Lecture 1: Stochastic Volatility and Local Volatility models
www.math.ku.dk/~rolf/.../Gatheral.1.pdf
University of Copenhagen
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by J Gatheral - Cited by 38 - Related articles
given by models based on Black-Scholes assumptions can be wildly wrong and dealers in such options are motivated to find models which can take the volatility
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local volatility surface - OpenGamma Developers
developers.opengamma.com/.../Local-Volatility-OpenGamma.pdf
by R White - Related articles
We present details of computing a local volatility surface from market data, then numer- ... this can also be solved analytically by the Black-Scholes-Merton option
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Black-Scholes) and the volatility surface to calculate the Greeks. ... such model, the local volatility model is probability the simplest extension of Black-Scholes.
Beyond Black-Scholes
www.columbia.edu/.../LocalStochasticJumpDiffusio...
Columbia University
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A Brief Review of Dupire’s Work For a given expiration T and current stock price S0 , the collection {C (S0,K,T);K ∈(0,∞)}of undiscounted option prices of different strikes yields the risk neutral density function ϕ of the final spot ST through the relationship C (S0,K,T) =Z∞ K dST ϕ(ST,T;S0) (ST −K) Differentiate this twice with respect to K to obtain
ϕ(K,T;S0) =
∂2C ∂K2 so the Arrow-Debreu prices for each expiration may be recovered by twice differentiating the undiscounted option price with respect to K. This process will be familiar to any option trader as the construction of an (infinite size) infinitesimally tight butterfly around the strike whose maximum payoff is one. Given the distribution of final spot prices ST for each time T conditional on some starting spot price S0, Dupire shows that there is a unique risk neu- tral diffusion process which generates these distributions.
ϕ(K,T;S0) =
∂2C ∂K2 so the Arrow-Debreu prices for each expiration may be recovered by twice differentiating the undiscounted option price with respect to K. This process will be familiar to any option trader as the construction of an (infinite size) infinitesimally tight butterfly around the strike whose maximum payoff is one. Given the distribution of final spot prices ST for each time T conditional on some starting spot price S0, Dupire shows that there is a unique risk neu- tral diffusion process which generates these distributions.
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