Local volatility

A Local volatility model, in mathematical finance and financial engineering, is one which treats volatility as a function of the current asset level and of time .

Formulation

In mathematical finance, the assets S_t which underlie financial derivatives, are typically assumed to follow stochastic differential equations of the type where r is the instantaneous risk free rate, giving an average local direction to the dynamics, and W is a Wiener process, representing the inflow of randomness into the dynamics. The amplitude of this randomness is measured instant by instant by the volatility σ_t.

When such volatility has a randomness of its own - often described by a different equation driven by a different W - the model above is called a stochastic volatility model. And when such volatility is merely a function of the current asset level S_t and of time t, we have a local volatility model.

"Local volatility" is thus a term used in quantitative finance to denote the set of diffusion coefficients, σ_t = σ(S_t,t), that are consistent with the set of market prices for all option prices on a given underlying. This model is used to calculate values of exotic options which are consistent with observed prices of vanilla options.

Development

The concept of a local volatility was originated by Emanuel Derman and Iraj Kani as part of the implied volatility tree model.^[1]

As described and implemented by Derman and Kani, the local volatility function models the instantaneous volatility to use at each node in a binomial options pricing model such that the tree will produce a set of option valuations that are consistent with the option prices observed in the market for all strikes and expirations.^[1] The Derman-Kani model was thus formulated with discrete time and stock-price steps. The key continuous-time equations used in local volatility models were developed by Bruno Dupire in 1994. Dupire's equation states

Use

Local volatility models are useful in any options market in which the underlying's volatility is predominantly a function of the level of the underlying, interest-rate derivatives for example. Time-invariant local volatilities are supposedly inconsistent with the dynamics of the equity index implied volatility surface,^[2] but see Crepey, S (2004). "Delta-hedging Vega Risk". Quantitative Finance 4. , who claims that such models provide the best average hedge for equity index options. Local volatility models are nonetheless useful in the formulation of stochastic volatility models.^[3]

Local volatility models have a number of attractive features. Because the only source of randomness is the stock price, local volatility models are easy to calibrate. Also, they lead to complete markets where hedging can be based only on the underlying asset. The general non-parametric approach by Dupire is however problematic, as one needs to arbitrarily pre-interpolate the input implied volatility surface before applying the method. Alternative parametric approaches have been proposed, notably the highly tractable mixture dynamical local volatility models by Damiano Brigo and Fabio Mercurio^[4][5].

Since in local volatility models the volatility is a deterministic function of the random stock price, local volatility models are not very well used to price cliquet options or forward start options, whose values depend specifically on the random nature of volatility itself.

References

1. Carol Alexander (2004). "Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects". Journal of Banking & Finance 28 (12).