Pricing Derivatives With Barriers in a Stochastic Interest Rate Environment
Journal of Economic Dynamics and Control, Vol. 32, No. 9, pp. 2903-2938, September 2008
32 Pages Posted: 26 Mar 2005 Last revised: 14 Apr 2010
Date Written: June 1, 2007
This paper develops a general valuation approach to price barrier options when the term structure of interest rates is stochastic. These products' barriers may be constant or stochastic, in particular we examine the case of discounted barriers (at the instantaneous interest rate). So, in practice, we extend Rubinstein and Reiner (1991), who give closed-form formulas for pricing barrier options in a Black and Scholes context, to the case of a Vasicek modeling of interest rates. We are therefore in the situation of pricing barrier options semi-explicitly or explicitly (depending on the shape of the barrier) with stochastic Vasicek interest rates. The model is illustrated with a specific contract, an up and out call with rebate, hence a typical barrier option. This example is merely here to show how any standard barrier option can be priced and its Greeks be obtained in such a context. The validity of the approximation is analyzed and the sensitivity to the barrier level and to discretization schemes are also derived.
Keywords: Change of Numéraire, HJM model, Barrier Option, Markovian Approximation
JEL Classification: C60, G10
Suggested Citation: Suggested Citation