COMP0041 (Applied Computational Finance), Due on April 29, 3pm(UTC+8)
1. This question is on the application of the Binomial option pricing model.

PKZ stock is currently trading at 100. Over three-months it will either go up by 6% or down by 5%. Interest rates are zero.

a. [25 marks] Using a two period binomial model to construct a delta- hedged portfolio, price a six month European call option on PKZ stock with a strike price of £105.

b. [3 Marks] Using your answer from the first part, together with the put-call parity, price a put option on the same stock with same strike and expiry.

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2. This question is on the Binomial method in the limit ?t ? 0.

[40 Marks] The binomial model for pricing options leads to the for- mula

V (S,t) = e?r?t [qV (US,t + ?t) + (1 ? q) V (DS,t + ?t)]

where

U = e? ? ?t, D = e??

? ?t, q =

er?t ?D U ?D

.

V (S,t) is the option value, t is the time, S is the spot price, ? is volatil- ity and r is the risk-free rate. By carefully expanding U,D,q as Taylor series in ?t or

? ?t (as appro-

priate) and then expanding V (US,t + ?t) and V (DS,t + ?t) as Taylor series in both their arguments, deduce that to O (?t) ,

?V

?t +

1

2 ?2S2

?2V

?S2 + rS

?V

?S ? rV = 0.

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3. This question is on probability and Monte Carlo

a. Consider theprobabilitydensity function p (x) fora randomvariable X given by

p (x) =

{ µ exp (?µx) x ? 0 0 x < 0 where µ (> 0) is a constant.

i. [15 Marks] Show that for this probability density function

E [ e?X ]