Economics homework help. 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
σ
2
t−1 + σt−1zt
,
with zt ∼ NID(0, 1), and σ
2
t = ω + α(σt−1zt − θσt−1)
2 + βσ2
t−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. Compare 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 observation ?
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 = 0.152
,
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.

Economics homework help