Compute the value of a standard call option using Monte Carlo simulation

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Consider a 1-year call option with strike price 100 on a non-dividend paying stock with the following parameters: Underlying stock price $ 100 Volatility 30% pa Riskless rate 1% (cc) pa Assume that the standard Black-Scholes assumptions apply, i.e. the underlying stock price follows a Geometric Brownian Motion and the interest rate is constant. Assume the year has 252 business days.

1. Standard call. Compute the value of a standard call option using Monte Carlo simulation with N = 10, 000 draws and compute a 95% confidence interval for the option price.1

2. Compute the value of a standard call option using Monte Carlo simulation with N = 10, 000 draws using Antithetic Variables and compute a 99% confidence interval for the option price. Is the confidence interval wider or narrower than that in the previous question?

3. Option with sweetener. Assume that the call has a “sweetener”: If the stock return after six months is negative (i.e., the stock price six months from now is lower than today) the final strike price of the option is 90 instead of 100. Compute the value of the call with the sweetener using Monte Carlo simulation with N = 10, 000 draws. Is it different from the price of a standard call (why)? Compute a 99% confidence interval for the option price.

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