An approximation formula for basket option prices under local stochastic volatility with jumps: An application to commodity markets

Abstract

a b s t r a c tThis paper develops a new approximation formula for pricing basket options in a localstochastic volatility model with jumps. In particular, the model admits local volatility functions and jump components in not only the underlying asset price processes, but also the volatility processes. To the best of our knowledge, the proposed formula is the first one which achieves an analytical approximation for the basket option prices under this type of the models.Moreover, in numerical experiments, we provide approximate prices for basket options on the WTI futures and Brent futures based on the parameters through calibration to the plain-vanilla option prices, and confirm the validity of our approximation formula.

Pricing Average and Spread Options Under Local-Stochastic Volatility Jump-Diffusion Models

“…The conditional expectations in (47) are calculated based on the formulas in Section B of the online appendix [31], as well as formulas in Appendix B of Shiraya -Takahashi [34].…”

“…Hereafter, we call this approximation the partial 3rd order (asymptotic) expansion. Then, for the numerical evaluation of conditional expectations appearing in E[f i,T ], we can use formulas in Appendix B of Shiraya -Takahashi [34], as well as Lemma B.1 of the online appendix [31] which is an extension of [34] and Takahashi et al [42]. Finally, we summarize an approximate pricing formula for average call options as the following theorem.…”

“…The previous papers (e.g. [32], [33], [34] and [35]) did not provide such a consideration. Moreover, we confirmed the consistency between the theoretical asymptotic errors in Theorem 3.2 with Appendix C and the numerical errors in Figures from 3 to 8 as well as the ones in Figures 9 and 10 showing stress tests with regard to parameters.…”

“…• Derived asymptotic expansions for approximating average option prices under multi-dimensional general diffusion processes with jumps, which is a necessary setup for applications in practice. To obtain this formula, we need new conditional expectation formulae with some approximation techniques, which are not shown in previous papers: We note that although LSV with jumps models for basket option is considered in [34], the formula derived by [34] cannot be applied to pricing average options.…”

“…Li [26] introduced the closed-form likelihood expansions for jump-diffusion models as an application in statistics. More recently, Shiraya -Takahashi [34], [35] obtained an analytical approximation formula for basket option prices with local-stochastic volatility with jump diffusion model. However, they have not provided asymptotic error or justifications for the approximation of non-smooth functions, and their conditional expectation formulae can not be directly applied to pricing the average options.…”