MATH3075 Assignment 1: Solutions
1. Single-period market model [12marks]
Consider a single-period market model M = (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(S_1) = 36q_1 + 20q_2 + 4q_3 = (1 + r)S_0 = 20 ] or, equivalently, [ EQ(S_1 – (1 + r)S_0) = 16q_1 – 16q_3 = 0. ] Let q2 = λ. Then $q_1 = q_3 = \frac{1 – \lambda}{2}$ where 0 < λ < 1. Hence [ M = { (q_1, q_2, q_3) q_1 = q_3 = , q_2 = , 0 < < 1 } ] The market model M is arbitrage-free since M ≠ ⌀. Moreover, it is incomplete since the uniqueness of a martingale measure for M 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 [ V_1(x, ) = (x – S_0)(1 + r) + S_1 = x(1 + r) + (S_1 – S_0(1 + r)) = xB_1 + (S_1 – S_0B_1) ] Then we solve the following equations [
] 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 [ V_0() = _0B_0 + _0S_0, V_1() = _0B_1 + _0S_1. ] 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 [
] 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.
(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 $q_1 = q_3 = \frac{1 – \lambda}{2}$, the risk-neutral valuation formula yields, for every 0 < λ < 1, [ p_{i0}(Y) = EQ(Y/B_1) = + – 3 = 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 in M.
Answer: [2 marks] To find a replicating strategy, we need to solve the following equations [
] The strategy $(\phi_0, \phi_1) = (\frac{11}{16}, \frac{1}{16})$ is a unique solution to the first two equations, but it does not satisfy the last one. Hence no replicating strategy for X exists in M.
(e) Find the range of arbitrage prices for X using the class M of all martingale measures for the model M.
Answer: [2 marks] We compute the range of prices for X consistent with the no-arbitrage principle. We have [ p_{i0}(X) = EQ(X/B_1) = + 4- 1 = 0.5(1 + ). ] 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 [ V_1(x, ) = (x – S_0)(1 + r) + S_1 = x(1 + r) + (S_1 – S_0(1 + r)) = xB_1 + (S_1 – S_0B_1) ] 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 [
] 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. Static hedging with options [8marks]
Consider a parametrised family of contingent claims with the payoff Y(α) at time T given by the following expression [ Y() = {, + 2|- S_T| – S_T} ] 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 continuous 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 [
]
We now consider the case α < 3β. We take again β = 1 and we obtain by taking, for instance, α = 2 [
]
It is readily seen that for α ≥ 3β the payoff Y(α) can be represented as follows [ Y() = 3P_T() + C_T() – C_T(+ ) ] whereas for 0 ≤ α < 3β we have that [ Y() = 3P_T() – 3P_T(- ) + C_T() – C_T(+ ). ] Notice that the second decomposition above gives Y(α) = 0 when α = 0 and the two decompositions of Y(α) coincide when α = 3β.
(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 obtained in part (a).
Answer: [2 marks] By the additivity property of arbitrage pricing, we obtain, for every 0 ≤ α ≤ 3β, [ p_{i0}(Y()) = 3P_0() – 3P_0(- ) + C_0() – C_0(+ ) ] and, for every α ≥ 3β, [ p_{i0}(Y()) = 3P_0() + C_0() – C_0(+ ). ] 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 necessarily 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(BT − 1X) when Q runs over the class M of all martingale measures for the model M.
(d) Consider a complete arbitrage-free market model M = (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 → 0pi0(Y(α)), limα → ∞pi0(Y(α)) and limα → 3βpi0(Y(α)).
Answer: [2 marks] We observe that the payoff Y(α) increases when α increases. Specifically, 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 → 0pi0(Y(α)) = 0 since limα → 0Y(α)(ω) = 0 for all ω ∈ Ω and thus [ 0 {}{Q M}EQ(B_T^{-1}Y()) {}{}B_T^{-1}Y()() = 0. ] Moreover, using (1) and (2) [ {}p{i0}(Y()) = 3P_0() + C_0() – C_0(4), {}p{i0}(Y()) = 3P_0() + C_0(). ]