financial economics

For simulation exercises, you may use any software or programming language of your choice, but Matlab or R should be easier to use.

t 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)

Payoff analysis:

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.

1

Question 2: Assume that the price process S

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:

dS1t S1t

= µ1dt + σ1dz1t

dS2t S

= 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:

dBt Bt

= rdt

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

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.