Plain Monte-Carlo

Motivation and basic principle

The calculation of the price of options can only be done analytically in some very special cases: the statistics of the underlying contract must be ‘simple enough’ (for example, the Black-Scholes log-normal process), and the option contract itself must not be too ‘exotic’. For example, ‘American’ options, that can be exercised at any time, cannot be priced exactly, even in the simple framework of the Black-Scholes model.

On the other hand, as emphasized throughout this book, the fluctuations of the underlying assets are in general non-Gaussian, with complex volatility-volatility and return-volatility correlations. Furthermore, real world options often involve clauses that are difficult to deal with analytically. It can therefore be compulsory to rely on numerical methods to price and hedge these options. Aversatile and commonly used method is the Monte-Carlo method, that we first explain in the context of ‘plain vanilla’ European options. Suppose that the interest rate r is zero, and that one chooses to Δ-hedge the option. In this case, the price of the option today is equal to the average pay-off of the option, over the detrended distribution of price returns (this price is called the ‘risk-neutral’ price, see the discussion in Section 15.2.2).

The most naive method to compute the option price would be to generate an ensemble of paths according to a certain (zero-drift) stochastic model, and to compute the average pay-off over this ensemble.