All questions should be attempted. Marks obtained in all solutions will count.
The use of an electronic calculator is permitted in this examination.
NOTE: In the questions which follow the current price of an asset (or similar instrument)
will often be denoted either by St or simply by S with the time subscript suppressed.
Reference may be made to the following definitions:
where K denotes the exercise price, r the riskless rate, σ the volatility and t is the time
to expiry. The Black-Scholes formula for pricing a European call is
C = SΦ(d1) Ke rtΦ(d2).
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1. (a) Suppose GameStop shares satisfy the following one-period model:
S(0, ωi) S(T, ωi)
ω1 $100 $400
ω2 $100 $25
WallStreetBetsTrader decides to buy GameStop shares at a price of $100 per share.
She is concerned about the share price crashing and so to hedge her shares she
also buys a put option on GameStop shares struck at K with expiration T where
$25 < K < $400. Assuming the interest rate r = 0:
(i) Calculate the risk-neutral probability measure Q = (q1, q2) corresponding to each
of the outcomes in .
(ii) Using the No-Arbitrage Theorem, determine whether there exists an arbitrage
opportunity.
(iii) Let WallStreetBetsTrader’s portfolio be H(t, ωi) for 0 < t < T. Calculate the
gain or loss for this portfolio at time T for each ωi
in terms of K.
(iv) Prove that the ratio of gains to losses for H(T, ωi) is constant for all K. Express
this ratio with respect to the probability measure Q. Give an intuitive explanation
for why this ratio is constant.
(v) The financial regulator decides to ban short-selling in GameStop shares. What
are the consequences for derivative traders in terms of replicating portfolios and
market completeness
(b) Using an arbitrage argument derive a relationship between the prices of an
American call option and an American put option with the same strike K and ex_x005f piration T on a stock with no dividends.
(c) Give an example of why you may choose to exercise an American option early.
MATH0031 CONTINUED
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2. (a) Consider a multi-period binomial model with T time-steps of length t.
Derive the value of a put option struck at K in terms of the risk-neutral proba bilities Q = (qU , qD). You must give a clear definition for each variable required for
this formula.
(b) Let C(K, t) be a European call option with strike K and expiration T. Using
the binomial model with two time-steps, show by example how a trader constructs
a self-financing trading strategy at each time-step to remain risk-neutral.
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3. (a) An Asian option is a path-dependent option where the payoff is calculated using
the average asset value for each path ω. So an Asian call option with strike price
K has a payoff function given by,
A(T, ω) = max (( 1
T + 1
T
X
t=0
St(ω)
) K, 0
)
where St(ω) denotes the asset price at time t for a given path ω. The asse
t S satisfies
the following filtration (Pt)
T
t=0, where r = 0 and the prices are in USD.
ω S(0) S(1y) S(2y)
ω1 12 15 16
ω2 12 15 13
ω3 12 9 13
ω4 12 9 7
(i) Define the conditional probability and the conditional expectation associated
with Pt
.
(ii) Calculate the conditional probabilities for each edge of the tree. Deduce the
risk-neutral measure Q.
(iii) Show that for all paths ωi
:
EQ[S(2y)|P1](ωi) = S1(ωi)
and
EQ[S(2y)|P0](ωi) = S0(ωi).
(iv) Hence or otherwise derive the value of a two year Asian call option with strike
price K = $12 written on S.
(b) (i) Is the value of the Asian call option above greater or less than the value of
a two year European call option with strike price K = $12 Justify your answer.
(ii) A Geometric Asian option calculates the geometric average price of the asset S
to construct the payoff above. Is the Geometric Asian call option struck at K = $12
worth more or less than the Asian call option calculated in (a)(iv) What about if
they were puts instead of calls Explain why.
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4. (a) Let the process (B(t))t> 0 be a standard Brownian motion and suppose c > 0 is a
constant. Let τ be a stopping time such that B(τ ) = c and B(u) < c for all u < τ .
(i) Explain what is meant by saying τ is a stopping time.
(ii) Define
Z(u) = ( B(u), for 0 6 u < τ,
2c B(u), for u > τ.
(1)
Describe the Brownian motion Z(u) in relation to B(t). Show that Z(u) is also a
Brownian motion and calculate its mean and variance.
(b) W = W(t) is standard Brownian motion. Let f(S, t) be a function of two
variables (continuously twice differentiable in S and once in t) obeying the stochastic
differential equation:
dS = μdt + σdW
(i) Find an expression for
I(k) = Z
0
T
(W(t))k
dW(t)
that does not involve It o integrals.
(ii) Calculate the mean and variance of I(1). You may use without proof the fact
that E[W(T)
4
] = 3T
2
.
(iii) Evaluate
d(exp[S(t) t/2])
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5. Let V (S, t) denote the value at time t ≤ T of a European option when the price
of the underlying asset is S. Assume that the asset price process S(t) follows the
stochastic equation
dS = μSdt + σSdW
where W = W(t) is a standard Brownian motion, μ, σ are constants and r is a
constant riskless interest rate applicable throughout the life of the option.
(a) Discuss market environments where the Black-Scholes model is less applicable.
(b) Suppose that S is a foreign exchange spot rate on USDJPY where USD has an
interest rate given by rUSD and JPY has an interest rate given by rJP Y . By hedg ing the delta, derive a form of the Black-Scholes equation satisfied by the function
V (S, t).
(c) A European binary call option with strike price K and expiration T has a
payoff function given by
0 if S(T) < K
1 if S(T) > K
Using Feynman-Kac’s formula, solve the Black-Scholes equation in the case of a
European binary call option.
MATH0031 END OF PAPER
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