Weighted Local Time for Fractional Brownian Motion and Applications to Finance

Abstract

A Meyer-Tanaka formula involving weighted local time is derived for fractional Brownian motion and geometric fractional Brownian motion. The formula is applied to the study of the stop-loss-start-gain (SLSG) portfolio in a fractional Black-Scholes market. As a consequence, we obtain a fractional version of the Carr-Jarrow decomposition of the European call and put option prices into their intrinsic and time values.

Key Words:

• Asymptotic behavior
• Chaos expansion
• Fractional Brownian motions
• Local time

Mathematics Subject Classification:

• Primary 60H05
• 60H10

Acknowledgment

We are grateful to Knut Aase, Michel Emery, and John van der Hoek for their invaluable discussions. Partially supported by the National Science Foundation under Grant No. EPS-9874732 and matching support from the State of Kansas. Salopek's work is partially supported by NSERC Canada grant number 203232-98 York University.