Foreign Exchange Option Symmetry

by Kholodnyi, Valery A.; Price, John F.

• ISBN13: 9789810233624
• ISBN10: 9810233620
• Format: Hardcover
• Copyright: 1998-03-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

This book studies the actual financial phenomena underlying the evaluation of financial derivatives, which is today virtually identified with and even replaced by the study of the mathematical aspects of stochastic calculus as a model for such phenomena. It adopts the view that the study of financial phenomena is on the brink of a revolution similar to that of quantum physics in the 1920s. History has shown that virtually all the major revolutions in physics were made through recognizing the presence of an inherent symmetry in underlying phenomena. In this volume, a fundamental symmetry in a foreign exchange market that associates financially equivalent options on opposite sides of the market is introduced. This symmetry holds in a general foreign exchange market environment. In particular, it requires no assumptions to be made on the nature of a probability distribution for exchange rates -- not even on the existence of such a distribution. The practical applications of the symmetry are significant and far reaching. They range from the detection of a new type of true arbitrage and a screen for consistency of option pricing models, to the reduction of the cost of software development. The symmetry introduced is not restricted to foreign exchange markets but is also valid for any financial markets.

Table of Contents

Preface	v	(12)
List of Figures	xvii
1 Introduction	1	(12)
I Financial Matters	13	(76)
2 Market Environment	15	(18)
2.1 The Framework of a Market Environment	16	(3)
2.2 The Binomial Market Environment	19	(8)
2.3 The Black and Scholes Market Environment	27	(3)
2.4 Relationship Between the Binomial and Black and Scholes Market Environments	30	(3)
3 Symmetry in a Foreign Exchange Market	33	(22)
3.1 The Kelvin Transform	34	(5)
3.2 Symmetry for European Options	39	(5)
3.3 The Intervention Condition	44	(4)
3.4 Symmetry for Bermudan Options	48	(3)
3.5 Symmetry for American Options	51	(4)
4 Further Symmetries	55	(16)
4.1 Symmetry for Barrier Options	56	(8)
4.2 Symmetry for Greek Letters	64	(2)
4.3 Symmetry in the Exchange-Rate Homogeneous Market Environment	66	(5)
5 Options with Consistently Smoothed Payoffs	71	(12)
5.1 The Importance of Smooth Payoffs	71	(2)
5.2 The Continuous Strike Range Call and Put Options	73	(3)
5.3 The Continuous Strike Range Call and Put Options and the Symmetry	76	(1)
5.4 The Consistently Smoothed Call and Put Payoffs and the Symmetry	77	(3)
5.5 Symmetry for Options with the Consistently Smoothed Call and Put Payoffs	80	(3)
6 Applications	83	(6)
6.1 Detecting Arbitrage	83	(1)
6.2 Selecting a Financially Equivalent Portfolio	83	(1)
6.3 Detecting Inconsistent Option Pricing Models	84	(1)
6.4 Instant Valuation of Foreign Counterparts	84	(1)
6.5 Instant Evaluation of Greek Letters	85	(1)
6.6 Approximation Schemes	85	(1)
6.7 Reduced Cost of Software Development	86	(1)
6.8 Guiding the Choice of Options with Improved Hedging Features	87	(1)
6.9 Validity of the Symmetry in any Financial Market	87	(2)
II Mathematical Matters	89	(38)
7 Validity of the Symmetry Relationships for European Options	91	(6)
7.1 Direct Proof of the Symmetry Relationships for European Call and Put Options	91	(1)
7.2 European Options: General Case	92	(1)
7.3 European Call and Put Options	93	(1)
7.4 Symmetry in the Binomial Market Environment	94	(1)
7.5 Symmetry in the Black and Scholes Market Environment	95	(2)
8 Validity of the Symmetry Relationships for Bermudan and American Options	97	(18)
8.1 Bermudan Options: General Case	97	(3)
8.2 Bermudan Call and Put Options	100	(2)
8.3 Bermudan Options in the Restricted Binomial Market Environment	102	(1)
8.4 American Options: General Case	103	(2)
8.5 American Call and Put Options	105	(1)
8.6 The Semilinear Evolution Equation for American Options	106	(4)
8.7 Approximation of American Options by Bermudan Options	110	(3)
8.8 Symmetry at all Levels of Approximation	113	(2)
9 Validity of the Symmetry Relationships for Barrier Options	115	(8)
9.1 Barrier Options: General Case	115	(3)
9.2 Barrier Call and Put Options	118	(5)
10 Validity of the Symmetry Relationships for Options with Consistently Smoothed Payoffs	123	(4)
10.1 The Consistently Smoothed Call and Put Payoffs	123	(2)
10.2 Options with the Consistently Smoothed Call and Put Payoffs	125	(2)
Bibliography	127	(3)
Index	130