Semester 2, 2021
CONFIDENTIAL EXAM PAPER
SCHOOL OF MATHEMATICS AND STATISTICS
MATH3975: FINANCIAL DERIVATIVES (Advanced)
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MATH3975 Semester 2, 2021 Page 2 of 5
1. [20 marks] Single-period market model
Consider a single-period market modelM = (B, S) on the space Ω = {ω1, ω2, ω3}.
We assume that the savings account B equals B0 = 1, B1 = 1 + r = 2 and the
stock price S is given by S0 = 11 and S1 = (S1(ω1), S1(ω2), S1(ω3)) = (24, 20, 16).
The real-world probability P is such that P(ωi) = pi > 0 for i = 1, 2, 3.
(a) Find the class M of all martingale measures for the modelM and check if
the market modelM is arbitrage-free and complete.
(b) Show that the contingent claim X = (8, 6, 4) is attainable and compute its
arbitrage price pi0(X) using two methods:
– the replicating strategy for X,
– the risk-neutral valuation formula.
(c) Consider the contingent claim Y = (4, 2, 3).
– Find the range of arbitrage prices for Y inM. Is the claim Y attainable
inM
– Find the minimal initial endowment x for which there exists a portfolio
(x, ) with V0(x, ) = x and such that the inequality V1(x, )(ωi) ≥ Y (ωi) is
satisfied for i = 1, 2, 3.
(d) Consider the extended market M = (B, S1, S2) where S1 = S and S2 is an
additional risky asset given by: S21 = Y = (4, 2, 3) and S20 = 1.35.
– Find a unique martingale measure Q for the extended market M =
(B, S1, S2).
– Compute the price of the claim Z = ( 2, 5, 3) in the extended market
M = (B, S1, S2). Is the claim Z attainable in M
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MATH3975 Semester 2, 2021 Page 3 of 5
2. [20 marks] CRR model: European contingent claim.
Consider the CRR model of stock price S with T ≥ 1 periods and parameters d, u
satisfying d < 1 + r < u where r is the one-period interest rate. We denote by P
a unique martingale measure for the discounted stock price = B 1S.
(a) Consider the binary call and put options with expiry date T and payoffs
C T (K) := 1[K,∞)(ST ) =
1 if ST ≥ K,0 if ST < K,
and
P T (K) := 1[0,K)(ST ) =
0 if ST ≥ K,1 if ST < K,
respectively. By examining the sum of the payoffs C T (K) and P T (K), find
the put-call parity relationship for binary options for any strike K > 0.
(b) Consider the binary call option with the payoff C T (K) at expiry date T and
strike K = S0(1 + r)T . Compute the arbitrage price C 0(K) for this option at
time t = 0 using the risk-neutral valuation formula.
(c) Find a unique probability measure P on (Ω,FT ) such that the process S 1B
is a martingale under P with respect to the filtration FS and show that the
following equality holds for any European contingent claim X and any date
t = 0, 1, . . . , T
pit(X) = St EP
(
XS 1T | Ft
)
.
(d) Let Y and Z be two European contingent claims with maturity T . Assume
that the equality piU(Y ) = piU(Z) holds for some date U such that 0 < U < T .
Does this assumption imply that pit(Y ) = pit(Z) for every U < t ≤ T
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MATH3975 Semester 2, 2021 Page 4 of 5
3. [20 marks] CRR model: American contingent claim.
Consider the CRR model with T = 2 and S0 = 45, Su1 = 49.5, Sd1 = 40.5. Assume
that the interest rate is negative, specifically, r = 0.05. Consider an American
claim Xa with the reward process gt = (St Kt)+ where K0 = 40,
K1(ω) = 35.5 for ω ∈ {ω1, ω2}, K1(ω) = 38.5 for ω ∈ {ω3, ω4}.
and K2 = 36.45.
(a) Find the unique martingale measure P on (Ω,F2) and compute the price pro-
cess Ca for the American claim Xa using the recursive relationship, which
holds for t = 0, 1,
pit(X
a) = max
{
gt, Bt EP
(
B 1t+1pit+1(X
a) | Ft
)}
.
Find the rational exercise time τ 0 for the holder of the American claim Xa.
(b) Find the replicating strategy for the American claim Xa up to the rational
exercise time τ 0 and check that the wealth V ( ) of the replicating strategy
matches the price computed in part (b).
(c) Find the early exercise premium for the American claim Xa.
(d) Determine whether the discounted arbitrage price B 1pi(Xa) is a super-
martingale or a submartingale under P with respect to the filtration F. Find
a probability measure Q on (Ω,F2) under which the process B 1pi(Xa) is a
martingale with respect to the filtration F.
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MATH3975 Semester 2, 2021 Page 5 of 5
4. [20 marks] Black-Scholes model: European contingent claim.
Assume that the stock price S is governed under the martingale measure P 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.
The savings account B is given by Bt = ert for all t ∈ R+.
Let K and L be arbitrary real numbers such that 0 < L < K. Consider a Euro-
pean claim with the payoff X at time T given by the following expression
X = max
(|ST K|, K L).
(a) Sketch the profile of the payoff X as the function of the stock price ST and
find the decomposition of this payoff in terms of cash and payoffs of stan-
dard call and put options with expiry date T and various strikes.
(b) Using the decomposition from part (a), compute the arbitrage price and
the replicating portfolio at time t = 0 for the claim X. Take for granted
the Black-Scholes pricing formulae for call and put options and respective
hedge ratios.
(c) Show that the arbitrage price of X at time t = 0 is a monotone function of
the parameter L and compute the limits limL→0 pi0(X) and limL→K pi0(X).
(d) Find the arbitrage price at time 0 of the claim Y = S2T with maturity T
using the property that the process Mα, which is given by
Mαt = exp
(
αWt 1
2
α2 t
)
, t ∈ [0, T ],
is a martingale under P for any choice of α ∈ R.
End of examination
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