### Abstract

After the celebrated Black-Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics motivated numerous works during the 1980s and 1990s. In particular, a lot of attention has been paid to stochastic volatility models in which the volatility is randomly fluctuating driven by an additional Brownian motion. We have shown in [Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000; Internat. J. Theoret. Appl. Finance, 13 (2000), pp. 101-142] that, in the presence of a separation of time scales between the main observed process and the volatility driving process, asymptotic methods are very efficient in capturing the effects of random volatility in simple robust corrections to constant volatility formulas. From the point of view of PDEs, this method corresponds to a singular perturbation analysis. The aim of this paper is to deal with the nonsmoothness of the payoff function inherent to option pricing. We present the case of call options for which the payoff function forms an angle at the strike price. This case is important since these are the typical instruments used in the calibration of pricing models. We establish the pointwise accuracy of the corrected Black-Scholes price by using an appropriate payoff regularization which is removed simultaneously as the asymptotics is performed.

Original language | English (US) |
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Pages (from-to) | 1648-1665 |

Number of pages | 18 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 63 |

Issue number | 5 |

DOIs | |

State | Published - Jun 2003 |

### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Keywords

- Mathematical finance
- Option pricing
- Singular perturbations
- Stochastic volatility

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## Cite this

*SIAM Journal on Applied Mathematics*,

*63*(5), 1648-1665. https://doi.org/10.1137/S0036139902401550