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

See all articles by Carole Bernard

Carole Bernard

Grenoble Ecole de Management; Vrije Universiteit Brussel (VUB)

Olivier Le Courtois

EM Lyon (Ecole de Management de Lyon) - Department of Economics, Finance, Control

Francois Quittard-Pinon

EMLYON Business School

Date Written: June 1, 2007

Abstract

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

Bernard, Carole and Le Courtois, Olivier Arnaud and Quittard-Pinon, Francois, Pricing Derivatives With Barriers in a Stochastic Interest Rate Environment (June 1, 2007). Journal of Economic Dynamics and Control, Vol. 32, No. 9, pp. 2903-2938, September 2008. Available at SSRN: https://ssrn.com/abstract=686705

Carole Bernard

Grenoble Ecole de Management ( email )

12, rue Pierre Sémard
Grenoble Cedex, 38003
France

Vrije Universiteit Brussel (VUB) ( email )

Pleinlaan 2
http://www.vub.ac.be/
Brussels, 1050
Belgium

Olivier Arnaud Le Courtois

EM Lyon (Ecole de Management de Lyon) - Department of Economics, Finance, Control ( email )

23, av. Guy de Collongue
69134 Ecully Cedex
France

Francois Quittard-Pinon (Contact Author)

EMLYON Business School ( email )

23, Avenue Guy de Collongue
69134, Ecully
France

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