Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Breeden, D.T., Litzenberger, R.H.: Price of state-contingent claims implicit in option prices. Journal of Business 51 (4), 621–651 (1978)
Brigo, D., Mercurio, F.: Interest rate models, theory and practice. Springer Finance, 2001
Bertsimas, D., Popescu, I.: On the relation between option and stock prices: a convex optimization approach. Operations Research 50 (2), 358–374 (2002)
Black, F., Scholes, M.: The pricing of options and corporate liabilities. Journal of Political Economy 81, 637–659 (1973)
Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, 2004
d'Aspremont, A.: Interest rate model calibration using semidefinite programming. Applied Mathematical Finance 10 (3), 183–213 (2003)
Duffie, D.: Dynamic asset pricing theory. 2nd ed., Princeton University Press, Princeton, N.J., 1996
El Karoui, N., Quenez, M.C.: Programmation dynamique et évaluation des actifs contingents en marchés incomplets. Comptes Rendus de l'Académie des Sciences de Paris, Série I 313, 851–854 (1991)
El Karoui, N., Quenez, M.C.: Dynamic programming and pricing of contingent claims in an incomplete market. SIAM Journal of Control and Optimization 33, 29–66 (1995)
Helgason, S.: The Radon transform. 2nd ed., Birkhauser, Boston, 1999 Progress in mathematics (Boston, Mass.) ; vol. 5
Hettich, R., Kortanek, K.O.: Semi-infinite programming: Theory, methods and applications. SIAM Review 35 (3), 380–429 (1993)
Hobson, D., Laurence, P., Wang, T.H.: Static arbitrage upper bounds for the prices of basket option. Working paper (2004)
Henkin, G., Shananin, A.: Bernstein theorems and Radon transform, application to the theory of production functions. American Mathematical Society: Translation of mathematical monographs 81, 189–223 (1990)
Karatzas, I., Shreve, S.E.: Methods of mathematical finance. Applications of mathematics, vol. 39, Springer, New York, 1998
Laurent, J.P., Leisen, D.: Building a consistent pricing model from observed option prices. In: Quantitative Analysis in Financial Markets Avellaneda, M. (ed.) World Scientific Publishing, 2000
Laurence P., Wang, T.H.: Sharp upper and lower bounds for basket options. To appear in Applied Mathematical Finance (2003)
Laurence P., Wang, T.H.: What's a basket worth? Risk, 2004
Rebonato, R.: Interest-rate options models. Financial Engineering, Wiley, 1998
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
d'Aspremont, A., Ghaoui, L. Static arbitrage bounds on basket option prices. Math. Program. 106, 467–489 (2006). https://doi.org/10.1007/s10107-005-0642-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-005-0642-z