Mathematics homework help. This assignment will account for 2 credits (20% of the 10 credits) as part of the replacement assessment for this module. You should answer all questions (total 20 marks).

You may consult textbooks and journal articles, but these must be fully referenced in

your submission. Any material from the course (notes, examples, worksheets, past papers) can be used without reference.

The deadline for this assignment is 4pm (BST) Thursday 7th May 2020 and

must be submitted on Blackboard without your name, but with your university

ID number. A link for submission will be set up under the Assessment and Feedback

tab. You will need to submit the coursework as a single file in pdf format. Your

solution should not take more than 6 sides of A4 paper, and can be hand written or

typeset with latex. Please explain clearly your working, and any assumptions that are

made.

Consider an option V (S1, S2, t) where the two underlyings S1 and S2 have volatilities

σ1 and σ2 respectively, and pay a continuous dividend at a rate D1, D2 respectively; the

risk-free interest rate is r.

(i) Write one or two sentences to justify the use of the following stochastic processes

to model underlyings:

dSi = (µi − Di)Sidt + σiSidWi

where the dWi are Wiener processes, such that

E[dW2

i

] = dt, E[dW1dW2] = 0.

[2 marks]

(ii) By considering a portfolio (using the dSi

in (i) above)

Π = V (S1, S2, t) − ∆1S1 − ∆2S2,

find the choices of ∆1 and ∆2 for which the portfolio is perfectly hedged.

[2 marks]

(iii) Equating the hedged portfolio in (ii) above, show that the option value is determined from

∂V

∂t +

1

2

σ

2

1S

2

1

∂

2V

∂S2

1

+

1

2

σ

2

2S

2

2

∂

2V

∂S2

2

+ (r − D1)S1

∂V

∂S1

+ (r − D2)S2

∂V

∂S2

− rV = 0.

[2 marks]

1

Consider now the case of an exchange option, where you can swap one asset for

another, whose payoff at expiry (t = T) is

V (S1, S2, t = T) = max(S1 − S2, 0).

The boundary conditions are:

V → 0 as S1 → 0

V → S1e

−D1(T −t)

as S2 → 0

V → S1e

−D1(T −t)

as S1 → ∞

(iv) Starting from the PDE derived in (iii) above, and assuming a solution of the form

V (S1, S2, t) = S2H(ξ, t),

where ξ =

S1

S2

, show that H satisfies the PDE

∂H

∂t +

1

2

σˆ

2

ξ

2

∂

2H

∂ξ2

+ (D2 − D1)ξ

∂H

∂ξ − D2H = 0,

where you are to determine ˆσ.

[6 marks]

(v) Write down all the boundary conditions for H.

[4 marks]

(vi) Show that the solution (in the usual notation) for V is

V (S1, S2, t) = S1e

−D1(T −t)N

ˆd1

− S2e

−D2(T −t)N

ˆd2

,

where you are to determine ˆd1 and ˆd2.

[4 marks]

END OF ASSESSM