THE AUSTRALIAN NATIONAL UNIVERSITY RESEARCH SCHOOL OF FINANCE, ACTUARIAL STUDIES AND STATISTICS FINM3007: Advanced Derivatives Pricing and Applications Semester 2, 2024 Assignment 2 Due: 11 October 2024, 11:59:59 PM Total Marks: 32 General instructions. Submit your answers as a single PDF file on Wattle. To ensure full marks, include working out and explanations. For some questions, you may want to use computer software. When this occurs, your written explanation should be sufficient to explain your answer without having to read code. Attach the cover sheet with your submission. Please check that your submission on Wattle can be opened and the content of the file is complete and correct. Question 1. Let c(K,T ) be the price of a European call with strike price K and maturity time T on a stock. By the definition of the derivative, ∂2c(K,T ) ∂K2 = lim δ→0 c(K − δ, T )− 2c(K,T ) + c(K + δ, T ) δ2 . Suppose that call options with all strike prices K > 0 are traded and assume the risk-free rate r is constant. (a) Consider a portfolio that holds 1 δ2 , − 2 δ2 , 1 δ2 units of European calls with strike pricesK−δ, K, K+δ and maturity time T , respectively. By con- sidering the payoff, show that the value of the portfolio is nonnegative at time T . [2 marks] (b) Suppose that the implied risk-neutral distribution of the stock price at time T calculated by the Breeden-Litzenberger formula is negative for some values. Using (a), explain how an arbitrage strategy can be constructed. [3 marks] 1 Question 2. Suppose the risk-free rate is r = 0.05 and the implied volatili- ties on a stock with current price S0 = 100 with strike prices K and maturity times T satisfy σI(K,T ) = (0.1 + 0.4T )e −(K/100)+1. In the local volatility model, the stock price S = (St)t∈[0,T ∗] satisfies dSt St = rdt+ σ(St, t)dZt, t ∈ [0, T ∗], under the risk-neutral measure P̂, where σ is the local volatility function. Note that the finite difference approximation with stepsize δ for the first and second order derivatives of f are ∂f(x) ∂x ≈ f(x+ δ 2 )− f(x− δ 2 ) δ , ∂2f(x) ∂x2 ≈ f(x− δ)− 2f(x) + f(x+ δ) δ2 , respectively. (a) Using the Dupire formula and finite difference approximations with stepsize 0.1 for derivatives with respect to K and with stepsize 0.001 for derivatives with respect to T , find the value σ(109.5, 0.125) of the local volatility surface σ(K,T ). [4 marks] (b) Recall that we can approximate S by the Euler scheme S(i+1)∆t − Si∆t = rSi∆t∆t+ σ(Si∆t, i∆t)Si∆t∆Zi, i = 0, 1, . . . , n− 1, where ∆Zi ∼ N(0,∆t) are iid and ∆t = T ∗/n. We set n = 5. In order to approximately evaluate the local volatility surface σ in the Euler scheme, the following table gives σ(K,T ) computed using the Dupire formula in the same way as (a) on a grid where the values are computed at the midpoints of the grid (except at the endpoints of K where the grid has been extended to 0 and ∞), and where A = σ(109.5, 0.125) from (a). T K (0, 92) [92, 99) [99, 106) [106, 113) [113,∞) [0, 0.05) 0.152799 0.130798 0.113750 0.099506 0.095336 [0.05, 0.1) 0.197973 0.172181 0.149667 0.130093 0.113211 [0.1, 0.15) 0.244338 0.212461 0.184647 A 0.139612 [0.15, 0.2) 0.290171 0.252262 0.219200 0.190515 0.165706 [0.2, 0.25] 0.335778 0.291842 0.253544 0.220332 0.191618 2 Now consider a fixed-strike lookback call on the stock with strike price K = 110 and maturity time T ∗ = 0.25, which pays off cfix-lT ∗ = (MT ∗ −K)+, where MT ∗ = maxs∈[0,T ∗] Ss is the maximum of the stock price. Find the payoff of the option for the sample path of S approximated by the above Euler scheme with ∆Z0 = −0.069474, ∆Z1 = −0.403733, ∆Z2 = −0.224149, ∆Z3 = 0.169977, ∆Z4 = 0.416593. [3 marks] (c) Repeat (b) using the random variables ∆Z0 = −0.247006, ∆Z1 = −0.092446, ∆Z2 = 0.022566, ∆Z3 = 0.154839, ∆Z4 = −0.079953. instead. [1 mark] (d) Suppose that by repeating the procedure in (b) and (c) 8 more times, we obtain the following payoffs: 3.551903, 0, 0, 0, 14.186861, 11.632845, 0, 0. On the basis of these 10 simulation results (1 from (b), 1 from (c), 8 from (d)), approximate the price of the lookback call at time 0. [2 marks] Question 3. Let V be the price process of a ¥ asset which pays dividend yield q, and S be the ¥/$ exchange rate process. Assume the bivariate Black- Scholes model holds so that (V, S) follows bivariate GBM with correlation ρV S. Let σV and σS be the volatility of V and S, respectively. The domestic currency is $ with risk-free rate rd, and the foreign currency is ¥ with risk-free rate rf . Both risk-free rates are constant. A foreign currency European call option struck in the domestic currency with $ strike price Kd and maturity time T pays off cT = $(VT/ST −Kf )+. Note that VT/ST is the value of the ¥ asset converted into $. 3 (a) By using either the fact that V/S is a $ traded asset, or Tutorial 6 Solutions, Question 3 (b) (you do not need to use both), or otherwise, show that under the $ risk-neutral measure P̂$, d(Vt/St) Vt/St = (rd − q)dt+ σdZt, where σ = √ σ2V − 2ρV SσV σS + σ2S and Zt is a P̂$-SBM. [3 marks] (b) By using the Black-Scholes formula with appropriate substitutions or otherwise, find the $ price of the foreign currency call struck in the domestic currency at time 0. [3 marks] Question 4. Suppose that under the risk-neutral measure P̂, a stock price S = (St)t∈[0,T ] follows St = S0e mt+V (t), where V ∼ VGP(b∗, µ∗, σ∗) is a variance gamma process. Suppose the risk- free rate is r and S0. (a) Consider a European call on the stock with strike price K and ma- turity time T . Assuming that differentiation and integration can be interchanged, show that the delta at time 0 is ∆ = e(m−r)T (b∗)b ∗T Γ(b∗T ) ∫ ∞ 0 N(d1(S ′ 0, r ′, g, σ∗))gb ∗T−1e−(b ∗−r′)g dg, where S ′0 = S0e mT , r′ = µ∗ + 1 2 (σ∗)2, d1(S0, r, T, σ) = log(S0/K) + (r + 1 2 σ2)T σ √ T . [2 marks] (b) Suppose that r = 0.05, S0 = 100, and the risk-neutral parameters are b∗ = 0.6, µ∗ = −0.05, σ∗ = 0.2. Consider a European call on this 4 stock with maturity T = 2 and strike K = 105. By approximating the integral in (a) using the Riemann sum∫ ∞ 0 f(g) dg ≈ n−1∑ i=0 f ( xi + xi+1 2 ) (xi+1 − xi), where n = 10, xi = i, i = 0, 1, . . . , n, find the delta of the call ∆. [4 marks] Question 5. Let (Ft)t∈[0,T ] be a filtration on the σ-field F , and let Z = (Zt)t∈[0,T ] be a SBM under P such that Zt is Ft-measurable. We want to prove a version of Girsanov’s theorem which is used to obtain the EMM for the standard Black-Scholes model. In this case, it is enough to consider γ ∈ R as a constant. (a) By computing the cumulant generating function of Z under the Esscher measure Qh, show that Z ∼ BM(h, 1) under Qh. [3 marks] (b) Define a new process ZQ = (ZQt )t∈[0,T ] by ZQt = γt+ Zt. Define a new probability measure by Q(A) = E[e−γZT− 1 2 γ2T I(A)], A ∈ F . By applying (a) in the case where h = −γ and writing out the definition of the corresponding Esscher measure, show that ZQ is a SBM under Q. [2 marks] 5 51作业君版权所有