Path integral approach to the pricing of timer options with the Duru-Kleinert time transformation

TL;DR

This paper introduces a path integral method using Duru-Kleinert time transformation to derive closed-form pricing formulas for timer options under stochastic volatility models, including 3/2 and Heston models.

Contribution

It applies the Duru-Kleinert path integral approach to derive new closed-form pricing formulas for timer options in stochastic volatility frameworks.

Findings

Closed-form formulas for perpetual timer call options.

Closed-form formulas for finite time-horizon timer options.

Application of Morse and Kratzer potentials in pricing models.

Abstract

In this paper, a time substitution as used by Duru and Kleinert in their treatment of the hydrogen atom with path integrals is performed to price timer options under stochastic volatility models. We present general pricing formulas for both the perpetual timer call options and the finite time-horizon timer call options. These general results allow us to find closed-form pricing formulas for both the perpetual and the finite time-horizon timer options under the 3/2 stochastic volatility model as well as under the Heston stochastic volatility model. For the treatment of timer option under the 3/2 model we will rely on the path integral for the Morse potential, with the Heston model we will rely on the Kratzer potential.