ECON 643: Financial Economics II Assignment 2: Due on 27/11/2018
For simulation exercises, you may use any software or programming language of your choice, but Matlab or R should be easier to use.
Question 1: A fund manager has a well-diversified portfolio that mirrors the performance of the S&P500 and is worth $360 million. The value of the S&P500 is 1,200, and the portfolio manager would like to buy insurance against a reduction of more than 5% in the value of the portfolio over the next 6 months. The risk-free rate is 6% per annum. The dividend yield on both the portfolio and the S&P500 is 3%, and the volatility of the index is 30% per annum.
(a) If the fund manager buys traded European put options, how much would the insurance cost?
(b) Explain carefully alternative strategies open to the fund manager involving traded European call options, and show that they lead to the same result.
(c) If the fund manager decides to provide insurance by keeping part of the portfolio in risk-free securities, what should the initial position be?
(d) If the fund manager decides to provide insurance by using 9-month index futures, what should the initial position be?
Question 2: Assume that the price process St of an underlying asset follows the GARCH(1,1) dynamics:
rt ≡ ln St St−1
= r + λσt−1 − 1
2 σ2t−1 + σt−1zt,
with zt ∼ NID(0, 1), and σ2t = ω + α(σt−1zt − θσt−1)2 + βσ2t−1. Using GARCH parameters ω = 0.00001524, α = 0.1883, β = 0.7162, θ = 0, and λ = 0.007452,
simulate the GARCH call option price with a strike price of 100 and 20 days to maturity. Assume r = 0.02/365 and assume that today’s stock price is 100. Assume today’s variance is 0.00016. Com- pare the GARCH price with the BS price using a daily variance of 0.00016 as well. (Indication: Make sure to derive and show the risk-neutral dynamics to simulate from.)
Question 3: (Pricing of Outperformance Option) We consider a European-style option written on the S&P500 Index (denoted S1) and the MSCI index (denoted S2) which payoff at maturity T is given by:
C(S1(T ), S2(T )) = max(aS1(T )− bS2(T ), 0)
1. By choosing a suitable numeraire, show that the option payoff is merely the payoff of a call option on a new underlying with a given strike. Give the expression for the new underlying in function of S1 and S2 as well as the strike value.
2. Similarly, by choosing a suitable numeraire, show that the option payoff is merely the payoff of a put option on a new underlying with a given strike. Give also the expression for the new underlying in function of S1 and S2 as well as the strike value.
Pricing: Consider the case of the put option derived above. We assume a Black-Scholes world where S1 and S2 have geometric Brownian motion dynamics as follow:
= µ1dt+ σ1dz1t
= µ2dt+ σ2dz2t
where µ1 and µ2 are respectively the expected return of the S&P500 and the FTSE indices, σ1 and σ2 their respective volatilities, and z1t and z2t are correlated Brownian motions in the sense that dz1 · dz2 = ρdt. Moreover, you may assume that there is a money market account denotes Bt and paying a risk-free rate r:
1. Using risk-neutral pricing, derive an analytical formula for the price of the option.
2. What do you observe in this formula regarding the risk-free rate ? Can you explain your obser- vation ?
Monte-Carlo implementation and convergence analysis: Assume that: S10 = 2725, S20 = 2050, a = 10%, b = 10%, µ1 = 12%, µ2 = 9%, r = 3%, σ1 = 20%, σ2 = 10%, ρ = 0.4 and T = 1 (1year).
1. Implement a Monte-Carlo pricing engine to price the option.
2. Implement also an analytical pricing engine for the option.
3. Study the convergence of the Monte-Carlo pricing to the true price (the analytical price).
4. How could you improve this convergence (give briefly an overview of some techniques that could be used for doing so) ?
Question 4: Assume that the risk-neutral dynamics of the price S of an underlying asset is given by:
dS = rSdt+ √ V Sdz1
dV = µV dt+ ξV dz2,
where z1 and z2 are two Wiener processes with instantaneous correlation ρ.
Let fi, (i = 1, 2) be the value the value of the down-and-in (put) and the up-and-out (call) barrier options on S with barrier level Hi, strike price Xi and maturity Ti, (i = 1, 2).
1. What is the payoff of each option at their respective maturity dates.
2. By numerical simulation, give the price of each option when Ti = 0.5 (6 months), H1 = 95 and H2 = 105. Consider the current stock price S0 = 100 and the parameters values: V0 = 0.15
2, µ = 0, the risk-free rate r = 3%, ξ = 1 and ρ = 0.2. (Give the value of B and n simulated.)
3. Obtain the prices when S follows a geometric Brownian motion with volatility σ = 0.15
4. Compare the prices obtained at 2. and 3.