MATH3075: FINANCIAL DERIVATIVES (Mainstream) Semester 2, 2021
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Semester 2, 2021
CONFIDENTIAL EXAM PAPER
MATH3075: FINANCIAL DERIVATIVES (Mainstream)
1. [20 marks] Single-period market model
Consider a single-period market model M = (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 model M and check if the market model M is arbitrage-free and complete.
(b) Show that the contingent claim X = (8, 6, 4) is attainable and compute its arbitrage price π0 (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 in M. Is the claim Y attainable in M?
– Find the minimal initial endowment x for which there exists a portfolio (x, ϕ) with V0 (x, ϕ) = x and such that the inequality V1 (x, ϕ)(ωi ) > Y (ωi ) is satisfied for i = 1, 2, 3.
(d) Consider the extended market = (B, S1 , S2 ) where S1 = S and S2 is an additional risky asset given by: S1(2) = Y = (4, 2, -3) and S0(2) = 1.35.
– Find a unique martingale measure for the extended market = (B, S1 , S2 ).
– Compute the price of the claim Z = (-2, 5, 3) in the extended market = (B, S1 , S2 ). Is the claim Z attainable in ?
2. [20 marks] CRR model: European contingent claim
Consider the CRR model with S0 = 100, u = 1.4, d = 1.1 and r = 0.25. Let X be a European contingent claim with maturity date T = 2 and the payoff X given by
X = (S2 - S1 + 4)1{S2 一S1>15} =
(a) Show explicitly that the contingent claim X is path-dependent and compute the arbitrage price process for X using the relationship, which holds for
t = 0, 1,
πt (X) = Bt E ╱ │ rt 、
where is a unique martingale measure for the CRR model.
(b) Find the replicating strategy ϕ for the claim X and verify if the equality Vt (ϕ) = πt (X) is satisfied for t = 0, 1, 2.
(c) Find a unique probability measure on (Ω , r2 ) such that the process BS一1
π0 (Y) = S0 E ╱ 、.
(d) Let Y and Z be two European contingent claims with maturity T in the general CRR model with any number T of periods. Assume that the equality πU (Y) = πU (Z) holds for some date U such that 0 < U < T. Does this assumption imply that πt (Y) = πt (Z) for every U < t < T?
3. [20 marks] CRR model: American contingent claim
Consider the CRR model with the horizon date T = 2 and S0 = 8, S1(u) = 11, S1(d) = 7. We assume that the interest rate r = 0. Let X α be an American claim with the maturity date T = 2 and the reward process (gt , t = 0, 1, 2) where g0 = g1 = 12 and the random payoff g2 equals g2 = (14, 10, 10, 18), that is,
g2 (S1(u), S2(uu)) = 14, g2 (S1(u), S2(ud)) = g2 (S1(d), S2(du)) = 10, g2 (S1(d), S2(dd)) = 18.
(a) Find the unique martingale measure on (Ω , r2 ) and compute the price process Cα for this option using the recursive relationship, which holds for
t = 0, 1,
πt (Xα ) = max {gt , Bt E ╱ │rt 、}
with terminal condition πT (Xα ) = g2 . Find the rational exercise time τ0(*) of this claim by its holder.
(b) Find the replicating strategy ϕ for the issuer of the American claim X α up to the rational exercise time τ0(*) and verify if the equality Vt (ϕ) = πt (Xα ) holds up to time τ0(*) .
(c) Find the early exercise premium for the American claim X α .
(d) Determine whether the arbitrage price π(Xα ) is a supermartingale or a submartingale under with respect to the filtration F. Find a probabil- ity measure Q on (Ω , r2 ) under which the process π(Xα ) is a martingale with respect to the filtration F.
4. [20 marks] Black-Scholes model: European contingent claim
We assume that the stock price S is governed under the martingale measure by the Black-Scholes stochastic differential equation
dSt = St ╱r dt + σ dWt、, S0 > 0,
where σ > 0 is the stock price volatility and r is the instantaneous interest rate. Consider a European claim X with maturity T and payoff X = IK - ST I - LST where K > 0 and L > 0 are real numbers.
(a) Show that the payoff X satisfies X = PT (K) + CT (K) - LST . We henceforth assume that the strike K = S0 erT . Using the Black-Scholes pricing formulae for call and put options, find an explicit expression for the arbitrage price π0 (X).
(b) Compute the hedge ratio at time 0 for the claim X . Examine the sign of the hedge ratio when L = 0.
(c) Find a unique value of L such that π0 (X) = 0 and show that it satisfies 0 < L < 2. Compute the limits lim T →0 π0 (X) and limg →& π0 (X).
(d) Assume that r = 0. Show that the process S2 is a submartingale under , that is, the inequality E ╱ Ss(2) I rt ) > St(2) holds for all dates 0 < t < s.
2022-10-13