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Home arrow Business & Finance arrow Frequently Asked Questions in Quantitative Finance

Models and Equations

Equity, Foreign Exchange and Commodities

The lognormal random walk

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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