On the short-maturity behaviour of the implied volatility skew for random strike options and applications to option pricing approximation

Abstract

In this paper, we propose a general technique to develop first- and second-order closed-form approximation formulas for short-maturity options with random strikes. Our method is based on a change of numeraire and on Malliavin calculus techniques, which allow us to study the corresponding short-maturity implied volatility skew and to obtain simple closed-form approximation formulas depending on the derivative operator. The numerical analysis shows that these formulas are extremely accurate and improve some previous approaches for two-asset and three-asset spread options such as Kirk’s formula or the decomposition method presented in Alòs et al. [Energy Risk, 2011, 9, 52–57]. This methodology is not model-dependent, and it can be applied to the case of random interest rates and volatilities.

Keywords:

• Spread options
• Girsanov’s Theorem
• Kirk’s formula
• Malliavin calculus
• Derivative operator in the Malliavin calculus sense
• Skorohod integral

JEL Classifications:

• G12
• G13
• 91B28
• 91B70
• 60H07

Acknowledgements

The authors thank the referees for their careful reading of the paper and for their useful suggestions for improvements.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Supported by the [grant number ECO2011-288755] and [grant number MEC FEDER MTM 2009-08869]. Partially supported by the CONACyT [grant number 220303].