ASSIGNMENT 2
MATH3975 Financial Derivatives (Advanced)
Due by 11:59 p.m. on Sunday, 22 October 2023
1. [10 marks] Path-dependent American claim. LetM = (B,S) be the CRR model with
r = 0 and the stock price S satisfying S0 = 4, Su1 = 5.5, Sd1 = 3.5. Consider a path-dependent
American claim with maturity T = 2 and the reward process g defined as follows: g0 =
5.5, g1 = 6 and the random variable g2 is given by
g2(S
u
1 , S
uu
2 ) = 8, g2(S
u
1 , S
ud
2 ) = 4, g2(S
d
1 , S
du
2 ) = 5, g2(S
d
1 , S
dd
2 ) = 9.
(a) Let P˜ be the probability measure under which the process S/B is a martingale. Com-
pute the arbitrage price process (pit(Xa), t = 0, 1) for the American claim using the
recursive relationship
pit(X
a) = max
{
gt, Bt EP˜
(
pit+1(X
a)
Bt+1
∣∣∣Ft)}
with the terminal condition pi2(Xa) = g2. Find the rational exercise time τ∗0 of this
claim by its holder.
(b) Find the replicating strategy ϕ for the claim up to the random time τ∗0 and check that
the equality Vt(ϕ) = pit(Xa) is valid for all t ≤ τ∗0 .
(c) Determine whether the arbitrage price process (pit(Xa); t = 0, 1, 2) is either a martin-
gale or a supermartingale under P˜ with respect to the filtration F.
(d) Find a probability measureQ on the space (Ω,F2) such that the arbitrage price process
(pit(X
a); t = 0, 1, 2) is a martingale underQwith respect to the filtration F and compute
the Radon-Nikodym density of Q with respect to P˜ on (Ω,F2).
(e) Let P̂ be a probability measure under which the process B/S is a martingale. Define
the process (pit(Xa), t = 0, 1) through the recursive relationship
pit(X
a) = max
{
gt, St EP̂
(
pit+1(X
a)
St+1
∣∣∣Ft)}
with pi2(Xa) = g2. Is it true that the equality pit(Xa) = pit(Xa) holds for all t = 0, 1, 2?
Justify your answer but do not perform any computations with numbers.
2. [10 marks] Gap option. We place ourselves with the setup of the Black-Scholes market
model M = (B,S) with a unique martingale measure P˜. Let the real numbers α and
β satisfy > 0. Consider the gap option with the payoff at maturity date T given by the
following expression
X = h(ST ) = (ST − β)+1{ST≥α}.
(a) Sketch the graph of the function g(ST ) and show that the inequality pit(X) < Ct(β) is
valid for every 0 ≤ t < T where Ct(β) is the Black-Scholes price of the standard call
option with strike β.
(b) Show that the payoff of the gap option can be decomposed into the sum of the payoff
CT (α) of the standard call option with the strike price α and α− β units of the digital
option with the payoff DT (α) = 1{ST≥α}.
(c) Compute the arbitrage price pit(X) at time t for the gap option. Take for granted the
Black-Scholes formula for the standard call option.
(d) Assume that S0 6= α. Find the limit limT→0 pi0(X). Explain your result.
(e) Find the limit limσ→∞ pit(X) for a fixed 0 ≤ t < T and compare with the limits
limσ→∞Ct(β) and limσ→∞Ct(α). Explain your findings.