MATH3075/3975
Financial Derivatives
School of Mathematics and Statistics
University of Sydney
Semester 2, 2020
Tutorial sheet 11
Background: Chapter 5 – The Black-Scholes Model.
Exercise 1 Consider the Black-Scholes model M = (B, S) with the initial
stock price S0 = 9, the continuously compounded interest rate r = 0.01 per
annum and the stock price volatility equals σ = 0.1 per annum.
(a) Using the Black-Scholes call option pricing formula
C0 = S0N
(
d+(S0, T )
)
−Ke−rTN
(
d−(S0, T )
)
compute the price C0 of the European call option with strike price
K = 10 and maturity T = 5 years.
(b) Using the Black-Scholes put option pricing formula
P0 = Ke
−rTN
(
− d−(S0, T )
)
− S0N
(
− d+(S0, T )
)
compute the price P0 for the European put option with strike price
K = 10 and maturity T = 5 years.
(c) Does the put-call parity relationship
C0 − P0 = S0 −Ke−rT
hold?
(d) Recompute the prices of call and put options for modified maturities
T = 5 months and T = 5 days.
(e) Explain the observed pattern of call and put prices when the time to
maturity goes to zero.
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Exercise 2 Assume that the stock price S is governed under the martingale
measure P̃ by the Black-Scholes stochastic differential equation
dSt = St
(
r dt+ σ dWt
)
where σ > 0 is a constant volatility and r is a constant short-term interest
rate. Let 0 < L < K be real numbers. Consider the contingent claim with
the payoff X at maturity date T > 0 given as X = min
(
|ST −K|, L
)
.
(a) Sketch the profile of the payoff X as the function of the stock price ST
at maturity date T and find the decomposition of the payoff X in terms
of the payoffs of standard call and put options with different strikes.
(b) Compute the arbitrage price πt(X) at any date t ∈ [0, T ]. Take for
granted the Black-Scholes pricing formulae for European call and put
options.
(c) Find the limits of the arbitrage price limL→0 π0(X) and limL→∞ π0(X).
(d) Find the limit of the arbitrage price limσ→∞ π0(X).
Exercise 3 We consider the call option pricing functions, that is, the func-
tions c : R+ × [0, T ]→ R and v : R+ × [0, T ]→ R such that Ct = v(St, t) =
c(St, T − t) for all t ∈ [0, T ] where Ct is the Black-Scholes price of the call
option.
(a) Show that v satisfies the terminal condition v(s, T ) = (s − K)+ in
the sense that lim t→T v(s, t) = (s −K)+. Equivalently, the function c
satisfies the initial condition lim t→0 c(s, t) = (s−K)+.
(b) (MATH3975) Show by direct computations that the pricing function
v satisfies the Black-Scholes PDE. To this end, compute the partial
derivatives vs, vss and vt (for answers, see Section 5.5 in the course
notes). Write down the PDE satisfied by the function c and the initial
condition.
Exercise 4 (MATH3975) Consider the stock price process S under the Black
and Scholes assumption, that is,
St = S0 exp
((
r −
1
2
σ2
)
t+ σWt
)
where W is the Wiener process under the martingale measure P̃.
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(a) Show that Ŝt := e
−rtSt is a martingale under P̃ with respect to the
filtration F = (Ft)t≥0 generated by the stock price process S. Hint: Use
the property that St
Ss
is independent of Fs for 0 ≤ s < t.
(b) Compute the expectation EP̃(St) and the variance Var P̃(St) of the stock
price under the martingale measure P̃ using the martingale property of
Ŝ under P̃.
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