MATH39032 Mathematical Modelling of Finance
TAKE-HOME COURSEWORK
This assignment will account for 2 credits (20% of the 10 credits) as part of the re- placement 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 pa- pers) 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 ?1 and ?2 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+ ?iSidWi
where the dWi are Wiener processes, such that
E[dW 2i ] = dt, E[dW1dW2] = 0.
[2 marks]
(ii) By considering a portfolio (using the dSi in (i) above)
? = 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 deter- mined from
?V
?t + 1
2 ?21S
2 1
?2V
?S21 + 1
2 ?22S
2 2
?2V
?S22 + (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(?, t),
where ? = S1 S2
, show that H satisfies the PDE
?H
?t + 1
2 ??2?2
?2H
??2 + (D2 ?D1)?
?H
?? ?D2H = 0,
where you are to determine ??.
[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
( d?1 ) ? S2e?D2(T?t)N
( d?2 ) ,
where you are to determine d?1 and d?2.
[4 marks]
END OF ASSESSMENT
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