Black–Scholes in a CEV random environment

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Abstract

Classical (Itô diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential Lévy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see Tankov in Pricing and hedging in exponential Lévy models: review of recent results. Paris-Princeton Lecture Notes in Mathematical Finance, Springer, Berlin, 2010 for an overview), and more recently rough volatility models (Alòs et al. in On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch 11(4):571–589, 2007, Fukasawa in Asymptotic analysis for stochastic volatility: martingale expansion. Finance Stoch 15:635–654, 2011). We suggest here a different route, randomising the Black–Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Lévy models and fractional stochastic volatility models.

Keywords

Volatility asymptotics Random environment Forward smile Large deviations 

Mathematics Subject Classification

60F10 91G99 41A60 

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK

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