The most common and simplest model is the lognormal random walk:

The Black-Scholes hedging argument leads to the following equation for the value of non-path-dependent contracts,

The parameters are volatility a, dividend yield D and risk-free interest rate r. All of these can be functions of S and/or t, although it wouldn't make much sense for the risk-free rate to be S dependent.

This equation can be interpreted probabilistically. The option value is

where St is the stock price at expiry, time T, and the expectation is with respect to the risk-neutral random walk

When a, D and r are only time dependent we can write down an explicit formula for the value of any non-path-dependent option without early exercise (and without any decision feature) as

where

The r parameters represent the 'average' of the parameters from the current time to expiration. For the volatility parameter the relevant average is the root-mean-square average, since variances can be summed but standard deviations (volatilities) cannot.

The above is a very general formula which can be greatly simplified for European calls, puts and binaries.

Multi-dimensional lognormal random walks

There is a formula for the value of a European non-path-dependent option with payoff of Payoff(S1,..., Sd)at time T :

X is the correlation matrix and there is a continuous dividend yield of Di on each asset.

Stochastic volatility

If the risk-neutral volatility is modelled by

where X is the market price of volatility risk, with the stock model still being

with correlation between them of p, then the option-pricing equation is

This pricing equation can be interpreted as representing the present value of the expected payoff under risk-neutral random walks for both S and a. Sofora call option, for example, we can price via the expected payoff

For other contracts replace the maximum function with the relevant, even path-dependent, payoff function.

Hull & White (1987)

Hull & White considered both general and specific volatility models. They showed that when the stock and the volatility are uncorrelated and the risk-neutral dynamics of the volatility are unaffected by the stock (i.e. p — Xq and q are independent of S) then the fair value of an option is the average of the Black--Scholes values for

the option, with the average taken over the distribution of o2.

Square-root model/Heston (1993)

In Heston's model

where v = a2. This has arbitrary correlation between the underlying and its volatility. This is popular because there are closed-form solutions for European options.

/2 model

where v = a2. Again, this has closed-form solutions.

GARCH-diffusion

In stochastic differential equation form the GARCH(1,1) model is

Here v = a2.

Ornstein-Uhlenbeck process

With y = ln v, v = a2,

This model matches real, as opposed to risk-neutral, data well.

Asymptotic analysis

If the volatility of volatility is large and the speed of mean reversion is fast in a stochastic volatility model,

with a correlation p, then closed-form approximate solutions (asymptotic solutions) of the pricing equation can be found for simple options for arbitrary functions p — Xq and q.In the above model the e represents a small parameter. The asymptotic solution is then a power series in e1/2.

Schonbucher's stochastic implied volatility

Schonbucher begins with a stochastic model for implied volatility and then finds the actual volatility consistent, in a no-arbitrage sense, with these implied volatilities. This model calibrates to market prices by definition.

Jump diffusion

Given the jump-diffusion model

the equation for an option is

£[•] is the expectation taken over the jump size. If the logarithm of J is Normally distributed with standard deviation a' then the price of a European non-path-dependent option can be written as

where

and

and Vbs is the Black-Scholes formula for the option value in the absence of jumps.

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