MATH_3075_3975_s12

7212019 MATH_3075_3975_s12

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MATH30753975

Financial Mathematics

Week 12 Solutions

Exercise 1 We consider the Black-Scholes model M = (B S ) with theinitial stock price S 0 = 9 the continuously compounded interest rate r = 001per annum and the stock price volatility σ = 01 per annum

1 Using the Black-Scholes call option pricing formula

C 0 = S 0N 1048616

d+(S 0 T )1048617minusKeminusrT N

1048616dminus(S 0 T )

1048617

we compute the price C 0 of the European call option with strike priceK = 10 and maturity T = 5 years We find that

d+(S 0 T ) = minus013578 dminus(S 0 T ) = minus035938

and thus C 0 = 059285

2 Using the Black-Scholes put option pricing formula

P 0 = K eminusrT N 1048616minus dminus(S 0 T )

1048617minus S 0N 1048616minus d+(S 0 T )

1048617we find that the price P 0 = 110514

3 The put-call parity relationship holds since

C 0minusP 0 = 059285minus 110514 = minus051229 = 9minus 10eminus005 = S 0minusKeminusrT

4 We now recompute the prices of call and put options for modified ma-turities T = 5 months and T = 5 days

bull We note that 5 months is equivalent to T = 0416667 and thus


Hence C 0 = 0015315 and P 0 = 0973735

bull We note that 5 days is equivalent to T = 0013699 and thus


Hence C 0 = 149E minus 21 and P 0 = 099863

bull The call option (respectively put option) price decreases to zero(respectively increases to K minus S 0 = 1) when the time to matu-rity tends to zero This is related to the fact that S 0 lt K andthus for short maturities it is unlikely (respectively very likely)that the call option (respectively put option) will be exercised atexpiration

1

7212019 MATH_3075_3975_s12


Exercise 2 Assume that the stock price S is governed under the martingale

measure P by the Black-Scholes stochastic differential equation

dS t = S t1048616

r dt + σ dW t1048617

where σ gt 0 is a constant volatility and r is a constant short-term interestrate Let 0 lt L lt K be real numbers Consider the contingent claim withthe payoff X at maturity date T gt 0 given as

X = min1048616|S T minusK | L

1048617

1 It is easy to sketch the profile of the payoff X as the function of thestock price S T The decomposition of this payoff in terms of payoffs of

standard call and put options reads

X = L minus C T (K minus L) + 2C T (K )minus C T (K + L)

Note that other decompositions are possible

2 The arbitrage price πt(X ) satisfies for every t isin [0 T ]

πt(X ) = Leminusr(T minust) minus C t(K minus L) + 2C t(K )minus C t(K + L)

3 We will now find the limits of the arbitrage price limLrarr0 π0(X ) andlimL

rarrinfinπ0(X ) We observe the payoff X increases when L increases

Hence the price π0(X ) is also a increasing function of L Moreover

limLrarr0

π0(X ) = minusC 0(K ) + 2C 0(K )minus C 0(K ) = 0

By analysing the payoff X when L tends to infinity we obtain

limLrarrinfin

π0(X ) = P 0(K ) + C 0(K )

4 To find the limit lim σrarrinfin π0(X ) we observe that

limσ

rarrinfin

d+(S 0 T ) = infin limσ

rarrinfin

dminus(S 0 T ) = minusinfin

so that

limσrarrinfin

N 1048616

d+(S 0 T )1048617

= 1 limσrarrinfin

N 1048616

dminus(S 0 T )1048617

= 0

Hence the price of the call option satisfies for all strikes K isin R+

limσrarrinfin

C 0(K ) = S 0

This in turn implies that lim σrarrinfin π0(X ) = Leminusr(T minust)

2

7212019 MATH_3075_3975_s12


Exercise 3 (MATH3975) We consider the stock price process S given by

the Black and Scholes model

1 We will first show that S t = eminusrtS t is a martingale with respect tothe filtration F = (F t)tge0 generated by the stock price process S Weobserve that this filtration is also generated by W Using the propertiesof the conditional expectation we obtain for all s le t

EP

1048616S t

F s1048617 = EP

983080S s eσ(W tminusW sminus 1

2σ2(tminuss)) F s

983081= S s eminus

1

2σ2(tminuss)

EP

1048616eσ(W tminusW s) | F s

1048617= S s eminus

1

2σ2(tminuss)

EP


1048617= S s eminus

1

2σ2(tminuss) EP1048616eσ(W tminusW s)1048617

where in the last equality we used the independence of increments of the Wiener process Recall also that W t minus W s =

radic tminus s Z where Z sim

N (0 1) and thus

EP

1048616S t | F s

1048617 = S s eminus

1

2σ2(tminuss)

EP

1048616eσradic tminussZ 1048617

Let us finally recall that if Z sim N (0 1) then for any real number a

EP

1048616eaZ

1048617 = e

1

2a2

By setting a = σradic

t

minuss we obtain

EP

1048616S t | S u u le s


1

2σ2(tminuss) e

1

2σ2(tminuss) = S s

which shows that S is a martingale under P

2 To compute the expectation EP(S t) we observe that

EP(S t) = ert EP

(S t) = ert EP(S 0) = ertS 0 = ertS 0

To compute the variance VarP(S t) we recall that

Var P(S t) = EP(S 2

t)minus 1048667

EP(S t)

10486692

where in turnEP

(S 2t

) = S 20e2rt EP

983131e2σW tminusσ2t983133

= S 20e2rteσ2tEP

983131e2σW tminus 1

2(2σ)2t

983133

= S 20e2rteσ2tEP

983131eaZ minus

1

2a2

983133= S 20e2rteσ

2t

where we denote a = σradic

t Hence Var P(S t) = S 20e2rt1048616

eσ2t minus 1

1048617

3

7212019 MATH_3075_3975_s12


Exercise 4 (MATH3975) We denote by v the Black-Scholes call option pri-

cing that is the function v R+ times [0 T ] rarr R such that C t = v(S t t) for allt isin [0 T ]

1 We wish to check that the pricing function of the European call optionsatisfies the Black-Scholes PDE

partv

partt +

1

2 σ2s2

part 2v

parts2 + rs

partv

parts minus rv = 0 forall (s t) isin (0infin)times (0 T )

with the terminal condition v(s T ) = (sminusK )+

To this end we use the partial derivatives from Section 55 in the course

notes (note that v(s t) = c(s T minus t))

vs(s t) = N (d+(s T minus t))

vss(s t) = n(d+(s T minus t))

sσradic

T minus t

vt(s t) = minus sσ

2radic

T minus tn(d+(s T minus t))minusKreminusr(T minust)N (dminus(s T minus t))

where n is the density function of the standard normal distributionHence

minus sσ

2radic

T minus tn(d+(s T minus t)) minusKreminusr(T minust)N (dminus(s T minus t))

+ 1

2 σ2s2

n(d+(s T minus t))

sσradic

T minus t+ rsN (d+(s T minus t))minus rv(s t) = 0

where we have also used the equality

v(s t) = sN (d+(s T minus t))minusKeminusr(T minust)N (dminus(s T minus t))

2 We need to show that for every s isin R+

limtrarrT

v(s t) = (sminusK )+

For this purpose we observe that d+(s K ) and dminus(s K ) tend to infin(respectively minusinfin) when t rarr T and s gt K (respectively s lt K )Consequently N (d+(s K )) and N (dminus(s K )) tend to 1 (respectively 0)when t rarr T and s gt K (respectively s lt K ) This in turn impliesthat v(s T ) tends to either sminusK or 0 depending on whether s gt K ors lt K The case when s = K is also easy to analyse

4

7212019 MATH_3075_3975_s12


Exercise 2 Assume that the stock price S is governed under the martingale

measure P by the Black-Scholes stochastic differential equation

dS t = S t1048616

r dt + σ dW t1048617

where σ gt 0 is a constant volatility and r is a constant short-term interestrate Let 0 lt L lt K be real numbers Consider the contingent claim withthe payoff X at maturity date T gt 0 given as

X = min1048616|S T minusK | L

1048617

1 It is easy to sketch the profile of the payoff X as the function of thestock price S T The decomposition of this payoff in terms of payoffs of

standard call and put options reads

X = L minus C T (K minus L) + 2C T (K )minus C T (K + L)

Note that other decompositions are possible

2 The arbitrage price πt(X ) satisfies for every t isin [0 T ]

πt(X ) = Leminusr(T minust) minus C t(K minus L) + 2C t(K )minus C t(K + L)

3 We will now find the limits of the arbitrage price limLrarr0 π0(X ) andlimL

rarrinfinπ0(X ) We observe the payoff X increases when L increases

Hence the price π0(X ) is also a increasing function of L Moreover

limLrarr0

π0(X ) = minusC 0(K ) + 2C 0(K )minus C 0(K ) = 0

By analysing the payoff X when L tends to infinity we obtain

limLrarrinfin

π0(X ) = P 0(K ) + C 0(K )

4 To find the limit lim σrarrinfin π0(X ) we observe that

limσ

rarrinfin

d+(S 0 T ) = infin limσ

rarrinfin

dminus(S 0 T ) = minusinfin

so that

limσrarrinfin

N 1048616

d+(S 0 T )1048617

= 1 limσrarrinfin

N 1048616

dminus(S 0 T )1048617

= 0

Hence the price of the call option satisfies for all strikes K isin R+

limσrarrinfin

C 0(K ) = S 0

This in turn implies that lim σrarrinfin π0(X ) = Leminusr(T minust)

2

7212019 MATH_3075_3975_s12





EP

1048616S t

F s1048617 = EP


2σ2(tminuss)) F s

983081= S s eminus

1

2σ2(tminuss)

EP


1048617= S s eminus

1

2σ2(tminuss)

EP


1048617= S s eminus

1




N (0 1) and thus

EP

1048616S t | F s


1

2σ2(tminuss)

EP



EP

1048616eaZ

1048617 = e

1

2a2


t

minuss we obtain

EP

1048616S t | S u u le s


1

2σ2(tminuss) e

1

2σ2(tminuss) = S s



EP(S t) = ert EP



Var P(S t) = EP(S 2

t)minus 1048667

EP(S t)

10486692

where in turnEP

(S 2t

) = S 20e2rt EP


= S 20e2rteσ2tEP


2(2σ)2t

983133

= S 20e2rteσ2tEP

983131eaZ minus

1

2a2

983133= S 20e2rteσ

2t



eσ2t minus 1

1048617

3

7212019 MATH_3075_3975_s12





partv

partt +

1

2 σ2s2

part 2v

parts2 + rs

partv







sσradic

T minus t

vt(s t) = minus sσ

2radic



minus sσ

2radic


+ 1

2 σ2s2

n(d+(s T minus t))

sσradic





limtrarrT



4

7212019 MATH_3075_3975_s12





EP

1048616S t

F s1048617 = EP


2σ2(tminuss)) F s

983081= S s eminus

1

2σ2(tminuss)

EP


1048617= S s eminus

1

2σ2(tminuss)

EP


1048617= S s eminus

1




N (0 1) and thus

EP

1048616S t | F s


1

2σ2(tminuss)

EP



EP

1048616eaZ

1048617 = e

1

2a2


t

minuss we obtain

EP

1048616S t | S u u le s


1

2σ2(tminuss) e

1

2σ2(tminuss) = S s



EP(S t) = ert EP



Var P(S t) = EP(S 2

t)minus 1048667

EP(S t)

10486692

where in turnEP

(S 2t

) = S 20e2rt EP


= S 20e2rteσ2tEP


2(2σ)2t

983133

= S 20e2rteσ2tEP

983131eaZ minus

1

2a2

983133= S 20e2rteσ

2t



eσ2t minus 1

1048617

3

7212019 MATH_3075_3975_s12





partv

partt +

1

2 σ2s2

part 2v

parts2 + rs

partv







sσradic

T minus t

vt(s t) = minus sσ

2radic



minus sσ

2radic


+ 1

2 σ2s2

n(d+(s T minus t))

sσradic





limtrarrT



4

7212019 MATH_3075_3975_s12





partv

partt +

1

2 σ2s2

part 2v

parts2 + rs

partv







sσradic

T minus t

vt(s t) = minus sσ

2radic



minus sσ

2radic


+ 1

2 σ2s2

n(d+(s T minus t))

sσradic





limtrarrT



4

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MATH_3075_3975_s12