Local volatility

A local volatility model, in mathematical finance and financial engineering, is one that treats volatility as a function of both the current asset level S t {displaystyle S_{t}} and of time t {displaystyle t} . As such, a local volatility model is a generalisation of the Black-Scholes model, where the volatility is a constant (i.e. a trivial function of S t {displaystyle S_{t}} and t {displaystyle t} ). A local volatility model, in mathematical finance and financial engineering, is one that treats volatility as a function of both the current asset level S t {displaystyle S_{t}} and of time t {displaystyle t} . As such, a local volatility model is a generalisation of the Black-Scholes model, where the volatility is a constant (i.e. a trivial function of S t {displaystyle S_{t}} and t {displaystyle t} ). In mathematical finance, the asset St that underlies a financial derivative, is typically assumed to follow a stochastic differential equation of the form where r t {displaystyle r_{t}} is the instantaneous risk free rate, giving an average local direction to the dynamics, and W t {displaystyle W_{t}} is a Wiener process, representing the inflow of randomness into the dynamics. The amplitude of this randomness is measured by the instant volatility σ t {displaystyle sigma _{t}} . In the simplest model i.e. the Black-Scholes model, σ t {displaystyle sigma _{t}} is assumed to be constant; in reality, the realized volatility of an underlying actually varies with time. 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 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 to denote the set of diffusion coefficients, σ t = σ ( S t , t ) {displaystyle sigma _{t}=sigma (S_{t},t)} , that are consistent with market prices for all options on a given underlying. This model is used to calculate exotic option valuations which are consistent with observed prices of vanilla options. The concept of a local volatility 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 key continuous-time equations used in local volatility models were developed by Bruno Dupire in 1994. Dupire's equation states There exist few known parametrisation of the volatility surface based on the Heston model (Schönbucher, SVI and gSVI) as well as their de-arbitraging methodologies.