Bismut–Elworthy–Li formula for subordinated Brownian motion applied to hedging financial derivatives

Abstract

The objective of the paper is to extend the results in Fournié, Lasry, Lions, Lebuchoux, and Touzi (1999), Cass and Fritz (2007) for continuous processes to jump processes based on the Bismut–Elworthy–Li (BEL) formula in Elworthy and Li (1994). We construct a jump process using a subordinated Brownian motion where the subordinator is an inverse $ \alpha $-stable process $ (L_t)_{t \ge 0} $ with $ (0,\, 1] $. The results are derived using Malliavin integration by parts formula. We derive representation formulas for computing financial Greeks and show that in the event when $ L_t \equiv t $, we retrieve the results in Fournié et al. (1999). The purpose is to by-pass the derivative of an (irregular) pay-off function in a jump-type market by introducing a weight term in form of an integral with respect to subordinated Brownian motion. Using MonteCarlo techniques, we estimate financial Greeks for a digital option and show that the BEL formula still performs better for a discontinuous pay-off in a jump asset model setting and that the finite-difference methods are better for continuous pay-offs in a similar setting. In summary, the motivation and contribution of this paper demonstrates that the Malliavin integration by parts representation formula holds for subordinated Brownian motion and, this representation is useful in developing simple and tractable hedging strategies (the Greeks) in jump-type derivatives market as opposed to more complex jump models.

Keywords

• stable distributions
• malliavin calculus
• integration by parts
• subordinated brownian motion
• bismut–elworthy–li formula