Binomial tree pricing model
Closely following the derivation of Black and Scholes, John Cox, Stephen Ross and Mark Rubinstein developed the original version of the binomial options pricing model. It models the dynamics of the option’s theoretical value for discrete time intervals over the option’s life. The model starts with a binomial tree of discrete future possible underlying stock prices. By constructing a riskless portfolio of an option and stock (as in the Black-Scholes model) a simple formula can be used to find the option price at each node in the tree. This value can approximate the theoretical value produced by Black Scholes, to the desired degree of precision. However, the binomial model is considered more accurate than Black-Scholes because it is more flexible; e.g., discrete future dividend payments can be modeled correctly at the proper forward time steps, and American options can be modeled as well as European ones. Binomial models are widely used by professional option traders. The Trinomial tree is a similar model, allowing for an up, down or stable path; although considered more accurate, particularly when fewer time-steps are modelled, it is less commonly used as its implementation is more complex.
Monte Carlo models
For many classes of options, traditional valuation techniques are intractable because of the complexity of the instrument. In these cases, a Monte Carlo approach may often be useful. Rather than attempt to solve the differential equations of motion that describe the option’s value in relation to the underlying security’s price, a Monte Carlo model uses simulation to generate random price paths of the underlying asset, each of which results in a payoff for the option. The average of these payoffs can be discounted to yield an expectation value for the option. Note though, that despite its flexibility, using simulation for American styled options is somewhat more complex than for lattice based models.
Finite difference models
The equations used to model the option are often expressed as partial differential equations (see for example Black–Scholes PDE). Once expressed in this form, a finite difference model can be derived, and the valuation obtained. A number of implementations of finite difference methods exist for option valuation, including: explicit finite difference, implicit finite difference and the Crank-Nicholson method. A trinomial tree option pricing model can be shown to be a simplified application of the explicit finite difference method. Although the finite difference approach is mathematically sophisticated, it is particularly useful where changes are assumed over time in model inputs – for example dividend yield, risk free rate, or volatility, or some combination of these – that are not tractable in closed form.
Other models
Other numerical implementations which have been used to value options include finite element methods. Additionally, various short rate models have been developed for the valuation of interest rate derivatives, bond options and swaptions. These, similarly, allow for closed-form, lattice-based, and simulation-based modelling, with corresponding advantages and considerations.