MATH 5816
Continuous Time Financial Modelling
Term 3 2020
TUTORIAL QUESIONS
Important skills to demonstrate in this assignment:
• applying Itoˆ formula and various stochastic analysis tools
• learn more about Asian options and American options
• working with Black Scholes type questions and see how robust the model is.
Question 1 An asset price St is a geometric Brownian motion under the market measure P . Define
YT = exp
( 1
T
∫ T
0
lnStdt
)
.
Suppose that an Asian call option has the payoff (YT −K)+ at time T . Find an explicit formula for
the price of such an option at time zero.
Question 2 The digital put option with strike K at time T has payoff
PT =
{
0 ST ≥ K
1 ST < K.
Find the Black Scholes price for a digital put. What is the put call parity for digital options?
Question 3 (Shreve’s Exercise 9.3 on page 400 - Change in volatility caused by change of numer-
arire)
Let S(t) and N(t) be the prices of two assets, denominated in a common currency, and let σ and
ν denote the volatilities, which we assume are constant. We assume also that the interest rate r is
constant. Then
dS(t) = r S(t)dt+ σ S(t) dW˜1(t),
dN(t) = r N(t)dt+ ν N(t) dW˜3(t)
where W˜1(t) and W˜3(t) are Brownian motions under the risk neutral measure P˜ . We assume that
these Brownian motions are correlated, with dW˜1(t) dW˜3(t) = ρdt for some constant ρ.
(a) Show that S(N) = S(t)N(t) has volatility γ =
√
σ2 − 2ρσν + ν2. In other words, show that there
exists a Brownian motion W˜4 under P˜ such that
dS(N)(t)
S(N)(t)
= (Something) dt+ γdW˜4(t).
(b) Show how to construct a Brownian motion W˜2(t) under P˜ that is independent of W˜1(t) such
that dN(t) may be written as
dN(t) = r N(t)dt+ ν N(t) [ρ dW˜1(t) +
√
1− ρ2 dW˜2(t)].
(c) Using Shreve’s Theorem 9.2.2 (see below), determine the volatility vector of S(N)(t). In other
words, find a vector (v1, v2) such that
dS(N)(t) = S(N)(t)[v1dW˜
(N)
1 (t) + v2 dW˜
(N)
2 (t)],
where W˜1(t) and W˜2(t) are independent Brownian motions under P˜ (N). Show that√
v21 + v
2
2 =
√
σ2 − 2ρσν + ν2.
Shreve’s Theorem 9.2.2: Change of risk-neutral measure
Let S(t) and N(t) be the prices of two assets, denominated in a common currency, and let σ(t) =
(σ1(t), . . . , σd(t)) and ν(t) = (ν1(t), . . . , νd(t)) denote their respective volatility vector processes:
d(D(t)S(t)) = D(t)S(t)σ(t) · dW˜ (t), d(D(t)N(t)) = D(t)N(t)ν(t) · dW˜ (t).
Here D(t) is the discount process: D(t) = e−
∫ t
0 R(u)du, where R(t) is an adapted interest rate
process, 0 ≤ t ≤ T .
Take N(t) as the numeraire, so the price S(t) becomes S(N)(t) = S(t)N(t) . Under the measure P˜
(N),
the process S(N)(t) is a martingale. Moreover
dS(N)(t) = S(N)(t)[σ(t)− ν(t)] · dW˜ (N)(t).
Question 4 This is a example of a two factor model. A financial forward contract is called a quanto
product if it is denominated in a currency other than that in which it is traded.
AMP, an Australian company has an AUD denominated stock price that we denote by {St}t≥0. For
an USD investor, a quanto forward contract on AMP stock with maturity T has payoff (ST −K)
converted into USD according to some prearranged exchange rate. This is, the payoff will be
E(ST −K) for some preagreed E, where ST is the AUD asset price at time T .
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Assume that there is a riskless cash bond in each of the USD and AUD markets, but you have two
random processes to model, the stock price St and the exchange rate, that is the value of one AUD
in USD which you will denote by {Et}.
Then the Black-Scholes quanto model is
USD bondBt = ert
AUD bondDt = eut
AUD asset priceSt = S0 exp(νt+ σ1W 1t ),
Exchange rateEt = E0 exp(λt+ ρσ2W 1t +
√
1− ρ2σ2W 2t )
where W 1t and W 2t are independent P -Brownian motions and r, u, ν, λ, σ1, σ2 and ρ are constants.
In this model, the volatilities of St and Et are σ1 and σ2 respectively and {W 1t , ρW 1t +
√
1− ρ2W 2t }
is a pair of correlated Brownian motions with correlation coefficient ρ. There is no extra generality
in replacing the expressions for St and Et by
St = S0 exp(νt+ σ11W˜
1
t + σ12W˜
2
t )
Et = E0 exp(λt+ σ21W˜
1
t + σ22W˜
2
t )
for independent Brownian motions W˜ 1t and W˜ 2t .
(a) What is the value of K that makes the value at time zero of the quanto forward contract
zero?
(b) A quanto call option write on AMP stock is worth E(ST −K)+ USD at time T , where ST is
the AUD stock price. Assuming this B-S quanto model, find the time zero price of the option
and the replicating portfolio.
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