A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility. As such, it is a generalisation of the Black–Scholes model). Local volatility models are often compared with stochastic volatility models but depends also on a new "global" randomness coming from an additional random component.

Formulation

In mathematical finance, the asset St that underlies a financial derivative is typically assumed to follow a stochastic differential equation of the form

1. ,

is the instantaneous risk free rate is a Wiener process. In the simplest model i.e. the Black–Scholes model is assumed to be constant, or at most a deterministic function of time; in reality, the realised volatility of an underlying actually varies with time and with the underlying itself.

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 underlying asset level St and of time t, we have a local volatility model. The local volatility model is a useful simplification of the stochastic volatility model.

"Local volatility" is thus a term used in quantitative finance, that are consistent with market prices for all options on a given underlying, yielding an asset price model of the type

This model is used to calculate exotic option valuations which are consistent with observed prices of vanilla options.

Development

The concept of a local volatility fully consistent with option markets was developed when Bruno Dupire and Emanuel Derman and Iraj Kani noted that there is a unique diffusion process consistent with the risk neutral densities derived from the market prices of European options.

Derman and Kani described and implemented a local volatility function to model instantaneous volatility. They used this function at each node in a binomial options pricing model. The tree successfully produced option valuations consistent with all market prices across strikes and expirations. The Derman-Kani model was thus formulated with discrete time and stock-price steps. (Derman and Kani produced what is called an "implied binomial tree"; with Neil Chriss they extended this to an implied trinomial tree. The implied binomial tree fitting process was numerically unstable.)

The key continuous-time equations used in local volatility models were developed by Bruno Dupire in 1994. Dupire's equation states

In order to compute the partial derivatives, there exist few known parameterizations of the implied volatility surface based on the Heston model: Schönbucher, SVI and gSVI. Other techniques include mixture of lognormal distribution and stochastic collocation.

Derivation

governed by the risk neutral SDE

satisfies the forward Kolmogorov equation (also known as Fokker–Planck equation)

.

Because of the Martingale pricing is

and replacing in the formula for the price of a call option and rearranging terms

twice

yields

using the Forward Kolmogorov equation

integrating by parts the first integral once and the second integral twice

Parametric local volatility models

Dupire's approach is non-parametric. It requires to pre-interpolate the data to obtain a continuum of traded prices and the choice of a type of interpolation. As an alternative, one can formulate parametric local volatility models. A few examples are presented below.

Bachelier model

The Bachelier model has been inspired by Louis Bachelier's work in 1900. This model, at least for assets with zero drift, e.g. forward prices or forward interest rates under their forward measure, can be seen as a local volatility model

1. .

. As interest rates turned negative in many economies, the Bachelier model became of interest, as it can model negative forward rates F through its Gaussian distribution.

Displaced diffusion model

This model was introduced by Mark Rubinstein. For a stock price, it follows the dynamics

it is immediate to see that Y follows a standard Black-Scholes model

is a geometric Brownian motion, it has a lognormal distribution. it follows that the asset S is allowed to take negative values with positive probability. This is useful for example in interest rate modelling, where negative rates have been affecting several economies.

CEV model

The constant elasticity of variance model (CEV) is a local volatility model where the stock dynamics is, under the risk neutral measure and assuming no dividends,

so that in this case

The model is at times classified as a stochastic volatility model, although according to the definition given here, it is a local volatility model, as there is no new randomness in the diffusion coefficient. This model and related references are shown in detail in the related page.

The lognormal mixture dynamics model

This model has been developed from 1998 to 2021 in several versions by Damiano Brigo, Fabio Mercurio and co-authors. Carol Alexander studied the short and long term smile effects.. In the Black Scholes model the price of a European non-path-dependent option is obtained by integration of the option payoff against this lognormal density at maturity. The basic idea of the lognormal mixture dynamics model. When modelling a stock price, Brigo and Mercurio build a local volatility model

included in the mixture. The instantaneous volatility is defined as

1. or more in detail

.'s with weights

. The same holds for put options and all other simple contingent claims. The same convex combination applies also to several option greeks, and a possible shift parameter, allows one to reproduce most market smiles. The model has been used successfully in the equity, FX, and interest-rate markets.

. This can be avoided, and the smile allowed to be more general, by combining the mixture dynamics and displaced diffusion ideas, leading to the shifted lognormal mixture dynamics.

's in the mixture components that are time dependent, so as to calibrate the smile term structure. An extension of the model where the different mixture densities have different means has been studied, while preserving the final no arbitrage drift in the dynamics. A further extension has been the application to the multivariate case, where a multivariate model has been formulated that is consistent with a mixture of multivariate lognormal densities, possibly with shifts, and where the single assets are also distributed as mixtures, reconciling modelling of single assets smile with the smile on an index of these assets. A second application of the multivariate version has been triangulation of FX volatility smiles.. Technically, it can be shown that the local volatility lognormal mixture dynamics is the Markovian projection of the uncertain volatility model.

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, but see Crepey (2004), who claims that such models provide the best average hedge for equity index options, and note that models like the mixture dynamics allow for time dependent local volatilities, calibrating also the term structure of the smile. Local volatility models are also useful in the formulation of stochastic volatility models.

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. Numerous calibration methods are developed to deal with the McKean-Vlasov processes including the most used particle and bin approach. Also, they lead to complete markets where hedging can be based only on the underlying asset. As hinted above, the general non-parametric approach by Dupire is problematic, as one needs to arbitrarily pre-interpolate the input implied volatility surface before applying the method. Alternative parametric approaches with a rich and sound parametrization, as the above tractable mixture dynamical local volatility models, can be an alternative. 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. In such cases, stochastic volatility models are preferred.

References