Abstract
In this work, we extend the applicability of regret minimization to pricing financial instruments, following the work of [11]. More specifically, we consider pricing a type of exotic option called a fixed-strike lookback call option. A fixed-strike lookback call option has a known expiration time, at which the option holder has the right to receive the difference between the maximal price of a stock and some pre-agreed price. We derive upper bounds on the price of these options, assuming an arbitrage-free market, by developing two-way trading algorithms. We construct our trading algorithms by combining regret minimization algorithms and one-way trading algorithms. Our model assumes upper bounds on the absolute daily returns, overall quadratic variation, and stock price, otherwise allowing for fully adversarial market behavior.
This research was supported in part by the Google Inter-university center for Electronic Markets and Auctions, by a grant from the Israel Science Foundation, by a grant from United States-Israel Binational Science Foundation (BSF), and by a grant from the Israeli Ministry of Science (MoS). This work is part of Ph.D. thesis research carried out by the first author at Tel Aviv University.
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References
Black, F., Scholes, M.: The pricing of options and corporate liabilities. Journal of Political Economy 81(3), 637–654 (1973)
Blum, A., Kalai, A.: Universal portfolios with and without transaction costs. Machine Learning 35(3), 193–205 (1999); special issue for COLT 1997
Borodin, A., El-Yaniv, R., Gogan, V.: Can we learn to beat the best stock. Journal of Artificial Intelligence Research 21, 579–594 (2004)
Cesa-Bianchi, N., Lugosi, G.: Prediction, Learning, and Games. Cambridge University Press, New York (2006)
Cesa-Bianchi, N., Mansour, Y., Stoltz, G.: Improved second-order bounds for prediction with expert advice. Machine Learning 66(2-3), 321–352 (2007)
Chou, A., Cooperstock, J., El-Yaniv, R., Klugerman, M., Leighton, T.: The statistical adversary allows optimal money-making trading strategies. In: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 467–476 (1995)
Conze, A., Viswanathan: Path dependent options: the case of lookback options. Journal of Finance 46(5), 1893–1907 (1991)
Cover, T.M.: Universal portfolios. Mathematical Finance 1(1), 1–29 (1991)
Cover, T.M., Ordentlich, E.: Universal portfolios with side information. IEEE Transactions on Information Theory 42(2), 348–363 (1996)
Dawid, A.P., de Rooij, S., Shafer, G., Shen, A., Vereshchagin, N., Vovk, V.: Insuring against loss of evidence in game-theoretic probability. Statistics and Probability Letters 81(1), 157–162 (2011)
DeMarzo, P., Kremer, I., Mansour, Y.: Online trading algorithms and robust option pricing. In: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, pp. 477–486. ACM, New York (2006)
El-Yaniv, R., Fiat, A., Karp, R.M., Turpin, G.: Optimal search and one-way trading online algorithms. Algorithmica 30(1), 101–139 (2001); a preliminary version appeared in FOCS (1992)
Györfi, L., Lugosi, G., Udina, F.: Nonparametric kernel-based sequential investment strategies. Mathematical Finance 16(2), 337–357 (2006)
Hazan, E., Kale, S.: On stochastic and worst-case models for investing. In: Advances in Neural Information Processing Systems (NIPS), vol. 22, pp. 709–717 (2009)
Helmbold, D.P., Schapire, R.E., Singer, Y., Warmuth, M.K.: On-line portfolio selection using multiplicative updates. In: Proc. 13th International Conference on Machine Learning, pp. 243–251. Morgan Kaufmann, San Francisco (1996)
Kakade, S., Kearns, M.J., Mansour, Y., Ortiz, L.E.: Competitive algorithms for vwap and limit order trading. In: ACM Conference on Electronic Commerce, pp. 189–198. ACM, New York (2004)
Kalai, A., Vempala, S.: Efficient algorithms for universal portfolios. Journal of Machine Learning Research 3, 423–440 (2002)
Lorenz, J., Panagiotou, K., Steger, A.: Optimal algorithms for k-search with application in option pricing. Algorithmica 55(2), 311–328 (2009)
Merton, R.C.: Theory of rational option pricing. Bell Journal of Economics and Management Science 4(1), 141–183 (1973)
Ordentlich, E., Cover, T.M.: The cost of achieving the best portfolio in hindsight. Mathematics of Operations Research 23(4), 960–982 (1998)
Singer, Y.: Switching portfolios. In: UAI, pp. 488–495 (1998)
Vovk, V., Watkins, C.: Universal portfolio selection. In: Proceedings of the Eleventh Annual Conference on Computational Learning Theory, pp. 12–23 (1998)
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Gofer, E., Mansour, Y. (2011). Regret Minimization Algorithms for Pricing Lookback Options. In: Kivinen, J., Szepesvári, C., Ukkonen, E., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 2011. Lecture Notes in Computer Science(), vol 6925. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24412-4_20
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DOI: https://doi.org/10.1007/978-3-642-24412-4_20
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