These models use various forms of time series analysis to estimate current and future expected actual volatility. They are typically based on some regression of volatility against past returns and they may involve autoregressive or moving-average components. In this category are the GARCH type of models. Sometimes one models the square of volatility, the variance, sometimes one uses high-low-open-close data and not just closing prices, and sometimes one models the logarithm of volatility. The latter seems to be quite promising because there is evidence that actual volatility is lognormally distributed. Other work in this area decomposes the volatility of a stock into components, market volatility, industry volatility and firm-specific volatility. This is similar to CAPM for returns.

Deterministic models

The simple Black-Scholes formulae assume that volatility is constant or time dependent. But market data suggests that implied volatility varies with strike price. Such market behaviour cannot be consistent with a volatility that is a deterministic function of time. One way in which the Black-Scholes world can be modified to accommodate strike-dependent implied volatility is to assume that actual volatility is a function of both time and the price of the underlying. This is the deterministic volatility (surface) model. This is the simplest extension to the Black-Scholes world that can be made to be consistent with market prices. All it requires is that we have a (S, t), and the Black-Scholes partial differential equation is still valid. The interpretation of an option's value as the present value of the expected payoff under a risk-neutral random walk also carries over. Unfortunately the Black-Scholes closed-form formula are no longer correct. This is a simple and popular model, but it does not capture the dynamics of implied volatility very well.

Stochastic volatility

Since volatility is difficult to measure, and seems to be forever changing, it is natural to model it as stochastic. The most popular model of this type is due to Heston. Such models often have several parameters which can either be chosen to fit historical data or, more commonly, chosen so that theoretical prices calibrate to the market. Stochastic volatility models are better at capturing the dynamics of traded option prices better than deterministic models. However, different markets behave differently. Part of this is because of the way traders look at option prices. Equity traders look at implied volatility versus strike, FX traders look at implied volatility versus delta. It is therefore natural for implied volatility curves to behave differently in these two markets. Because of this there have grown up the sticky strike, sticky delta, etc., models, which model how the implied volatility curve changes as the underlying moves.

Poisson processes

There are times of low volatility and times of high volatility. This can be modelled by volatility that jumps according to a Poisson process.

Uncertain volatility

An elegant solution to the problem of modelling the unseen volatility is to treat it as uncertain, meaning that it is allowed to lie in a specified range but whereabouts in that range it actually is, or indeed the probability of being at any value, are left unspecified. With this type of model we no longer get a single option price, but a range of prices, representing worst-case scenario and best-case scenario.

References and Further Reading

Avellaneda, M, Levy, A & Paráis, A 1995 Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance 2 73-88

Derman, E & Kani, I 1994 Riding on a smile. Risk magazine 7 (2) 32-39 (February)

Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18-20 (January)

Heston, S 1993 A closed-form solution for options with stochastic volatility with application to bond and currency options. Review of Financial Studies 6 327-343

Javaheri, A 2005 Inside Volatility Arbitrage. John Wiley & Sons Ltd

Lewis, A 2000 Option Valuation under Stochastic Volatility. Finance Press

Lyons, TJ 1995 Uncertain volatility and the risk-free synthesis of derivatives. Applied Mathematical Finance 2 117-133

Rubinstein, M 1994 Implied binomial trees. Journal of Finance 69 771-818

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

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