Barrier Option Pricing by Branching Processes

Abstract

This paper examines the pricing of barrier options when the price of the underlying asset is modeled by a branching process in a random environment (BPRE). We derive an analytical formula for the price of an up-and-out call option, one form of a barrier option. Calibration of the model parameters is performed using market prices of standard call options. Our results show that the prices of barrier options that are priced with the BPRE model deviate significantly from those modeled assuming a lognormal process, despite the fact that for standard options, the corresponding differences between the two models are relatively small.

Large deviations for a Poisson random indexed branching process

“…Results in this direction can be found for instance in Mitov and Mitov (2007, 2011), Mitov et al (2010, 2009a, Williams (2001) and Wu (2012).…”

Large deviations for Lotka-Nagaev estimator of a randomly indexed branching process

“…In applied direction, a formula for the fair price of an European call option was derived in [13]. Later on, [14] obtained a formula for the fair price of an up-and-out call option.…”

Limit theorems for a supercritical Poisson random indexed branching process

“…Results on the subcritical case were presented in [10]. In a more applied direction, Mitov and Mitov [7] derived an equation for the fair price of a European call option based on modeling the underlying stock price by this process with a Poisson subordinator and with a geometric offspring distribution. Subsequently, an equation for the fair price of an up-and-out call option, a particular form of a barrier option, was derived in [9].…”