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微积分代写-MATH373401

时间：2021-04-17

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.

Page 1 of 4 Turn the page over

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 <∞.

Page 2 of 4 Turn the page over

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).

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 Itoˆ’s formula for tB2t to rewrite Xt in a different form.

Page 3 of 4 Turn the page over

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 ).

Page 4 of 4 End.

学霸联盟

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.

Page 1 of 4 Turn the page over

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 <∞.

Page 2 of 4 Turn the page over

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).

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 Itoˆ’s formula for tB2t to rewrite Xt in a different form.

Page 3 of 4 Turn the page over

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 ).

Page 4 of 4 End.

学霸联盟