Description
In this project you will value call options via Monte Carlo simulation using a
GARCH stochastic volatility model. Take initial stock price S0 = 100, risk-free
rate r = 5% per year, time to expiration T = 0.5 years, and use the model of
Chapter 16, §3, to generate the stock price paths. The starter code StoVol.cpp
will generate illustrative paths for you. Note that in this code the long-run
volatility is 30% annually and the stochastic volatility starts off at 35% annually
— use these values in your simulations. The values of α, β, and γ are those that
maximum likelihood estimation produced in Chapter 16 for Exxon Mobil.
1. Use your model to value a call option with strike K = 0. The payoff at
time T is max(ST −0, 0) = ST , so this “option” replicates the payoff of the stock
at time T. Its value today should therefore be S0 (taken to be 100). If it is not,
there’s a mistake somewhere in your code.
2. Value call options on XOM with strikes of 60, 70, 80, 90, 100, 110, 120,
130, 140, 150 and 160.
3. The code ImpliedVol.cpp computes a call option’s implied volatility in
the Black-Scholes (constant volatility) framework. Once you have computed the
value of these eleven call options in the GARCH stochastic volatility framework,
calculate their B-S implied volatility. Plot the results with the option’s strike
on the horizontal axis and the implied volatility on the vertical axis. What do
you notice? Sanity check: the B-S implied volatilities should all be between 30
and 35. Use this as the vertical scale.
