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 δt → 0.

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

V (S,t) = e−rδt [qV (US,t + δt) + (1 − q) V (DS,t + δt)]

where

U = eσ √ δt, D = e−σ

√ δt, q =

erδt −D U −D

.

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

√ δt (as appro-

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

∂V

∂t +

1

2 σ2S2

∂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 [ eθX ]

=

( 1 −

θ

µ

)−1 Hint: You may assume µ > θ in obtaining this result.

ii. [20 Marks] By expanding (

1 − θ

µ

)−1 as a Taylor series, show

that

E [xn] = n!

µn , n = 0, 1, 2, ....

iii. [15 Marks] Hence calculate the skew and kurtosis for X.

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b. [32 Marks] An Exchange Option gives the holder the right to exchange one asset for another. The discounted payoff for this contract V is

V = e−rT max (S1 (T) −S2 (T) , 0) .

The option price is then given by θ = E [V ] where

Si (t) = Si (0) e (r−12σ

2 i )t+σiφi

√ t

for i = 1, 2, and φi ∼ N (0, 1) with correlation coeffi cient ρ. Youmayassumethatauniformrandomnumbergenerator isavail- able. Use a Cholesky factorisation method to show(

φ1 φ2

) =

( 1 0

ρ √

1 −ρ2

)( x1 x2

) ,

where ( x1 x2

) is a vector of independent N (0, 1) variables and

has the same distribution as ( φ1 φ2

) .

Give a Monte Carlo simulation algorithm that makes use of anti- thetic variates for the estimation of θ.

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4. This question is on finite differences

a. [30 Marks] Consider a forward difference operator, ∆, such that

∆V (S) = V (S + h) −V (S) , (4.1)

where h is an infinitessimal. By introducing the operators

D ≡ ∂

∂S ; D2 ≡

∂2

∂S2

show that ∆ ≡ ehD −1 (4.2)

where 1 is the identity operator. Hint: start by doing a Taylor expansion on V (S + h) . By rearranging (4.2) show that

D = 1

h

( ∆ −

∆2

2 +

∆3

3 −

∆4

4 + O

( ∆5 ))

.

Hence obtain the second order approximation for ∂V

∂S .

b. [20 Marks] The Speed of an option V (S,t) is the sensitivity of its gamma Γ, to changes in the underlying stock price S and written

Speed = ∂Γ

∂S = ∂3V

∂S3

Given that

Γ ∼ V mn−1 − 2V mn + V mn+1

δS2

where theoption V (S,t) = V (nδS,mδt) canbeexpressed infinite difference form as V mn , derive the speed in the form

1

δS3 [ a1V

m n+2 + a2V

m n+1 + a3V

m n + a4V

m n−1 ] ,

where theconstants ai (i = 1, 2, 3, 4) shouldbegiven. Hint: Con- sider 3 Taylor series expansions

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