Module Code: MATH373401
Module Title: Stochastic Calculus for Finance c UNIVERSITY OF LEEDS
School of Mathematics Semester Two 201920
Exam information:
There are 4 pages to this exam.
There will be 2.5 hours to complete this exam (+0.5 hours to upload your solutions
online).
Answer all questions.
The numbers in brackets indicate the marks available for each subquestion.
The total number of marks is 100.
You must show all your calculations.
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Module Code: MATH373401
1. This question carries 35 marks.
(a) Let W = (W (t))t≥0 be a Brownian motion. Calculate the mean of the following
random variables, citing any properties you use.
(i) [2 marks] (W (5) W (2))W (1);
(ii) [2 marks] W 2(4);
(iii) [3 marks] W (3)W (5).
(b) State whether the following statements on conditional expectation are true or false
(here F ,G are sigma-algebras, X, Y are random variables and a, b are real numbers)
(i) [2 marks] If X is F -measurable then E[X|F ] = X;
(ii) [2 marks] If X is independent of Y then E[XY |F ] = XE[Y |F ];
(iii) [2 marks] If F G then E[E[X|F ]|G] = E[X|F ];
(iv) [2 marks] E[a+ bX|F ] = a+ bE[X|F ];
(v) [2 marks] If F is the trivial sigma-algebra then E[X|F ] = E[X].
(c) [8 marks] Let (Ω,F ,P) be a probability space and W = (W 1,W 2) = (W 1t ,W 2t )t≥0
be a 2-dimensional Brownian motion. Let (Ft)t≥0 be the filtration generated by
W . Let M = (Mt)t∈[0,T ] be defined by Mt = exp{ 5t+W 1t + 3W 2t }.
Prove that M is a martingale with respect to (Ft)t≥0.
(d) [10 marks] Let W = (Wt)t≥0 be a 1-dimensional Brownian motion and F = (Ft)t≥0
be its natural filtration. Let X = (Xt)t≥0 be defined by Xt = ln(1 + (Wt)2) for all
t ≥ 0.
Find the function f(x) such that the process Mt = Xt Yt for t ∈ [0, T ] is an
F -martingale, with Yt =
∫ t
0
f(Ws)ds.
Hint: you may use the fact that E(supt∈[0,T ](Wt)2) = c <∞.
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Module Code: MATH373401
2. This question carries 28 marks.
(a) [4 marks] Let (Xt)t≥0 be an Ito process of the form dXt = μ(t)dt + σ(t)dWt for
some μ ∈ L1(0, T ) and σ ∈ L2(0, T ).
Apply Ito ’s formula to Yt = g(t,Xt) for g(t, x) = e
x + t sin(x) to write Yt as an
Ito process.
(b) [5 marks] Let W be a one dimensional Brownian motion and F = (Ft)t≥0 its
natural filtration. Let the process G = (G(s))s≥0 be defined as G(s) = W 2s for
s ∈ [0, T ].
Show that G is an element of L2(0, T ).
Hint: use the fact that for X ～ N(0, σ2) then E[X4] = 3σ4.
(c) [10 marks] Let ξ1 and ξ2 be bounded random variables on (Ω,F ,P). Let W be a
one dimensional Brownian motion and F = (Ft)t≥0 its natural filtration. Consider
the process G = (G(t))t∈[0,T ] defined by
G(t) = ξ11
[0,
T
2
)
(t) ξ21
[
T
2
,T )
(t).
Assume that ξ1 ∈ F0 and ξ2 ∈ FT/2.
Show that G ∈ L2(0, T ) and then calculate the stochastic integral ∫ T
0
G(t)dW (t).
Show all your calculations.
(d) [9 marks] Let (Bt)t≥0 be a one dimensional Brownian motion and let
Xt =
∫ t
0
B2sds tB2t
for all t ≥ 0.
Calculate E[X2t ] for any fixed t ≥ 0. You must show all your calculations and cite
any results you use.
Hint: Use Ito ’s formula for tB2t to rewrite Xt in a different form.
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Module Code: MATH373401
3. This question carries 37 marks.
(a) Consider the SDE satisfied by a generalised Geometric Brownian motion{
dS(t) = r(t)S(t)dt+ σ(t)S(t)dWt
S(0) = 1
(1)
for all t ≥ 0, where r(t), σ(t) are given continuous deterministic functions.
(i) [3 marks] Using Ito ’s formula derive the solution of SDE (1).
(ii) [6 marks] Calculate the mean and variance of ln(S(t)), for a given fixed t.
(iii) [4 marks] Derive the distribution of ln(S(t)), for a given fixed t.
Hint: The stochastic integral of a deterministic function is Gaussian
(b) Consider a financial market where the risk-free rate is a deterministic function of
time, r(t), and there is a risky asset S that follows the generalised Geometric
Brownian motion dynamics (1). Consider a European call option on S with strike
K and maturity T .
(i) [4 marks] Write the price V (0) at t = 0 of the the call option using the risk
neutral pricing formula.
(ii) [6 marks] Using the distribution for S(T ) found in part (a) item (iii), derive
an explicit expression for the price V (0).
Note: Your formula should only depend on r(t), σ(t), T,K and the distribution of
S(T ), no expectation E should be involved here. You should however leave your
formula in terms of integrals.
(c) [14 marks] Consider a market with a stochastic interest rate r and a stock S with
dynamics {
dr(t) = (a br(t))dt+ c√r(t)dBt,
dS(t) = r(t)S(t)dt+ σS(t)dWt,
for some constants a, b, c, σ. Assume that the Brownian motions W and B are
independent.
Let v(t, r(t), S(t)) denote the price at time t of a financial derivative with payoff
h(S(T )) at time T .
Derive (heuristically) the PDE satisfied by v(t, r, s) by using the fact that the
process (Mt)t≥0 is a martingale, where Mt is given by
Mt = e
∫ t0 r(u)duv(t, r(t), S(t)).
Show all steps in your derivation.
Hint: You need to apply two-dimensional Ito ’s formula. You can assume without
proof that v ∈ C1,2([0, T ]×R2) and that all processes appearing in your derivation
are in L2(0, T ).
Page 4 of 4 End.
