Static arbitrage bounds on basket option prices

We consider the problem of computing upper and lower bounds on the price of an European basket call option, given prices on other similar options. Although this problem is hard to solve exactly in the general case, we show that in some instances the upper and lower bounds can be computed via simple closed-form expressions, or linear programs. We also introduce an efficient linear programming relaxation of the general problem based on an integral transform interpretation of the call price function. We show that this relaxation is tight in some of the special cases examined before.

[1]  A. Ramm,et al.  The RADON TRANSFORM and LOCAL TOMOGRAPHY , 1996 .

[2]  Ioana Popescu,et al.  On the Relation Between Option and Stock Prices: A Convex Optimization Approach , 2002, Oper. Res..

[3]  Douglas T. Breeden,et al.  Prices of State-Contingent Claims Implicit in Option Prices , 1978 .

[4]  S. Helgason.  The Radon Transform , 1980 .

[5]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[6]  Jean-Paul Laurent,et al.  Building a Consistent Pricing Model from Observed Option Prices , 1998 .

[7]  A. d'Aspremont.  A Harmonic Analysis Solution to the Static Basket Arbitrage Problem , 2003, math/0309048.

[8]  C. Berg,et al.  Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions , 1984 .

[9]  M. C. Quenez,et al.  Programmation dynamique et évaluation des actifs contingents en marché incomplet , 1991 .

[10]  D. Brigo,et al.  Interest Rate Models , 2001 .

[11]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[12]  D. Brigo,et al.  Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit , 2001 .

[13]  Tai-Ho Wang,et al.  Static-arbitrage upper bounds for the prices of basket options , 2005 .

[14]  Tai-Ho Wang,et al.  Sharp Upper and Lower Bounds for Basket Options , 2004 .

[15]  Alexandre d'Aspremont.  Interest rate model calibration using semidefinite Programming , 2003, ArXiv.

[16]  D. Duffie.  Dynamic Asset Pricing Theory , 1992 .

[17]  Kenneth O. Kortanek,et al.  Semi-Infinite Programming: Theory, Methods, and Applications , 1993, SIAM Rev..

[18]  N. Karoui,et al.  Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market , 1995 .