Singular Perturbations in Option Pricing

Abstract

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 has been the motivation of numerous works during the Eighties and Nineties. In particular, a lot of attention has been paid to stochastic volatility models where the volatility is randomly fluctuating driven by an additional Brownian motion. We have shown in [2,3] 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 partial differential equations 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.1. Introduction. Stochastic volatility models in financial mathematics can be thought of as a Brownian-type particle (the stock price) moving in an environment where the diffusion coefficient is randomly fluctuating in time according to some ergodic (mean-reverting) diffusion process. We then have two Brownian motions, one driving the motion of the particle and the other driving the fluctuations of the medium. In the context of Physics there is no natural correlation between these two Brownian motions since they do not "live" in the same space. In the context of Finance they jointly define the dynamics of the stock price under its physical probability measure or an equivalent risk-neutral martingale measure. Correlation between them is perfectly natural. There are economic arguments for a negative correlation or leverage effect between stock price and volatility shocks, and from common experience and empirical studies, asset prices tend to go down when volatility goes up.The diffusion equation appears as a contingent claim pricing equation, its terminal condition being the payoff of the claim. We refer to [5] or [6] for surveys on stochastic volatility. When volatility is fast mean-reverting, on a time-scale smaller than typical maturities, one can perform a singular perturbation analysis of the pricing PDE. As we have shown in [2], this expansion reveals a first correction made of two terms: one is directly associated with the market price of volatility risk and the other is proportional to the correlation coefficient between the two Brownian motions involved. We refer to [2] for a detailed account of evidence of a fast scale in volatility and the use of this asymptotics to parametrize the evolution ...

Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration

“…The results from the smooth case proof are used in Appendix B, where we establish the accuracy of the approximation for options with payoffs which may have a finite number discontinuities in h or its derivatives. The proof that is given in Appendix B involves a regularization argument, which was used in [6] to establish the accuracy of the first order approximation with only a fast factor of volatility. 10.…”

“…6 Let χ(y, z) be a function that is at most polynomially growing, with χ(·, z) = 0 for all z. Then for q < 1 and z fixed, there existε > 0 and a polynomial C(y) such that…”

“…Proof This is an improved version of Lemma 5.2 in [6], where the proof consisted in an explicit computation of ∂ k η P Δ 0,0 in the case of a call payoff. Here, we aim at estimates which are uniform in Δ.…”

Calibration of a Stock's Beta Using Options Prices

“…We can deduce that Z S,ǫ is at most linearly growing in |y| and |z| and bounded uniformly in x, and we have: The demonstration of the accuracy of the approximation for a non smooth payoff h (as in the case of a call option) is derived in [14].…”

A Remark on a Singular Perturbation Method for Option Pricing Under a Stochastic Volatility Model

“…The theoretical justification of the first order approximation is argued in Fouque et al (2003a). The mathematical validity for the higher order approximation is given by theorem 1 in Conlon and Sullivan (2005).…”