Journal of Mathematical Finance, 2015, 5, 286-303
Published Online August 2015 in SciRes. http://www.scirp.org/journal/jmf
http://dx.doi.org/10.4236/jmf.2015.53025
How to cite this paper: Offen, E.R. and Lungu, E.M. (2015) Pricing a European Option in a Black-Scholes Quanto Market
When Stock Price Is a Semimartingale. Journal of Mathematical Finance, 5, 286-303.
http://dx.doi.org/10.4236/jmf.2015.53025
Pricing a European Option in a
Black-Scholes Quanto Market When
Stock Price Is a Semimartingale
E. R. Offen, E. M. Lungu
University of Botswana, Gaborone, Botswana
Email: offen@mopipi.ub.bw , lunguem@mopipi.ub.bw
Received 18 April 2015; accepted 26 July 2015; published 30 July 2015
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
We look at the price of the European call option in a quanto market defined on a filtered probability
space ( ) , , , Ω when the exchange rate is being modeled by the process { } =
0
exp
t t
E E H
where
t
H is a semimartingale. Precisely we look at an investor in a Sterling market who intends
to buy a European call option in a Dollar market. The market consists of a Dollar bond, Sterling
bond and and Sterling risky asset. We first of all convert the Sterling assets by using the exchange
rate
t
E and later on derive an integro-differential equation that can be used to calculate the
price on the option.
Keywords
Semimartingale, Hedging, Arbitrage, Contingent Claim
1. Introduction
This paper considers the European call option in the Black-Scholes market when the exchange-rate is a semimar-
tingale. Specifically, we consider a problem of a Dollar investor seeking to invest in a Sterling market. Theory
of exchange rates has been widely discussed (see [1]-[4]). Exchange rates change with time due to a number of
factors, such as changes in fiscal and monetary policies, interest rate differentials between two countries usually
resulting in revaluation or devaluation of currency. The main challenge is to construct a model which captures
the dynamics of exchange rate and its effect when investments are made in different currencies. A number of
models have been developed which are being modified to accommodate reality. Generally, exchange rate
models fall into two major categories: Those that treat the dynamics of exchange rate as a continuous process