Monte Carlo Methods For Option Pricing - Methodology

Methodology

In terms of theory, Monte Carlo valuation relies on risk neutral valuation. Here the price of the option is its discounted expected value; see risk neutrality and Rational pricing: Risk-neutral valuation. The technique applied then, is (1) to generate several thousand possible (but random) price paths for the underlying (or underlyings) via simulation, and (2) to then calculate the associated exercise value (i.e. "payoff") of the option for each path. (3) These payoffs are then averaged and (4) discounted to today. This result is the value of the option.

This approach, although relatively straightforward, allows for increasing complexity:

• An option on equity may be modelled with one source of uncertainty: the price of the underlying stock in question. Here the price of the underlying instrument is usually modelled such that it follows a geometric Brownian motion with constant drift and volatility . So:, where is found via a random sampling from a normal distribution; see further under Black–Scholes. Since the underlying random process is the same, for enough price paths, the value of a european option here should be the same as under Black Scholes. More generally though, simulation is employed for path dependent exotic derivatives, such as Asian options.

• In other cases, the source of uncertainty may be at a remove. For example, for bond options the underlying is a bond, but the source of uncertainty is the annualized interest rate (i.e. the short rate). Here, for each randomly generated yield curve we observe a different resultant bond price on the option's exercise date; this bond price is then the input for the determination of the option's payoff. The same approach is used in valuing swaptions, where the value of the underlying swap is also a function of the evolving interest rate. (Whereas these options are more commonly valued using lattice based models, as above, for path dependent interest rate derivatives – such as CMOs – simulation is the primary technique employed.) For the models used to simulate the interest-rate see further under Short-rate model; note also that "to create realistic interest rate simulations" Multi-factor short-rate models are sometimes employed.

• Monte Carlo Methods allow for a compounding in the uncertainty. For example, where the underlying is denominated in a foreign currency, an additional source of uncertainty will be the exchange rate: the underlying price and the exchange rate must be separately simulated and then combined to determine the value of the underlying in the local currency. In all such models, correlation between the underlying sources of risk is also incorporated; see Cholesky decomposition: Monte Carlo simulation. Further complications, such as the impact of commodity prices or inflation on the underlying, can also be introduced. Since simulation can accommodate complex problems of this sort, it is often used in analysing real options where management's decision at any point is a function of multiple underlying variables.

• Simulation can similarly be used to value options where the payoff depends on the value of multiple underlying assets such as a Basket option or Rainbow option. Here, correlation between assets is likewise incorporated.

• As required, Monte Carlo simulation can be used with any type of probability distribution, including changing distributions: the modeller is not limited to normal or lognormal returns; see for example Datar–Mathews method for real option valuation. Additionally, the stochastic process of the underlying(s) may be specified so as to exhibit jumps or mean reversion or both; this feature makes simulation the primary valuation method applicable to energy derivatives. Further, some models even allow for (randomly) varying statistical (and other) parameters of the sources of uncertainty. For example, in models incorporating stochastic volatility, the volatility of the underlying changes with time; see Heston model.