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he success of pricing models is often measured by the extent to which closed-form solutions of the Black-Scholes type are available for the
 

Summary: T
he success of pricing models is often measured by the extent to which
closed-form solutions of the Black-Scholes type are available for the
basic payouts. Analytic tractability is often crucial for the calibration
to market data and is helpful for implementing numerical algorithms for
exotics. Extensions of the Black-Scholes formula have been directed to-
wards three main model classes: local volatility models, which postulate a
deterministic relationship between the underlying state variable, time and
volatility; stochastic volatility models, which assume that the volatility fol-
lows a distinct but correlated process; and jump models, which can be re-
garded as limits of stochastic volatility models whereby the volatility can
occasionally be singularly large at some points in time, enough to cause
the underlying sample path to have discontinuous jumps.
As Dupire (1994) demonstrated, state-dependent volatility models are
able to reproduce arbitrage-free implied volatility surfaces. However, robust
estimations require either regularisations or settling on a parametric form for
the local volatility such as the constant elasticity of variance model in Cox
& Ross (1976), the quadratic volatility models in Rady (1997) and the more
comprehensive hypergeometric Brownian motions in Albanese et al (2001).
AwidelyadoptedjumpmodelisMadan'svariance-gammamodel(seeMadan,

  

Source: Albanese, Claudio - Department of Mathematics, King's College London

 

Collections: Mathematics