ASSIGNMENT 2
MATH3975 Financial Derivatives (Advanced)
Due by 11:59 p.m. on Sunday, 22 October 2023
1. [10 marks] Path-dependent American claim. Let M = (B, S) be the CRR model with r = 0 and the stock price S satisfying S0 = 4, S1u = 5.5, S1d = 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
g (Su,Suu) = 8, g (Su,Sud) = 4, g (Sd,Sdu) = 5, g (Sd,Sdd) = 9. 212 212 212 212
(a) Let Pe be the probability measure under which the process S/B is a martingale. Com- pute the arbitrage price process (πt(Xa), t = 0,1) for the American claim using the recursive relationship
a πt+1(Xa) πt(X)=max gt,BtEPe B Ft
with the terminal condition π2(Xa) = g2. Find the rational exercise time τ0∗ of this
claim by its holder.
(b) Find the replicating strategy φ for the claim up to the random time τ0∗ and check that
the equality Vt(φ) = πt(Xa) is valid for all t ≤ τ0∗.
(c) Determine whether the arbitrage price process (πt(Xa); t = 0,1,2) is either a martin-
gale or a supermartingale under Pe with respect to the filtration F.
(d) FindaprobabilitymeasureQonthespace(Ω,F2)suchthatthearbitragepriceprocess (πt(Xa); t = 0, 1, 2) is a martingale under Q with respect to the filtration F and compute the Radon-Nikodym density of Q with respect to Pe on (Ω, F2).
(e) Let Pb be a probability measure under which the process B/S is a martingale. Define the process (πet(Xa), t = 0, 1) through the recursive relationship
a πet+1 (X a ) πet(X)=max gt,StEPb S Ft
with πe2(Xa) = g2. Is it true that the equality πet(Xa) = πt(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 Pe. 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 πt(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 πt(X) at time t for the gap option. Take for granted the Black-Scholes formula for the standard call option.
(d) Assume that S0 ̸= α. Find the limit limT→0 π0(X). Explain your result.
(e) Find the limit limσ→∞ πt(X) for a fixed 0 ≤ t < T and compare with the limits lim σ→∞ Ct(β) and lim σ→∞ Ct(α). Explain your findings.
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