Abstract
We develop two new pricing formulae for European options. The purpose of these formulae is to better understand the impact of each term of the model, as well as improve the speed of the calculations. We consider the SABR model (with β=1) of stochastic volatility, which we analyze by tools from Malliavin Calculus. We follow the approach of Alos et al. (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call “Dyson series on the return’s idiosyncratic noise” yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor’s (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion. The second approach, which we call “Dyson series on the common noise” necessitates the calculation of only a one-dimensional integral, but the formula is more complex. This research consisted of both analytical derivations and numerical calculations. The latter show that our formulae are in general more exact, yet more time-consuming to calculate, than the first order expansion of Hagan et al. (2002).
Highlights
European options are traditionally priced and hedged by Black-Scholes [1] (1973) model, one of the natural extensions of the Black-Scholes model to make volatility stochastic
We follow the approach of Alòs et al (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation
The Hull-White formula is of practical use for Monte Carlo simulation of prices in a correlated stochastic volatility model since only one Brownian motion path has to be generated
Summary
European options are traditionally priced and hedged by Black-Scholes [1] (1973) model, one of the natural extensions of the Black-Scholes model to make volatility stochastic. In the first approach, which we call “Dyson series in the return’s idiosyncratic noise’’, we first apply a Dyson series in the idiosyncratic noise term Z and apply Yor’s [8] formula (1992) for the joint density of a Brownian motion and the time-integral of geometric Brownian motion to integrate with respect to the common noise term W. We note that Yor’s formula is used for pricing Asian options, but it is ideally suited to analyze realized volatility in the SABR model with β = 1, since volatility is a geometric Brownian motion. The first approach yields a complete series expansion but necessitates the calculation of a 7-dimensional integral Two of these dimensions come from the analytical expression of the joint density of a Brownian motion and the.
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