Pricing Path-Dependent Options with Discrete Monitoring under Time-Changed Lévy Processes

Abstract

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Lévy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic function of the intertemporal joint distribution of the time-changed Lévy process. Using the derived representation of the characteristic function, we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader and lookback options, all of which are discretely monitored.

Key Words:

• Time-changed Lévy process
• intertemporal joint distribution
• multivariate characteristic function
• Fourier transform
• path-dependent option

Notes

1 We are not concerned with the implementation of numerical algorithms for the multivariate Fourier transform. However, we acknowledge that there are a number of efficient algorithms for computational finance in the existing literature. For example, see Section 5 in Griebsch and Wystup (2011).

Additional information

Funding

Akira Yamazaki acknowledges the financial support of JSPS KAKENHI [Grant Number 24830087]. His work is also supported by the Research Institute for Innovation Management.