Monte-Carlo Option Pricing

Monte-Carlo Option Pricing

This exercise uses observation that an option price can be calculated as a discounted risk-neutral expectation.

Context

This observation suggests that we could value an option by sampling many thousands, say, of possible asset prices, at , calculating the payoffs, taking their expected value and discounting this at the risk-free rate to today in order to estimate the option’s current value. This game of chance does in fact work and in this task you will see how. It is an example of a simple Monte Carlo method: so-called because games of chance form an important part of that part of Monaco’s economy.

You have seen in lectures that the asset price It process, for with and , has at expiry, with the solution exp((r- In what follows we let be the payoff function for a plain vanilla European-style option written on a non-dividend paying underlying asset. In particular, if S is the underlying’s price at expiry then V(S) will give the value of the option at the same time.

Task:

· First make sure you know what the central limit theorem says. Then complete the following task.

Let... be a sequence of independent and identically distributed (i.i.d.) random variables each with mean and variance Define the sample mean,

Explain why, as M gets large, we have:

and

· The statement for the estimated option price becomes more accurate as (the central limit theorem). Explain why you can infer from this that the standard deviation of the estimate decreases with. Explain how and why we can expect that,

And show that this may also be written as,

And hence, explain why is an approximate 95% confidence interval for the true option price.

Where is the mean and is the variance of the current option price.

is the price and a standard deviation.