCK PRICES

∂G ∂G 1∂2G

(a∆t+b∆B)+ ∆t+ b2∆t+ higher terms in ∆B,∆t.

∂x ∂t 2 ∂x2

Example. (Application of Ito’s Lemma)

(1) Use Ito’s Lemma to show that

d(cid:0) B2(cid:1)

=dt+2B (dB ).

t t t

(2) Then use the result to deduce that

(cid:90) t 1 t

B dB = B2− .

s s 2 t 2

0

Let X be the standard Brownian Motion, i.e., it is the Ito process with dX =0×dt+1×dB

t t

and consider the process Y =G(X ,t), where G(x,t)=x2. We have

t t

∂G ∂2G ∂G

=2x, =2, =0

∂x ∂x2 ∂t

so Ito’s Lemma implies that

(cid:18) (cid:19)

1

d(B2)=dY = 2X×0+0+ 2×12 dt+2X×1 dB =dt+2B dB.

t 2 t

Integrating both sides of the equation above gives

(cid:90) t d(cid:0) B2(cid:1) =t+2(cid:90) t B dB ⇒(cid:90) t B dB = 1 B2− t .

s s s s s 2 t 2

0 0 0

Example. The stochastic process followed by forward stock prices

Consider a forward contract on stock paying no dividends maturing at time T;

let F(t) be its forward price at time t≥0:

F(t)=S(t)er(T−t),

where S(t) is the spot price of the stock at time t. Regard F as a function of s and t,

i.e., F =F(s,t)=ser(T−t):

∂F ∂2F ∂F

=er(T−t), =0 and =−rser(T−t).

∂s ∂s2 ∂t

Our model assumes that

dS =µSdt+σSdB

(i.e., that S follows Geometric Brownian Motion with drift µ and volatility σ)

so Ito’s Lemma implies that

(cid:16) (cid:17)

dF = er(T−t)µS−rSer(T−t) dt+er(T−t)σSdB =(µ−r)Fdt+σFdB,

i.e., F follows a geometric Brownian motion with drift µ−r.

Example. The stochastic process followed by the logarithm of stock prices

Let S be the spot price of a certain stock at time t and let G=G(s,t)=lns.

Since

∂G 1 ∂2G −1 ∂G

= , = and =0

∂s s ∂s2 s2 ∂t

and since dS =µSdt+σSdB

5. EXERCISES 45

Ito’s Lemma implies that

(cid:18)

1

σ2S2(cid:19)

σS

(cid:18) σ2(cid:19)

dG= µS− dt+ dB = µ− dt+σdB.

S 2S2 S 2

Corollary 14. Consider a fixed time T in the future and the spot price S at that time: the

T

logarithm of the proportional price change of the stock,

S

ln T =lnS −lnS,

S T

(cid:18) σ2(cid:19)

is normally distributed with expected value µ− T and variance σ2T.

2

Example. ConsiderthepriceS ofastockwithcurrentpriceS =£10,expectedannualreturn

0

ofµ=15%andannualpricevolatility(i.e.,standarddeviation)ofσ =20%. Thestockpricefollows

a geometric Brownian motion

dS =µSdt+σSdB =0.15Sdt+0.2SdB

and for any time T in the future the price of the stock at time T satisfies

lnS

∼N(cid:18)

lnS

+(cid:18)

µ−

σ2(cid:19) T,σ2T(cid:19)

=N(cid:0) ln10+0.13T,0.22T(cid:1)

.

T 0 2

So for example, one can say that with 90% confidence2 in 6 months,

0.2 0.2

ln10+0.065− √ ×1.645

1/2

2 2

i.e., with 90% confidence the logarithm of stock price in 6 months will be between approximately

2.135 and 2.6 and so the stock price itself will be between approximately £8.46 and £13.47 with

90% confidence.

5. Exercises

(1) If Y is the stochastic process Y =eBt, what is dY?

t t

(2) Letf(s)beafunctionforwhich(cid:82)t f(s)2dsexistsforallt≥0,andletX bethestochastic

0 t

(cid:82)t

process defined by X = f(s)dB . Explain why X is normally distributed with mean 0

t 0 s t

andvariance(cid:82)t f(s)2ds. (Hint: whatisthedistributionofasumofindependent,normally

0

distributed random variables?)

(3) Verify that Y =Y

e(cid:16) σBt−σ 22 t(cid:17)

is a solution of the SDE

t 0

dY =σYdB.

∂2u ∂u

(4) Let u(x,t) be a differentiable function satisfying the PDE +2 =0. Show that the

∂x2 ∂t

process Y =u(B ,t) satisfies

t t

∂u

dY = (B,t)dB.

∂x

(5) Show that

(cid:90) t 1 (cid:90) t

B2dB = B3− B ds.

s s 3 t s

0 0

2Youdon’thavetoworryaboutconfidenceintervalsfortheexam.

CHAPTER 6

The Black-Scholes pricing formulas

In this chapter we derive formulas for the prices of European call and put options.

1. The Black-Scholes differential equation

Lemma 15. Assume that a stock price S follows the Geometric Brownian motion

dS =µSdt+σSdB

where µ and σ are constants.

Let f =f(S,t) be the value at time t of any derivative contingent on the value of S at some t=T.

Assumef(s,t)istwicedifferentiablewithrespecttosanddifferentiablewithrespecttot. Theprocess

followed by f is

(cid:18) ∂f ∂f 1∂2f (cid:19) ∂f

df = µS+ + σ2S2 dt+ σSdB.

∂s ∂t 2 ∂s2 ∂s

Proof. Apply Ito’s Lemma with a(S,t)=µS and b(S,t)=σS. □

The discrete version of the equation is

(cid:18) ∂f ∂f 1∂2f (cid:19) ∂f

∆f = µS+ + σ2S2 ∆t+ σS∆B.

∂s ∂t 2 ∂s2 ∂s

where ∆B is a normally distributed random variable with zero mean and variance ∆t.

As in the discrete case (cf. the binomial tree chapter), we will construct a clever portfolio with

some number of shares (to be determined) so that no matter what happens to the price S of the

t

underlying asset (in the binomial tree it could either go up or down and assume only two possible

values), the value of this portfolio will be the same.

∂f

Consider a portfolio consisting of a variable quantity of shares and −1 derivatives; let Π be

∂s

∂f

the value of this portfolio, i.e., Π= S−f.

∂s

After a short period of time ∆t the value of the portfolio changes by

∆Π= ∂f ∆S−∆f = ∂f (µS∆t+σS∆B)−(cid:0)∂f µS+∂f +1∂2f σ2S2(cid:1) ∆t−∂f σS∆B =(cid:18) −1∂2f σ2S2− ∂f(cid:19) ∆t

∂s ∂s ∂s ∂t 2 ∂s2 ∂s 2 ∂s2 ∂t

Notice that ∆Π is non-stochastic!

Since the value of Π in the future is known with certainty, its value must be increasing at the

same rate as a risk-free deposit earning interest r:

∆Π=futureΠ−Πnow=er∆tΠ−Π

and for infinitesimal ∆t we obtain1

∆Π≈(1+r∆t)Π−Π⇒∆Π≈rΠ∆t.

1Taylor-expander∆t

47

48 6. THE BLACK-SCHOLES PRICING FORMULAS

If we substitute equations back we obtain the Black-Scholes differential equation:

Theorem 16.

∂f ∂f 1 ∂2f

+rS + σ2S2 =rf.

∂t ∂s 2 ∂s2

This is a partial differential equation (PDE).

2. Boundary conditions

Tomotivatethediscussionbelow,thinkofhavingtosolveastandardordinarydifferentialequa-

tion(ODE).Supposeforexamplethatwearegiventosolvex′(t)=1.Tosolvethis,simplyintegrate

in t to get x(t) = t+c. This means that for every value of c ∈ R, t+c is a solution to the above

ODE, i.e., we have an infinite family (parametrized by c) of solutions to the original ODE. To find

a specific solution, we need to impose additional requirements, e.g., that the value of the solution

x(t)atsomet=t willhavetobex(t )=x .Itiscommon(andforphysically-relevantmodelsalso

1 1 1

intuitive) to impose an initial condition (you might see the term “boundary condition” as well),

i.e., at time t = t = 0 : x(t ) = x . With this condition (plugging (t,x(t)) = (t ,x ) = (0,x ),c

0 0 0 0 0 0

is uniquely determined and the solution that satisfies both the differential equation and the initial

condition is x(t)=t+x .

0

To obtain prices (=specific solutions) from the Black-Scholes PDE we impose boundary condi-

tions, e.g., for a European call option with strike X and expiring at time T the boundary condition

isf(S,T)=max{S−X,0}forallS. ForaEuropeanputoptionwithstrikeX andexpiringattime

T the boundary condition is f(S,T)=max{X−S,0} for all S.

The theory of linear partial differential equations shows that we obtain a unique solution by

imposing a further boundary condition at S =0. Now if S =0 at any time 0≤t≤T, then S =0

t t

for all t ≤ T, and hence f(0,t) = e−r(T−t)f(0,T) which provides us with a boundary condition.

E.g., in the case of European call options, f(0,t)=0.

(For numerical purposes we may impose boundary conditions at S =∞).

Example. f(S,t)=ertS1−2r/σ2 is a solution of the Black-Scholes PDE.

Check:

∂f ∂f 2r

=rf(S,t), =(1−

)ertS−2r/σ2

,

∂t ∂s σ2

∂2f 2r 2r

=−(1− )

ertS−1−2r/σ2

∂s2 σ2 σ2

therefore,

∂f ∂f 1 ∂2f 2r 1 2r 2r

+rS + σ2S2 =rf(S,t)+rS(1− )ertS−2r/σ2 − σ2S2(1− ) ertS−1−2r/σ2

∂t ∂s 2 ∂s2 σ2 2 σ2 σ2

=rf(S,t)+r(1−

2r )ert(cid:2) S×S−2r/σ2 −S2×S−1−2r/σ2(cid:3)

σ2

=rf(S,t)+0.

3. THE BLACK-SCHOLES PRICING FORMULAS 49

ConsideraderivativeonacertainstockwithsinglepayoffattimeT

>0amountingtoS1−2r/σ2

,

T

where r is the constant interest rate. Assume that 1−2r/σ2 >0.

Find the value of the derivative at time 0≤t≤T.

There is a trick here, you will see why you do that at the very end of this argument. Instead

of a single derivative (the price of which we want to find), we shall consider a portfolio consisting

of erT (identical) derivatives, and write v(S,t) for the price of this portfolio; we have v(S ,T) =

T

erTS1−2r/σ2

=f(S ,T). Now,

T T

• both v(S,t) and f(S,t) are solutions of the Black-Scholes PDE2 and

• v andf havethesamevaluesattheboundary: v(0,t)=f(0,t)=0,andv(S,T)=f(S,T).

But there is only one solution of the Black-Scholes PDE satisfying a given boundary condition and

this forces v(S,t)=f(S,t)=ertS1−2r/σ2.

So the price of one derivative is e−rTertS1−2r/σ2 =e−r(T−t)S1−2r/σ2. For the last step we used

the linearity of the Black-Scholes PDE, if X is a solution, so is λX,∀λ∈R. (Here λ=erT)

3. The Black-Scholes pricing formulas

Theorem 17 (The Black-Scholes pricing formulas). Consider a European option at time t on

stock with spot price S , with strike price X and expiring at time T.

t

Let σ be the annual volatility of the stock, and r the T-year interest rate. Define

ln(S

/X)+(cid:0) r+σ2/2(cid:1)

(T −t)

t

d = √ ,

1

σ T −t

ln(S /X)+(cid:0) r−σ2/2(cid:1) (T −t) √

t

d = √ =d −σ T −t.

2 1

σ T −t

Then the price of the call option at time t is

c =S Φ(d )−Xe−r(T−t)Φ(d )

t t 1 2

and the price of the put option is

p =Xe−r(T−t)Φ(−d )−S Φ(−d ).

t 2 t 1

where Φ is the standard normal distribution function.

Proof. Sketch3

Define the function c(S,t)=S Φ(d )−Xe−r(T−t)Φ(d ). In view of the BS differential equation we

t 1 2

need to verify that

∂c ∂c 1 ∂2c

+rS + σ2S2 =rc,

∂t ∂s 2 ∂s2

lim c(S ,t)=max{S −X,0}

t T

t→T

lim c(S,t)=0.

S→0

The verification of both these statements is straightforward (but tedious!) A similar argument pro-

duces the value of p.

2for f we showed that by plugging it in, and for v because as the price of derivatives it has to satisfy the B-S

PDE

3MoredetailsabouttheproofcanbefoundintheAppendixofChapter13inJ.Hull’sbook“Options,Futures,

andotherDerivatives”.

50 6. THE BLACK-SCHOLES PRICING FORMULAS

□

TheBlack-Scholespricingformulasarearesultofano-arbitrageargument: ifviolateduseport-

folio Π to get a free lunch. The no-arbitrage argument was used after the proof of Lemma 15 when

we argued that if the price of the portfolio with spot value Π is to be known by certainty in T

years, its value would be erTΠ T years from now. (Make sure you understand exactly where the

no-arbitrage point is.)

In practice one cannot adjust Π continuously, and if there are trading charges, one cannot even

make frequent adjustments.

4. More exotic derivatives

Our derivation of the Black-Scholes PDE holds for prices of some derivatives whose payoffs

may depend not just on the underlying asset value at expiration time T, but also on prices before

expiration. In these cases one may need to restrict the region of definition of the solution of the

PDE and give appropiate boundary conditions for that region.

Example (Up an out barrier put options). Consider a derivative expiring at time T whose

underlying asset price at time t is S . This derivative is characterized by an additional parameter,

t

the barrier K > 0, and its payoff is as follows. If for any 0 ≤ t ≤ T, S > K the payoff is zero.

t

Otherwise, the payoff is identical to that of a corresponding European option with strike X

The price V(S,t) of such an option is a solution of the Black-Scholes PDE which satisfies the

boundary conditions V(K,t)=0 for all 0≤t≤T and V(S,T)=max{X−S,0}.

While it is not easy to find a solution, as soon as one finds a function of two variables, that

(i) satisfies the Black-Scholes PDE AND (ii) all the boundary conditions4, by uniqueness theory of

linear PDEs, this function has to be the price of the derivative we are seeking.

Example (Americanputoptions). ThepriceV(S,t)attimetofanAmericanoptionwhichhas

not been exercised is a solution of the Black-Scholes PDE which satisfies a complicated boundary

condition. For each time 0 ≤ t ≤ T there exists a lowest underlying price γ(t) below which the

option would be exercised.The boundary condition is then V(γ(t),t) = X −γ(t) for all 0 ≤ t ≤ T

and V(S,T)=max{X−S,0}.

Example (Path dependent options). The Black-Scholes PDE can describe the prices of com-

plicated, path-dependent options, i.e., of options whose values depend not only on the underlying

asset price, but on the underlying asset price during the life of the option.

For example, consider an up-and-out put option expring in T years with strike price X, and

barrier B. If the underlying asset price S goes above B for for 0 ≤ t ≤ T, the option expires

t

worthless. Otherwise, the option pays off maxX−S ,0 at time T.

T

If by time 0 ≤ t ≤ T the price S has not exceeded B, the price V(S,t) of this option is a

t

solution of the Black-Scholes PDE: the original argument still holds.

Howevertheboundaryconditionsneedtobemodified: weimposetheadditionalconditionV(B,t)=

0 for all 0≤t≤T.

4assumingthattheboundaryconditionsare“reasonable”–whichwillalwaysbethecaseinourcourse”

5. THE BLACK-SCHOLES PRICING FORMULAS: THE RISK NEUTRAL VALUATION APPROACH. 51

5. The Black-Scholes pricing formulas: the risk neutral valuation approach.

Recall the BS PDE :

∂f ∂f 1∂2f

+rS + σ2S2 =rf.

∂t ∂s 2 ∂s2

Notice that µ does not appear here! As we will see in more detail in the next chapters, the quantity

µ−r istheexcessreturnthatinvestorsdemandwheninvestinginanassetwhosevolatilitymeasure

is σ. We conclude that the risk-aversion of investors does not affect the value of derivatives given as

solutions of the Black-Scholes PDE. Since this formula holds regardless of amount of risk-aversion

and we might as well assume that investors are risk neutral.

Risk-neutralinvestorsonlycareabouttheexpectedreturnoftheirinvestments,andtheydonot

care about uncertainty regarding these returns. Hence all investments in a risk neutral world must

have the same expected return r equal to the risk-free interest rate r.

Definition (Useofarisk-neutralvaluationargument). Letf bethepriceofaderivativewhich

pays H(S ) for some function H at a future time T. In our risk-neutral world the stock price has

T

expected return r, the risk-free T-year interest rate. In a risk-neutral world the current value of the

derivative f(S,0) is the present value of the expected value of the derivative payoff at time T, i.e.,

f(S,0)=e−rTE(cid:101)(H(S T)),

where E(cid:101) denotes expected values in our risk-neutral world.

5.1. Example-Digital options. Consider a derivative on a stock, which at expiration time

T pays £1 if S ≤ a, for some positive number a, and zero otherwise. (These options are known

T

as digital or binary options.) Let the volatility of the stock price be σ and assume that all interest

rates are constant and equal to r. Apply a risk neutral valuation argument to show that, for any

0≤t≤T, the value of this derivative equals

(cid:32) ln(a/S)−(cid:0) r−σ2/2(cid:1)

(T

−t)(cid:33)

e−r(T−t)Φ √

σ T −t

where Φ is the cumulative distribution function of the standard normal distribution.

Solution:

We are assuming S follows the process

dS =µSdt+σSdB

forconstantsµandσ,andsoattime0≤t≤T,lnS isnormallydistributedwithmeanlnS+(µ−

√ T

σ2)(T −t) and standard deviation σ T −t5. In a risk neutral world we set µ=r and now lnS is

2 √ T

normally distributed with mean lnS+(r− σ2)(T −t) and standard deviation σ T −t.

2

The event S ≤ a is equivalent to the event lnS ≤ lna and so the probability in this risk

T T

(cid:18) (cid:19) (cid:18) (cid:19)

neutral world of the event S

T

≤a is Φ

lna−(lnS σ+ √(r T− −σ t22 )(T−t))

=Φ

lna/S− σ√(r− (Tσ −22 t) )(T−t))

. In our

risk neutral world the value of the derivative is the present value of the expected value of its payoff,

i.e.,

(cid:32) ln(a/S)−(cid:0) r−σ2/2(cid:1)

(T

−t)(cid:33)

e−r(T−t)Φ √

σ T −t

(Convince yourself that this is the correct expectation.)

5wehaveseenthisbeforeinChapter5,Corollary14.

52 6. THE BLACK-SCHOLES PRICING FORMULAS

5.2. Example-Back to European call/put options. Our assumptions imply that lnS is

T

(cid:18) σ2(cid:19)

normally distributed with mean lnS+ µ− T and variance σ2T; in our risk neutral valuation

2

argumentwesetµ=r. Risk-neutralinvestorsalsoexpectthecurrentvalueofthederivativef(S,0)

to be the expected value of the present value of the derivative payoff at time T, i.e.,

f(S,0)=e−rTE(cid:101)(H(S T)).

Consider the case where H(y)=max{y−X,0} with X being the strike price of the call option.

We have c=e−rTE(cid:101)(max{S

T

−X,0}).

Let ϕ be the density function of the lognormal random variable S in our risk neutral world.

T

(cid:90) ∞

c=e−rT (y−X)ϕ(y)dy.

X

Lemma 18.

(cid:90) ∞

(y−X)ϕ(y)dy =SerTΦ(d )−XΦ(d )

1 2

X

ln(S/X)+(r+σ2/2)T ln(S/X)+(r−σ2/2)T √

where d 1 = √ , d 2 = √ =d 1−σ T.

σ T σ T

Now we obtain

c=SΦ(d )−e−rTXΦ(d )

1 2

and a similar argument shows that the price of a European put option with strike price X is

p=Xe−rTΦ(−d )−SΦ(−d )

2 1

6. Volatility

[For your information only-not examinable]

The parameters European options are the spot price of the stock, the strike price, time to

expiration, the interest rate, and the volatility σ of the stock price.

The first four parameters are always known; but volatility is not directly observable. One can

estimatehistorical volatility byanalysingthetimeseriesconsistingofpricesofthestockatprevious

times in the past.

However, there are many problems involved with these estimates. One such problem is that

eventhoughourmodelforstockpricesassumesconstantvolatility,inpracticesomeperiodsoftime,

e.g., immediately after September 11th, 2001, are more volatile than others. So we might want to

give lower weights to more distant measurements.

Traders do not normally use historical volatility when applying the Black-Scholes pricing for-

mulas. Instead they use implied volatilities which are the value of the volatility parameter which

will produce the observed market price of a given option. This sounds like a circular argument, but

it is useful for example to produce prices of an option based on a similar one and to produce new

prices as the price of the underlying stock changes or as time progresses.

7. Exercises

(1) Find the price of a European call option, expiring in six months’ time, on a non-dividend

paying stock currently trading at £14, with strike price £15 and annual volatility of the

stock price 0.5. Assume the risk-free interest rate is 5%.

7. EXERCISES 53

(2) FindthepriceofaEuropeanputoption,expiringinthreemonths’time,onanon-dividend

payingstockcurrentlytradingat£10,withstrikeprice£9andannualvolatilityofthestock

price 1. Assume the risk-free interest rate is 8%.

(3) Let c (S,t) and c (S,t) be the prices at time t of two European call options on the same

1 2

non-dividend paying stock with price S, with same expiration T and with strike prices X

1

and X , respectively. Assume that X

2 1 2

(a) Explain why c −c is a solution of the Black-Scholes PDE.

1 2

(b) By considering c (S ,T)−c (S ,T) deduce that

1 T 2 T

0≤c (S ,t)−c (S ,t)≤(X −X )e−r(T−t).

1 t 2 t 2 1

(4) Consider an option on a stock which gives its holder at time T a European call option on

1

the stock whose strike price is S , the stock price at time T , and which expires at time

T1 1

T > T . These options are known as forward start options. Find a formula for the value

2 1

of this option at any time 0≤t≤T . (Hint: distinguish between the cases 0≤t≤T and

2 1

T ≤t≤T . Use the Black-Scholes pricing formula to find the value of the option at time

1 2

t=T .)

1

(5) Consider a derivative on certain stock which provides a single payoff at time T > 0, and

letf(S ,t)bethepriceofthatderivativewhere0≤t≤T andS isthesharepriceattime

t t

t. Explain why f(0,t)=e−r(T−t)f(0,T) for all 0≤t≤T where r is the constant interest

rate for all maturities.

(6) Show that two exact solutions of the Black-Scholes PDE are V(S,t) = kS, and V(S,t) =

kert, where k and r are constants. What do these solutions represent?

(7) (a) Show that

f(S,t)=ertS1−2r/σ2

is a solution of the Black-Scholes PDE.

(b) Consider a derivative on certain stock (whose price S, as always, follows the process

dS = Sµdt+SσdB ) which provides a single payoff at time T > 0 amounting to

S1−2r/σ2 where r is the constant interest rate. Assume that 1−2r/σ2 >0. Find the

T

value of the derivative at time 0≤t≤T.

(8) Find the price of an option similar to the digital option (see example in Section 5), with

the difference that the payoff is £1 if the stock price S at time T is at least a, for some

T

positive number a, and zero otherwise.

(9) A pay later option is an agreement between two parties in which:

• the first party will pay the second an amount equal to max{S −X,0} at time T

T

where S is the price of a certain stock at time T and X is a constant, and

T

• if S −X ≥0 the second party will pay an amount v to the first party.

T

Initially it costs nothing to enter this agreement, but it may be traded later. Let u(S,t)

be the price of entering this agreement as the second party at time t.

(a) Show that entering this agreement as the second party is equivalent to buying a

portfolio long one European call option on the stock with strike X and expiration T

and short v options as in the previous problem with a=X.

(b) Find a formula for u(S,t).

(c) Use the fact that v is the number which makes the portfolio above worthless at time

t=0 to deduce a formula for v.

(10) Let S be the price at time t of a certain non-dividend paying stock with annual volatility

t

σ. AderivativewillpaylnS atafuturetimeT. Applyariskneutralvaluationargument

T

54 6. THE BLACK-SCHOLES PRICING FORMULAS

to find the value v(S,t) of the derivative at time t < T. Verify that v(S,t) satisfies the

Black-Scholes PDE.

(11) Let c (S,t) and c (S,t) be the prices at time t ≥ 0 of two European call options on the

A B

same non-dividend paying stock with spot price S, with same strike price X and with

expirations T >T , respectively.

A B

(a) Explainwhythefunctionf(S,t)=c (S,t)−c (S,t)isasolutionoftheBlack-Scholes

A B

PDE.

(b) Show that f(S,t) is the price at time t of a European-style derivative with expiration

time T . What is the payoff function of this derivative?

B

(c) Use the fact that c A(S,t)≥S−Xe−r(TA−t) to deduce that the payoff function of the

derivative in (b) is non-negative.

(d) Show that c (S,t)−c (S,t)≥0 for all 0≤t≤T .

A B B

(12) Assume that a stock price S is given as the Ito process

dS =µSdt+σSdB

where µ and σ are constants. Assume also that all interest rates are non-stochastic and

equal to r. Let Φ be the cumulative standard normal distribution function. Let X >

ln(S/X)+(r+σ2/2)(T −t)

0 and define d(S,t) = √ . Assume henceforth that f(S,t) =

σ T −t

SΦ(d(S,t)) satisfies the Black-Scholes PDE.

∂f ∂f 1 ∂2f

+rS + σ2S2 =rf.

∂t ∂S 2 ∂S2

(a) Compute lim d(S,t). (Hint: your answer should depend on the values of S and

t→T, t

X.)

(b) Use (a) and the fact that Φ is a continuous function to compute lim Φ(d(S,t)).

t→T, t

(c) Assume that f(S,t) is continuous for all 0 ≤ t ≤ T and S ≥ 0. Compute f(0,t) for

any 0 ≤ t < T, and compute f(S ,T) for any S ≥ 0. (Hint: use the assumption to

T T

write these values as limits.)

(d) An asset-or-nothing option with strike price X and expiration time T pays owner at

time T: S if S >X, S /2 if S =X, and nothing otherwise. Show that the price

T T T T

of this option at time 0 ≤ t ≤ T is f(S ,t) where S is the stock price at time t.

t t

Explain in detail your reasoning.

CHAPTER 7

Portfolio Theory

Inthischapterweaimtofindoptimalinvestmentstrategies. Todosowefirstneedtounderstand

investors’ preferences, i.e., how does a person decide which investment is best?

Consider the following example involving three £1,000 one-year investments:

Portfolio A: Will be worth £1,100 with probability 1.

Portfolio B: Will be worth £1,000 with probability 1/2 and £1,300 with probability 1/2.

Portfolio C: Will be worth £500 with probability 1/10, £1,200 with probability 8/10 and

£3,000 with probability 1/10.

The expected returns are

1100−1000

r = =0.1,

A 1000

11000−1000 11300−1000

r = + =0.15

B 2 1000 2 1000

1 500−1000 8 1200−1000 1 3000−1000

r = + + =0.31

C 10 1000 10 1000 10 1000

Now consider the following investors:

Mr. X: Wants to sail around the world on a cruise costing £1,200 a year from now.

Ms. Y: Must repay her mortgage in a year and must have £1,100 to do so.

Dr. Z: Needs to buy a rare book worth £1,300.

The chances of Mr X. sailing around the world if he invests in investments A,B or C are 0, 1/2

and9/10respectively,soheshouldbeadvisedtoinvestinC.OnlyinvestmentAguaranteesareturn

sufficientforMs.Ytopayhermortgageandsheshouldchooseit. Theprobabilityoftheinvestments

being worth at least £1,300 after a year are 0, 1/2 and 1/10 respectively, so Dr. Z should invest in

portfolio B.

So different investors prefer different investments!

“highest expected returns”̸=“optimal”: the whole distribution of the returns needs to be taken

into account.

Instead of considering the whole distribution of the returns of an investment we will take into

account two parameters:

• the expected return which we denote r, and

• the standard deviation of the return which we denote σ.

Wenowrephraseourproblem: givenasetofinvestmentswhosereturnshaveknownexpectedvalues

and standard deviations, which one is “optimal”?

1. Axioms satisfied by preferences

Consider two investments A and B with expected returns r and r and standard deviation of

A B

returns σ and σ .

A B

The following assumptions look plausible:

55

56 7. PORTFOLIO THEORY

A1: Investors are greedy: If σ =σ and r >r investors prefer A to B.

A B A B

A2: Investors are risk averse: If r =r and σ >σ investors prefer B to A.

A B A B

A3: Transitivity of preferences: If investment B is preferable to A and if investment C is

preferable to B then investment C is preferable to A.

We are introducing a partial ordering ≺ on the points of the σ-r plane:

Notice that B is preferable to both A and C. Investments A and C are incomparable.

2. Indifference curves

Wedescribethepreferencesofaninvestorbyspecifyingthesetsofinvestmentswhichareequally

attractive to the given investor. We do so by defining an indifference curve of an investor: this is

a curve consisting of points (σ,r) for which investments with these expected returns and standard

deviation of returns are all equally attractive to our investor.

NoticethatassumptionsA1,A2andA3implythatthesecurvesmustbenon-decreasing. (Prove

this yourself!)

Consider hypothetical investors X,Y,Z and W with the following indifference curves.

3. PORTFOLIOS CONSISTING ENTIRELY OF RISKY INVESTMENTS 57

Investor W is risk neutral.

3. Portfolios consisting entirely of risky investments

Consider two investments A and B with expected returns r and r and standard deviation of

A B

returns σ and σ . We split an investment of £1 between the two investments: consider portfolio

A B

Π consisting of t units of investment A and 1−t units of investment B. We can do this for any t

t

and not just 0≤t≤1. For example, to construct portfolio Π we short sell £1 worth of B and buy

2

£2 worth of A, for a total investment of £1. Let A and B be the random variables representing the

annual return of investments A and B.

The variance of Π is given by

t

Var(Π ) = Var(tA+(1−t)B)

t

= Var(tA)+Var((1−t)B)+2Covar(tA,(1−t)B)

= t2Var(A)+(1−t)2Var(B)+2t(1−t)Covar(A,B)

(cid:112)

= t2Var(A)+(1−t)2Var(B)+2t(1−t)ρ(A,B) Var(A)Var(B)

= (tσ )2+2t(1−t)ρ(A,B)σ σ +((1−t)σ )2

A A B B

The shapes of these curves are concave:

Proposition 19. The curve in the σ-r plane given parametrically by

(cid:0)(cid:112) (cid:1)

(tσ )2+2t(1−t)ρ(A,B)σ σ +((1−t)σ )2,tr +(1−t)r

A A B B A B

for 0≤t≤1 lies to the left of the line segment connecting the points (σ ,r ) and (σ ,r ).

A A B B

58 7. PORTFOLIO THEORY

Proof. Since ρ(A,B)≤1,

(cid:112) (cid:112)

(tσ )2+2t(1−t)ρ(A,B)σ σ +((1−t)σ )2 ≤ (tσ )2+2t(1−t)σ σ +((1−t)σ )2

A A B B A A B B

(cid:113)

= (tσ +(1−t)σ )2

A B

= tσ +(1−t)σ .

A B

Theresultfollowsfromthefactthattheparametricequationofthelinesegmentconnectingthe

points (σ ,r ) and (σ ,r ) is

A A B B

{(tσ +(1−t)σ ,tr +(1−t)r ) |0≤t≤1}.

A B A B

□

4. The feasible set

Suppose now that there are many different investments A ,...,A available. We can invest our

1 n

one unit of currency by investing t in A for each 1 ≤ i ≤ n as long as

(cid:80)n

t = 1. What are all

i i i=1 i

possible pairs (σ,r) corresponding to these portfolios? This set of points in the σ-r plane is called

the feasible set.

5. Efficient portfolios

We now return to the main question in this chapter: which portfolios among all possible ones

should an investor satisfying axioms A1,A2 and A3 choose?

Definition. An efficient portfolio is a feasible portfolio that provides the greatest expected

return for a given level of risk, or equivalently, the lowest risk for a given expected return. (This is

also called an optimal portfolio.)

The efficient frontier is the set of all efficient portfolios.

Obviously, our investor should choose a portfolio along the efficient frontier!

6. DIFFERENT CHOICES OF PORTFOLIOS FOR DIFFERENT APPETITES FOR RISK 59

The feasible set is convex along the efficient frontier, in the sense that for any two portfolios A

andBinthefeasibleset,thereexistfeasibleportfoliosabovetheportfoliosinthesegmentconnecting

A and B.

Whichportfolioalongtheefficientfrontierwillourinvestorchoose? Thisiswhereriskpreferences

start playing a role.

6. Different choices of portfolios for different appetites for risk

ConsiderinvestorsX(withnorisktoleranceatall)andW(riskneutral)discussedbeforetogether

with investor U whose indifference curves are given below.

60 7. PORTFOLIO THEORY

If the indifference curves are not too badly behaved, e.g., if the indifference curves are the level

curvesofsomesmoothfunctionF(σ,r),thenweshouldexpecttheoptimalportfoliotobeatapoint

where the indifference curve is tangent to the efficient frontier. Otherwise, if it occurs at a point

where the indifference curve intersects the efficient frontier transversally, find an almost parallel

indifference curve very close to the original one and to its left.

Portfoliosonthiscurvearemorede-

sirable and, if we chose the second

indifference curve close enough to

theoriginalone,itwillalsointersect

the efficient frontier, and this inter-

section will correspond to a better

choice of portfolio than the one cor-

responding to the original point of

intersection.

7. Portfolios containing risk-free investments

We now add a risk-free investment B. Let r be its (expected) return. Since r is constant, its

B B

covariance with the returns of any other portfolio Π is zero so the portfolio Π consisting of t units

t

of currency invested in B and (1−t) units of currency invested in Π has expected return

E(tB+(1−t)Π)=tr +(1−t)r

B Π

(where we used B and Π to denote also the returns of the investments B and Π) and standard

deviation of return

(cid:112) (cid:112) (cid:112)

Var(tB+(1−t)Π)= Var((1−t)Π)= (1−t)2Var(Π)=|1−t|σ .

Π

The curve t (cid:55)→ (σ ,r ) for t ≤ 1 is a straight line passing through the points (0,r ) and

Πt Πt B

(σ ,r ) and all the points on or below such a line will be part of the feasible set.

Π Π

What happens to the efficient frontier? Consider the set S consisting of all the slopes s of lines

ℓ in the σ-r plane which pass through the point (0,r ) and intersect the feasible set. Let m be the

s B

supremum of S. Consider now the line ℓ which is above all the others: The line ℓ will either be

m m

tangent to the efficient frontier or asymptotic to it.

(We will see in Chapter 8 that, if we impose additional conditions on markets and investors, ℓ

m

cannot be an asymptote of the efficient frontier and so it is tangent to it.)

8. EXERCISES 61

This point of

tangency is called

the market port-

folio and we shall

denote the corre-

sponding portfolio

with M.

The new efficient

frontier, ℓ is

m

called the capital

market line.

We just proved the following:

Theorem 20. In the presence of a risk-free investment there exists an (essentially) unique

investment choice consisting entirely of risky investments which is efficient, namely, the market

portfolio.

Any other efficient investment is a combination of an investment in the market portfolio and in

the risk-free investment.

8. Exercises

(1) Consider the following two portfolios with present value £400:

Portfolio A: Is worth £400 with probability 1/2 and £500 with probability 1/2.

Portfolio B: Is worth £390 with probability 0.99 and £5000 with probability 0.01

(a) Calculate the expected returns and standard deviation of returns for these two port-

folios.

(b) Which portfolio would an investor who satisfies axioms A1, A2 and A3 choose?

(c) A bright student answered (b) as follows:

“Consider a person who has £400 and has to come up with £5000 to

preventhishousebeingrepossessed. Thatperson’sonlychanceistoinvest

in B.” Do you think this answer is correct? If so, does this contradict our

theory?

(2) ConsidertwoinvestmentsAandBwithidenticalexpectedreturnsrandstandarddeviation

of returns σ. Let ρ be the correlation between the returns of A and B.

(a) Describe the expected returns and standard deviation of returns of a portfolio Π

t

consisting of £t invested in A and £(1−t) invested in B.

(b) If you had to invest all your wealth in Π for some t, which one would you choose?

t

(c) If ρ=−1, what is the risk-free return?

(3) You want to invest £1000 so that on average you double your investment in a year. Given

thattherisk-freereturnis5%andthatthemarketportfoliohasanannualexpectedreturn

of 20% and standard deviation of return 10%, how much risk would you have to assume in

order to achieve your goal? Describe your investment in detail.

CHAPTER 8

The Capital Asset Pricing Model

Inthischapterweaimtofindthe“correct”priceoffinancialassets. Indoingsowewillintroduce

a new notion of correctness for prices: when we valued derivatives, correct prices were those which

created no arbitrage opportunities; in this chapter prices are correct if they are stable.

We make additional assumptions about markets and investors:

A4 Markets are in equilibrium: The total demand for any financial instrument equals its total

supply.

A5 Uniform horizon: All investors are investing for the same period of time.

A6 Homogeneity: All investors agree on the expected returns of investments, their standard

deviations of returns and the correlations between these returns. All investors can borrow

and lend unlimited amounts of money at the same uniform risk-free rate.

A7 No friction: There are no transaction costs and no taxes.

1. The market portfolio.

Themarketportfolioistheonlyefficientportfolioconsistingentirelyofriskyinvestments. What

is this portfolio?

Theorem 21. Let I ,I ,...,I be all risky investments, and assume their total market value

1 2 n

is w ,w ,...,w . Let W = w +w +···+w . The market portfolio consists of a portfolio of

1 2 n 1 2 n

w

investments of j in I for each 1≤j ≤n.

W j

Proof. Assume that there are m investors and that for any 1 ≤ k ≤ m investor k holds h

kj

worth of investment j.

Write H

=(cid:80)n

h for the total amount invested in risky investments for investor k. Axioms

k j=1 kj

A1, A2, A3, theresultsofChapter7andAxiomsA5, A6implythatallinvestorswillhavethesame

h h

proportion of each risky investment, i.e., kj = lj for every 1≤k,l≤m.

H H

k l

Axiom A4 implies that for each 1 ≤ j ≤ n, w =

(cid:80)m

h . (Notice w is the total supply

j k=1 kj j

of investment j while

(cid:80)m

h is its total demand; the two should be equal if the market is in

k=1 kj

equilibrium.) Summing over all investments gives W

=(cid:80)m

H .

k=1 k

63

64 8. THE CAPITAL ASSET PRICING MODEL

NowtheproportionofthemarketportfolioinvestedinI isthesameasanyinvestor’sproportion

j

of investment in I out of the total risky investments, so we can write this proportion as

j

h h

(cid:80)m

H

1j = 1j k=1 k

H H

(cid:80)m

H

1 1 k=1 k

(cid:16) (cid:17)

=

h 1j 1+(cid:80)m k=2 H Hk

1

(cid:80)m

H

k=1 k

h

+(cid:80)m

h

= 1j k=2 kj

W

(cid:80)m

h

= k=1 kj

W

w

= j

W

□

Stockmarketindices 1areapproximationsofmarketportfolios: thevaluesoftheseindicesarethe

weighted average price of a large set of stocks, where the weights are proportional to the proportion

of the total value of a stock as part of the total value of the whole set of stocks.

You might have also heard of tracker funds: these are investments that hold shares in the same

proportion as a given index, e.g., FTSE 100 trackers. The previous theorem says roughly that the

only risky investments in the portfolio of an investor who assumes axioms A1-A7 must be tracker

funds.

There are other types of investment funds, actively managed funds. These funds invest money

in stocks carefully chosen by spectacularly highly paid fund managers. These funds demand high

feesfrominvestorsinreturnforapplyingtheirtalentsinchoosingthewayinwhichthefund’sassets

will be invested. So you are asked to pay large fees to have axiom A6 broken: these fund managers

claim to possess knowledge which is not apparent to lesser investors.

There is a ongoing debate on whether these fund managers are worth these high fees.

2. The market price of risk

For any efficient investment A lying on the market line we have

r −r

r −r = M Bσ

A B σ A

M

r −r

where r is the risk-free interest rate and M is the market portfolio. We interpret M B as the

B σ

M

market price of risk: this slope measures how much more return investors demand for an increase of

one unit in the volatility of their returns.

We want a similar expression for the excess return above the risk-free interest rate for non-

efficient portfolios, e.g., individual stocks.

Theorem 22. For any portfolio A we have

r −r

r −r = M B Covar(A,M)

A B σ2

M

where r is the risk-free interest rate, M is the market portfolio and Covar(A,M) is the covariance

B

between the return of A and the return of M.

1e.g.,FTS100,theDowJonesIndustrialIndex,S&P500,Nikkei,CAC40,DAX,etc.

3. EXERCISES 65

Proof. Forany0≤t≤1,letportfolioΠ consistofaninvestmentoftinAandaninvestment

t

of 1−t in M. The curve c given by (σ ,r ) in the σ-r plane joins points A and M.

Πt Πt

c intersects the capital market line at M, and the capital market line must be tangent to c at M:

Otherwise c would cross the capital market line and we would have a portfolio above the capital

market line, contradicting the fact that the capital market line is the efficient frontier.

(cid:0)(cid:112)

Calculate the slope of c at M: c is given by t(cid:55)→ t2σ2 +2t(1−t)Covar(A,M)+(1−t)2σ2 ,

A M

(cid:1)

tr +(1−t)r as (0≤t≤1). Evaluate the derivative with respect to t

A M

(cid:32) (cid:33)

2tσ2 +(2−4t)Covar(A,M)−2(1−t)σ2

A M ,r −r

(cid:112) A M

2 t2σ2 +2t(1−t)Covar(A,M)+(1−t)2σ2

A M

at t = 0 to obtain the slope at point M: Covarr (A A− ,MrM

)−σ2

σ M. But this slope must be equal to the

r −r M r −r

slope of the capital market line, i.e., A M σ = M B and we can rearrange this

Covar(A,M)−σ2 M σ

M M

r −r

to obtain r −r = M B Covar(A,M).

A B σ2

M

□

Definition. The beta coefficient of a portfolio A is defined as

Covar(A,M)

β = .

σ2

M

The security market line is the linear relation between expected returns r and beta coefficients

β given by

r =r +(r −r )β.

B M B

We can restate the previous theorem: for any investment with expected return r and beta coef-

ficient β the point (β,r) lies on the security market line.

3. Exercises

(1) You are given the following data on three stocks and the market portfolio:

Expected Correlation with Standard deviation

return market portfolio of return

Stock 1 ? 0.9 20%

Stock 2 9% ? 18%

Stock 3 7% 0.7 ?

Market portfolio 10% 1 15%

The risk-free return is 5%. Give the equation of the capital market line and fill in the

missing entries in the table.

66 8. THE CAPITAL ASSET PRICING MODEL

(2) Consider a market with risk-free return 5% and two risky investments A and B. We are

given the following data:

Investment Expected return Standard deviation of return

A 10% 10%

B 15% 20%

We are also told that the correlation between the returns of A and B is ρ = 0.5. We

assume that the CAPM holds.

(a) Use the fact that the market portfolio is the unique portfolio which maximises

r −r

P B,

σ

P

as P ranges over all portfolios consisting entirely of risky investments, to find the

market portfolio in the market described above.

(b) You are a fund manager and are asked to invest £10,000,000 in a portfolio consisting

of investments A and B together with the risk free investment. The portfolio should

haveanexpectedreturnof12%andthelowestpossiblestandarddeviationofreturns.

Describe the portfolio.

(3) Describe all efficient portfolios whose beta-coefficient is 1.

Solutions

Chapter 1

(1) (a) If the continuous compounding rate is 8%, since each coupon is £5 (paid every year)

the price of the bond will be 5e−0.08∗1+5e−0.08∗2+5e−0.08∗3+105e−0.08∗4 ≈£89.06.

(b) If the rate is 5%, 5e−0.05∗1+5e−0.05∗2+5e−0.05∗3+105e−0.05∗4 ≈£99.55.

In both cases, we needed to figure out what is the present value of each payment, and

then sum them up.

(2) In this case interest is compounded continuously, so the formula we need is A =Aert.

t

Earning£1ininterestmeansthatthefinalamount(tyearsfromnow)willbe1,000,000+

1= £1,000,001. The interest is given to be 10%, or r =0.1,

1,000,000e0.1∗t =1,000,001.

Dividing and taking the natural logarithm on both sides:

(cid:18) (cid:19)

1,000,001

t=10×ln =0.00001 years, or ≈5.25 minutes.

1,000,000

(3) Let’s call the original amount we deposit, A. We want to find the interest rate r, so that

underannualcompounding(sodiscrete,notcontinuous),wedoubleourmoneyin10years.

Ingeneral,theformulagivingtheamountafternyears,fordiscretecompounding(mtimes

a year), with principal A is

(cid:16) r (cid:17)mn

A =A 1+ ,

t m

so in this case we have

(cid:16) r(cid:17)1∗10

2A=A 1+ ,

1

so

r =21/10−1≈0.0718, or r ≈7.18%

(4) Again, we want to use the formula A

=A(cid:0)

1+

r(cid:1)mn

, with r =0.1,n=2.

t m

• A

=100(cid:0)

1+

0.1(cid:1)1∗2

=121.00; here m=1 because interest is compounded annually

2 1

(i.e., once a year).

• A =

100(cid:0)

1+

0.1(cid:1)2∗2

= 121.55; here m = 2 because interest is compounded semi-

2 2

annually (i.e., twice a year).

(5) Call the initial investment P . Since we are working with annual compounding

0

P ·(1+0.12)2 =£1000⇒P ≈£797.2.

0 0

0.15

(6) At15%compoundeddailythedepositwillgrowto(1+ )1·365·1000≈£1,161.80after

365

one year.

If the interest is compounded semi-annually at 15.5%, after one year the value will be

0.155

(1+ )1·2·1000≈£1,161.01, which is less than the first option.

2

67

68 SOLUTIONS

(7) The loan will be repaid in 10 equal installments of £10,000 each (this is what we called P

in class). We are looking for the amount the bank is willing to pay us now, knowing that

they will get these 10 payments of P each, one at the end of every year (these payments

correspond to getting back part of the initial loan, along with interest). The question

can be restated to ask “how much is the bank willing to pay now, in order to receive £P

in 1,2,3,...,10 years from now.” In other words, we need to find the (10 different) present

values of these future payments. We saw in class that (under annual compounding), the

present value of a payment P in l years is P

(cid:0)

1+

r(cid:1)−l

, l=1,2,...,10.

1

The first payment they will receive will P in 1 year and the bank is willing to pay now

P

(cid:0)

1+

r(cid:1)−1

for it.

1

The second payment they will receive will be P in 2 years and the bank is willing to pay

now P

(cid:0)

1+

r(cid:1)−2

for it.

1

The third payment they will receive will be P in 3 years and the bank is willing to pay

now P

(cid:0)

1+

r(cid:1)−3

for it.

1

.

.

.

The tenth payment they will receive will be P in 10 years and the bank is willing to pay

now P

(cid:0)

1+

r(cid:1)−10

for it.

1

The amount L we are looking to find, is how much the bank is willing to pay now for

these payments, i.e., the sum of the above numbers:

10

L=(cid:88) 10,000(1+0.18)−l =44,941.

l=1

(8) (a)

P(t)=e−Y(t)t.

Y(0.25)=e−ln(99/100)/0.25 ≈4.02%

Y(0.5)=e−ln(97.8/100)/0.5 ≈4.45%

Y(1)=e−ln(95.5/100)/1 ≈4.60%

(b) To find Y(1.5) we solve

4e−Y(0.5)×0.5+4e−Y(1)×1+104e−Y(1.5)×1.5 =104.5

for the unknown Y(1.5). [[because coupons are worth £4.]] We have

4e−0.0402×0.5+4e−0.0445×1+104e−Y(1.5)×1.5 ≈104.5

and Y(1.5)=.0480=4.8%.

To find Y(2) we solve

6e−Y(0.5)×0.5+6e−Y(1)×1+6e−Y(1.5)×1.5+106e−Y(2)×2 =113

for the unknown Y(2). [[because coupons are now worht £6.]] We have

6e−0.0402×0.5+6e−0.0445×1+6e−0.0480×1.5+106e−Y(2)×2 ≈113

and Y(2)=0.0505=5.05%.

CHAPTER 2 69

(c) The forward rate r for the period between 0.5 and 1 years is

0.046×1−0.0445×0.5

r ≈ ≈.0476=4.76%<5

1−0.5

so we

• borrow £1000e−Y(0.5)×0.5 for a year at an interest rate of Y(1),

• deposit this money for 6 months at an interest rate of Y(0.5),

• after6monthswithwithdrawourdepositamountingto£1000e−Y(0.5)×0.5eY(0.5)×0.5 =

£1000,

• deposit this £1000 for six months at an interest rate of 5%,

• afteranadditional6monthswewithdrawthisdeposit,whosebalanceis1000e0.05×0.5

and

• werepaytheloanwetookinstep(1),andwhosebalancenowis£1000e−Y(0.5)×0.5eY(1)×1,

• we pocket the difference

1000e0.05×0.5−1000e−Y(0.5)×0.5eY(1)×1 ≈1.28.

(9) Ifthecurrentpricepislessthanv e−rT, borrowpforT yearsataninterestrateofr, and

T

use this cash to buy the asset. At time T, sell the asset for v , repay the balance of the

T

loan which is perT and pocket the difference

v −perT >v −v e−rTerT =0.

T T T

If the current price p is greater than v e−rT, those who own the asset will sell it and

T

lend the proceeds for T years earning an interest rate of r. At time T they will collect the

balance of their deposit, now worth perT, to buy the asset back and pocket the difference

perT −v >v e−rTerT −v =0.

T T T

So the only value of p which does not introduce arbitrage opportunities is p=v e−rT.

T

Chapter 2

(1) (a) The forward exchange rate is (F =Se(r−rf)T)

0.54e0.04−0.02 ≈0.55091.

(b) • Borrow $1e−0.02 for one year at an interest rate of 2%,

• buy £0.54e−0.02 and

• deposit this amount for one year earning an interest rate of 4%,

• wait one year,

• collect the balance of your deposit which is £0.54×e−0.02e0.04 ≈0.55091,

• have $1 delivered for £0.54,

• repay the the balance of your loan which amounts to $1,

• pocket 0.55091−0.54>0.

(2) Call the values of the portfolios now v0 and v0. If v0 < v0, short-sell B, buy A and

A B A B

pocket the difference v0 −v0 >0. Now wait until portfolio A is worth at least as much as

B A

B, sell A, buy back B (and maybe pocket some more money.) We just found an arbitrage

strategy, and since we assume these not to exist, we conclude that v0 ≥v0.

A B

Index

arbitrage,6 interest

forwardrate,16

betacoefficient,65

spotrate,14

Black-ScholesPDE,48

Itointegral,40

boundaryconditions,48

Itoprocess,41

Black-Scholespricingformula,5

Ito’sLemma,43

Black-Scholespriocingformulas,49

bond,13 marginaccount,19

coupons,13 margincall,19

facevalue,13 marketportfolio,61,63

maturitydate,13

no-arbitragepricing,6

zerocoupon,13

Brownianmotion,39

Option

geometric,41,47

forwardstart,53

option,25

concavity,57

American

derivative,19,31 call,25

discountcurve,14 expirationdate,25

put,25

efficientfrontier,58

strikeprice,25

expectedreturn,42

digital,51

European

feasibleset,58

call,25

forwardcontract,19

expirationdate,25

deliverydate,19

payoff,25

foreignexchange,22

put,25

longposition,19

strikeprice,25

maturitydate,19

underlyingasset,25

shortposition,19

forwardrateagreement,16

portfolio

futurescontract,19,22

efficient,58

closure,22

presentvalue,11

deliverydate,22

futuresprice,22 risk

longposition,22 appetite,59

markingtomarket,23 priceof,64

maturitydate,22 riskneutral

shortposition,22 investors,32,51

probability,32

indifferencecurve,56

valuation,32,51

interest,9

continuouscompounding,10 securitymarketline,65

discretecompounding,10 shortselling,19

71

72 INDEX

closure,20

spotprice,20

stochasticprocess,40

Stockmarketindices,64

volatility,28,42,49,52

historical,52

implied,52

yield,14

curve,14