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Computation of deterministic volatility surfaces

77 citations0/0 files storedThe Journal of Computational Finance

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Stochastic processes and financial applications Financial Markets and Investment Strategies Financial Risk and Volatility Modeling

Abstract

ABSTRACT The 'volatility smile' is one of the well-known biases of Black-Scholes models for pricing options. In this paper, the authors introduce a robust method of reducing this bias by pricing subject to a deterministic functional volatility ó = ó (S, t). This instantaneous volatility is chosen as a spline whose weights are determined by a regularized numerical strategy that approximately minimizes the difference between Black-Scholes vanilla prices and known market vanilla prices over a range of strikes and maturities; these Black-Scholes prices are calculated by solving the relevant partial differential equation numerically using finite element methods. The instantaneous volatility generated from vanilla options can be used to price exotic options where the skew and term structure of volatility are important, and the application to barrier options is illustrated.

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References

1973The Pricing of Options and Corporate Liabilities 1993A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options 1976Option pricing when underlying stock returns are discontinuous 1987The Pricing of Options on Assets with Stochastic Volatilities 1976The valuation of options for alternative stochastic processes 1994Implied Binomial Trees 1987The Pricing of Options on Assets with Stochastic Volatilities 1996Recovering Probability Distributions from Option Prices

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2006Spectral calibration of exponential Lévy models 2006Retrieving Lévy Processes from Option Prices: Regularization of an Ill-posed Inverse Problem 2015ESTIMATION OF STOCHASTIC VOLATILITY MODELS BY NONPARAMETRIC FILTERING 2007Retrieving Lévy processes from option prices : regularization of an ill-posed inverse problem 2001Dynamic hedging with a deterministic local volatility function model 2006

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