MATH3075 Assignment 1: Solutions 1. Single-periodmarketmodel [12marks] Consider a single-period market modelM = (B,S) on a sample space Ω = {ω1, ω2, ω3}. Assume that r = 3 and the stock price S = (S0, S1) satisfies S0 = 5 and S1 = (36, 20, 4). 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. Is the market model M arbitrage-free? Is this market model complete? Answer: [2 marks] We need to solve: q1 + q2 + q3 = 1, 0 < qi < 1 and (since 1 + r = 4) EQ(S1) = 36q1 + 20q2 + 4q3 = (1 + r)S0 = 20 or, equivalently, EQ(S1 − (1 + r)S0) = 16q1 − 16q3 = 0. Let q2 = λ. Then q1 = q3 = 1−λ2 where 0 < λ < 1. Hence M = { (q1, q2, q3) ∣∣ q1 = q3 = 1− λ 2 , q2 = λ, 0 < λ < 1 } . The market model M is arbitrage-free since M 6= ∅. Moreover, it is incomplete since the uniqueness of a martingale measure forM fails to hold. (b) Find the replicating strategy for the contingent claim Y = (10, 2,−6) and compute its arbitrage price pi0(Y ) at time 0 through replication. Answer: [2 marks] First solution. We may use a portfolio (x, ϕ) ∈ R2 and represent the wealth as follows: V0(x, ϕ) = x and V1(x, ϕ) = (x− ϕS0)(1 + r) + ϕS1 = x(1 + r) + ϕ ( S1 − S0(1 + r) ) = xB1 + ϕ ( S1 − S0B1 ) . Then we solve the following equations 4x+ 16ϕ = 10, 4x+ 0ϕ = 2, 4x− 16ϕ = −6. From the second equation, we obtain pi0(Y ) = x = 0.5 and thus from the first (or last) equation we get ϕ = 0.5. Second solution. The wealth process of a portfolio (ϕ00, ϕ10) satisfies V0(ϕ) = ϕ 0 0B0 + ϕ 1 0S0, V1(ϕ) = ϕ 0 0B1 + ϕ 1 0S1. Replication of a claim Y means that V1(ϕ)(ωi) = Y (ωi) for i = 1, 2, 3. Hence to find a replicating strategy for Y , we need to solve the following equations 4ϕ00 + 36ϕ 1 0 = 10, 4ϕ00 + 20ϕ 1 0 = 2, 4ϕ00 + 4ϕ 1 0 = −6. We obtain (ϕ00, ϕ10) = (−2, 0.5) and thus pi0(Y ) = x = ϕ00B0 + ϕ10S0 = −2 + 0.5 × 5 = 0.5. Hence at time 0 we need to buy 0.5 shares of stock. For this purpose, after receiving 0.5 units of cash from the buyer of the claim Y , we need to borrow two units of cash in the money market. 1 (c) Recompute pi0(Y ) using the risk-neutral valuation formula with an arbitrary martingale measure Q from the class M. Answer: [2 marks] For any 0 < q2 = λ < 1 and q1 = q3 = 1−λ2 , the risk-neutral valuation formula yields, for every 0 < λ < 1, pi0(Y ) = EQ(Y/B1) = 1 2 ( 5 1− λ 2 + λ− 3 1− λ 2 ) = 0.5. As expected, the price pi0(Y ) does not depend on λ, that is, on a choice of a martingale measure. (d) Check whether that the contingent claim X = (5, 4,−1) is attainable inM. Answer: [2 marks] To find a replicating strategy, we need to solve the following equations 4ϕ00 + 36ϕ 1 0 = 5, 4ϕ00 + 20ϕ 1 0 = 4, 4ϕ00 + 4ϕ 1 0 = −1. The strategy (ϕ0, ϕ1) = ( 11 16 , 1 16 ) is a unique solution to the first two equations, but it does not satisfy the last one. Hence no replicating strategy for X exists inM. (e) Find the range of arbitrage prices for X using the class M of all martingale measures for the modelM. Answer: [2 marks] We compute the range of prices for X consistent with the no-arbitrage principle. We have pi0(X) = EQ(X/B1) = 1 4 ( 5 1− λ 2 + 4λ− 1 1− λ 2 ) = 0.5(1 + λ). Since from part (c) we know that λ ∈ (0, 1), it is clear the range of prices pi0(X) consistent with the no-arbitrage principle is the open interval (0.5, 1). (f) Suppose that at time 0 you have sold the claim X for 2 units of cash. Show that there exists a hedge ratio ϕ such that the wealth V1(2, ϕ) at time 1 strictly dominates the payoff X, meaning that V1(2, ϕ)(ωi) > X(ωi) for i = 1, 2, 3. Answer: [2 marks] It suffices to give any example of a portfolio (x, ϕ) with the initial value x = 2 such that the inequality V1(x, ϕ)(ωi) > X(ωi) holds for i = 1, 2, 3. We may use the representation of the wealth at time t = 1 V1(x, ϕ) = (x− ϕS0)(1 + r) + ϕS1 = x(1 + r) + ϕ ( S1 − S0(1 + r) ) = xB1 + ϕ ( S1 − S0B1 ) . Since x = 2 and B1 = 4 so that S0B1 = 20, it suffices to find a number ϕ ∈ R such that the following inequalities are satisfied 8 + 16ϕ > 5, 8 + 0ϕ > 4, 8− 16ϕ > −1. For instance, we may take ϕ = 0.5. Then the wealth of the portfolio (x, ϕ) = (2, 0.5) at time t = 1 equals V1(2, 0.5) = (16, 8, 0) so it is clear that V1(2, 0.5)(ωi) > X(ωi) for i = 1, 2, 3. 2 2. Static hedging with options [8 marks] Consider a parametrised family of contingent claims with the payoff Y (α) at time T given by the following expression Y (α) = min ( α, β + 2|β − ST | − ST ) where a real number β > 0 is fixed and the parameter α is an arbitrary real number such that α ≥ 0. (a) For any fixed α ≥ 0, sketch the profile of the payoff Y (α) as a function of ST ≥ 0 and find a decomposition of Y (α) in terms of the payoffs of standard call and put options with maturity date T (do not use a constant payoff). Notice that a decomposition of Y (α) may depend on the value of the parameter α. Answer: [2 marks] It is easy to see that the payoff Y (α) is a piecewise linear and contin- uous function, which is nonnegative and bounded from above by α. We first consider the case α ≥ 3β. Let us take β = 1. Then for α ≥ 3β we obtain by taking, for instance, α = 4 β = 1 β + α = 5 1 2 3 4 ST Y (α) α ≥ 3β We now consider the case α < 3β. We take again β = 1 and we obtain by taking, for instance, α = 2 β = 1 β + α = 3 1 2 3 4 ST Y (α) α < 3β It is readily seen that for α ≥ 3β the payoff Y (α) can be represented as follows Y (α) = 3PT (β) + CT (β)− CT (β + α) whereas for 0 ≤ α < 3β we have that Y (α) = 3PT (β)− 3PT ( β − 13α ) + CT (β)− CT (β + α). Notice that the second decomposition above gives Y (α) = 0 when α = 0 and the two decom- positions of Y (α) coincide when α = 3β. 3 (b) Assume that call and put options with all strikes are traded at time 0 at some finite prices. For each value of α ≥ 0, compute the arbitrage price pi0(Y (α)) at time t = 0 for the claim Y (α) using the prices at time 0 of call and put options and a suitable decomposition ob- tained in part (a). Answer: [2 marks] By the additivity property of arbitrage pricing, we obtain, for every 0 ≤ α ≤ 3β, pi0(Y (α)) = 3P0(β)− 3P0 ( β − 13α ) + C0(β)− C0(β + α) (1) and, for every α ≥ 3β, pi0(Y (α)) = 3P0(β) + C0(β)− C0(β + α). (2) In particular, we deduce from (1) that pi0(Y (α)) = 0 when α = 0, which is obvious since Y (α) = 0 when α = 0. (c) For any α > 0, examine the sign of an arbitrage price of the claim Y (α) in any (not nec- essarily complete) arbitrage-free market model M = (B,S) with a finite state space Ω. Justify your answer. Answer: [2 marks] Since the payoff Y (α) is strictly positive for every value of ST (except for ST = β) the price pi0(Y (α)) should be strictly positive in any arbitrage-free market model M = (B,S) since otherwise an arbitrage opportunity would arise in the extended market model. You may use the argument that the range of prices for any contingent claim X coincides with the range of values of the expectation EQ(B−1T X) when Q runs over the class M of all martingale measures for the modelM. (d) Consider a complete arbitrage-free market modelM = (B,S) defined on some finite sample space Ω. Show that the arbitrage price of Y (α) at time t = 0 is a monotone function of the variable α ≥ 0 and find the limits limα→0 pi0(Y (α)), limα→∞ pi0(Y (α)) and limα→3β pi0(Y (α)). Answer: [2 marks] We observe that the payoff Y (α) increases when α increases. Specif- ically, if we consider the payoffs Y (α1) and Y (α2) corresponding to α1 and α2, respectively, where α1 < α2 then it is clear that Y (α1) ≤ Y (α2). Consequently, pi0(Y (α1)) ≤ pi0(Y (α2)) and thus the price pi0(Y (α)) is a nondecreasing function of the variable α. Furthermore, limα→0 pi0(Y (α)) = 0 since limα→0 Y (α)(ω) = 0 for all ω ∈ Ω and thus 0 ≤ lim α→0 sup Q∈M EQ ( B−1T Y (α) ) ≤ lim α→0 max ω∈Ω ( B−1T Y (α)(ω) ) = 0. Moreover, using (1) and (2) lim α→3β pi0(Y (α)) = 3P0(β) + C0(β)− C0(4β), lim α→∞pi0(Y (α)) = 3P0(β) + C0(β). 4 51作业君版权所有