In mathematical finance, the asset St that underlies a financial derivative is typically assumed to follow a stochastic differential equation of the form under the risk neutral measure, where
is the instantaneous risk free rate, giving an average local direction to the dynamics, and
"Local volatility" is thus a term used in quantitative finance to denote the set of diffusion coefficients,
The concept of a local volatility fully consistent with option markets was developed when Bruno Dupire[1] and Emanuel Derman and Iraj Kani[2] noted that there is a unique diffusion process consistent with the risk neutral densities derived from the market prices of European options.
The tree successfully produced option valuations consistent with all market prices across strikes and expirations.
The implied binomial tree fitting process was numerically unstable.)
The key continuous-time equations used in local volatility models were developed by Bruno Dupire[1] 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.
It requires to pre-interpolate the data to obtain a continuum of traded prices and the choice of a type of interpolation.
[5] For a stock price, it follows the dynamics where for simplicity we assume zero dividend yield.
To price a call option with strike K on S one simply writes the payoff
The model produces a monotonic volatility smile curve, whose pattern is decreasing for negative
it follows that the asset S is allowed to take negative values with positive probability.
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.
[7] The starting point is the basic Black Scholes formula, coming from the risk neutral dynamics
The original model has a regularization of the diffusion coefficient in a small initial time interval
has a unique strong solution whose marginal density is the desired mixture
The same convex combination applies also to several option greeks like Delta, Gamma, Rho and Theta.
, and a possible shift parameter, allows one to reproduce most market smiles.
[6][12] In the mixture dynamics model, one can show that the resulting volatility smile curve will have a minimum for K equal to the at-the-money-forward price
's in the mixture components that are time dependent, so as to calibrate the smile term structure.
[10] An extension of the model where the different mixture densities have different means has been studied,[12] 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, [13] reconciling modelling of single assets smile with the smile on an index of these assets.
Time-invariant local volatilities are supposedly inconsistent with the dynamics of the equity index implied volatility surface,[15] but see Crepey (2004),[16] 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.
[18] 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.
[19] 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.