Discrete Bismut formula: Conditional integration by parts and a representation for delta hedging process

Article type: Research Article

Authors: Akiyama, Naho^a | Yamada, Toshihiro^{a; b;}

Affiliations: [a] Hitotsubashi University, Tokyo, Japan | [b] Japan Science and Technology Agency (JST), Tokyo, Japan E-mail: toshihiro.yamada@r.hit-u.ac.jp

Correspondence: [*] Corresponding author: Toshihiro Yamada. Tel.: +81 42 580 8788; E-mail: toshihiro.yamada@r.hit-u.ac.jp

Abstract: The paper gives discrete conditional integration by parts formula using a Malliavin calculus approach in discrete-time setting. Then the discrete Bismut formula is introduced for asymmetric random walk model and asymmetric exponential process. In particular, a new formula for delta hedging process is obtained as an extension of the Malliavin derivative representation of the delta where the conditional integration by parts formula plays a role in the proof.

Keywords: Random walk, delta hedging, discrete Bismut formula, conditional integration by parts

DOI: 10.3233/RDA-202070

Journal: Risk and Decision Analysis, vol. 9, no. 1, pp. 1-9, 2023