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).
• 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 Itoˆ process of the form dXt = µ(t)dt + σ(t)dWt for
some µ ∈ L1(0, T ) and σ ∈ L2(0, T ).
Apply Itoˆ’s formula to Yt = g(t,Xt) for g(t, x) = e
x + t sin(x) to write Yt as an
Itoˆ 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).
(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 Itoˆ’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 Itoˆ’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 Itoˆ’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 ).
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