Malliavin Calculus for the Estimation of the U.S. Dollar/Euro Exchange Rate When the Volatility is Stochastic

Malliavin Calculus for the Estimation of the U.S. Dollar/Euro Exchange Rate When the Volatility is Stochastic

Ahmed Abutaleb (Cairo University, Egypt) and Michael Papaioannou (International Monetary Fund, USA)
Copyright: © 2006 |Pages: 31
DOI: 10.4018/978-1-59140-881-9.ch005
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Abstract

The tendency of exchange rates to fluctuate markedly and regularly is often referred as currency market volatility. The extent of currency market volatility is a major element of market risk. For financial transactions, volatility represents both costs and profit opportunities. Increased currency market volatility implies higher currency option premia and, therefore, higher hedging costs for investors and importers/exporters. However, for banks and other investment houses dealing in options, an increase in option prices may contribute to higher profits. It has been well established that the volatility of exchange rates changes with time. In recent years, various stochastic volatility models have been proposed in the literature that try to capture the exchange-rate volatility dynamics. In turn, several methods have been developed to estimate the parameters of such stochastic volatility models, with varying results. In this chapter, we propose another method for the estimation of the parameters of an exchange rate function when the volatility follows a stochastic process. Stochastic volatility is represented by a geometric Brownian motion. Using Malliavin calculus, we are able to find an explicit expression for the likelihood function of the observations. Numerical integration methods (Monte-Carlo simulations) and numerical optimization methods (generic algorithms) enable us to find an estimate for the unknown parameters and the volatility. This estimation method is then applied to the U.S. dollar/euro exchange rate. Specifically, first we formulate a U.S. dollar/euro exchange rate equation with a stochastic volatility model. We assume that the observed U.S. dollar/euro exchange rate follows a stochastic differential equation with random volatility, while the unobserved volatility follows a different stochastic differential equation. Then, we obtain the likelihood function of the observations by applying Malliavin calculus. The estimation of the unknown parameters is achieved through the maximization of the likelihood function. Using weekly U.S. dollar/euro exchange rates for the period April 28, 2000, to March 26, 2001, we obtain estimates of the parameters of the U.S. dollar/euro exchange rate function (i.e., the constant of the drift) and the assumed stochastic volatility model (i.e., the constants of the diffusion process). Application of the estimated model to out-of-sample data for the U.S. dollar/euro exchange rate shows a significantly high accuracy of the proposed method, as indicated by the very low root mean square error for the estimated exchange rate. This method can also be applied to other models of financial variables that follow similar processes.

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Table of Contents
Acknowledgments
Yutaka Kurihara , Sadayoshi Takaya , Nobuyoshi Yamori
Chapter 1
Irene Henriques, Perry Sadorsky
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Investment in IT and the Business Performance of Financial Companies
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Chapter 2
Nobuyoshi Yamori, Kozo Harimaya, Yoshihiro Asai
Although Japanese banks have suffered from huge, non-performing loans since the burst of the bubble, they have invested as much as possible in... Sample PDF
Recent Development of Information Technology in Japanese Banks
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Chapter 3
Yutaka Kurihara, Shigeaki Ohtsuka
Economic activity is always accompanied by payment. Payment systems, which are the subject of much recent discussion, are an indispensable part of... Sample PDF
Payment Systems of Financial Institutions: Current State and Future Prospects
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Chapter 4
Anita Ghatak
In this chapter, we assess the contribution of financial development to saving and economic growth in the UK in the 20th century. Financial... Sample PDF
Financial Innovation and Economic Growth: Some Further Evidence from the UK, 1900-2003
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Chapter 5
Ahmed Abutaleb, Michael Papaioannou
The tendency of exchange rates to fluctuate markedly and regularly is often referred as currency market volatility. The extent of currency market... Sample PDF
Malliavin Calculus for the Estimation of the U.S. Dollar/Euro Exchange Rate When the Volatility is Stochastic
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Chapter 6
Sadayoshi Takaya
This chapter focuses on the function of international currencies as foreign exchange vehicles, which has a character of the network externality. On... Sample PDF
Evolution of the Euro and Currency Competition in the Global ICT Age
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Chapter 7
Argyrios Volis
The purpose of this study is to explore the “revolution” that was caused by the rapid spread of information technology towards the development and... Sample PDF
Co-Integration of the International Capital Markets with the Use of Information Technology: The Case of Europe
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Chapter 8
Sadayoshi Takaya
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International Capital Movements, Currency Crisis, and ICT Innovation
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Chapter 9
Masayuki Susai
Highly developed IT technology can be the source of volatility spillover between markets located in other countries. In this chapter, we investigate... Sample PDF
Volatility Spillover Structure of Stock and Foreign Exchange Market between Korea, Japan, and Hong Kong
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Chapter 10
Andrew Marks
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Trade Liberalization and International Performance of Australian Manufacturing Industries and Its
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Chapter 11
Nobuyoshi Yamori, Nobuyoshi Nishigaki
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Recent Developments of Digital Cash Projects in Japan
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Chapter 12
Mariusz K. Krawczyk
Despite of amazing progress in information technology that has taken place in recent years, the electronic money failed to live to the expectations... Sample PDF
Money is What Money Does: Prospects for an Electronic Money Payment System in Japan
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Chapter 13
Takeshi Kobayashi
This chapter examined the relationship between Japanese life insurers’ investment in IT stocks and conventional financial statistics, such as ROE... Sample PDF
Investment in IT Stocks by Japanese Life Insurers
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Chapter 14
Waranya Atchariyachanvanich, Kanokwan Atchariyachanvanich
This chapter reviews the roles of information technology (IT) from two perspectives. First, from the macroeconomic perspective, the IT revolution... Sample PDF
The Roles of IT in the Conduct of Modern Monetary Policy
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Chapter 15
Yutaka Kurihara, Akio Fukushima
The use of local currency has been spreading gradually since the 1990s. It has been introduced by nonprofit organizations (NPOs) and similar groups... Sample PDF
The Role and Future of Local Currency and IT
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Chapter 16
Carolyn Currie
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Integrity and Security in the E-Century
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Chapter 17
Takashi Kubota
This chapter introduces the two newly emerging issues in the C2C and B2B area in the Japanese IT laws: (a) anti-fraud measures in Internet auctions... Sample PDF
Legal Concerns Against Auctions and Securities Conventions: A Japanese Perspective
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Glossary of Terms
About the Authors