Minimax Estimation of Value-at-Risk under Hedging of an American Contingent Claim in a Discrete Financial Market

Authors

  • Alexey I. Soloviev Lomonosov Moscow State University

Abstract

The game problems between seller and buyer of an American contingent claim relate to large scale problems because a number of buyer's strategies grows overexponentially. Therefore, decomposition of such games turns out to be a fundamental problem. In this paper we prove the existence of a minimax monotonous (in time) strategy of the seller in a loss minimization problem considering value-at-risk measure of loss. The given result allows to substantially decrease a number of constraints in the original problem and lets us turn to an equivalent mixed integer problem with admissible dimension.

Keywords:

decision making under uncertainty, value-at-risk, scenario tree, stopping time, hedging

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References

Aho, A.V. and N. J.A. Sloane (1973). Some doubly exponential sequences. Fibonacci Quart., 11, 429–437.

Black, F. and M. Scholes (1973). The Pricing of options and corporate liabilities. J. Polit. Econ., 81(3), 637–654.

Camci, A. and M.C. Pinar (2009). Pricing American contingent claims by stochastic linear programming. Optimization, 58(6), 627–640.

Chicago Mercantile Exchange. The Standart Portfolio Analysis of Risk (SPAN) performance bond system at the Chicago Mercantile Exchange. Technical specification, (1999).

F¨ollmer, H. and P. Leukert (1999). Quantile hedging. Financ. Stoch., 3(3), 251–273.

F¨ollmer, H. and A. Schied (2011). Stochastic Finance: An Introduction in Discrete Time. Walter de Gruyter: Berlin, Germany.

Harrison, J.M. and D.M. Kreps (1979). Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory, 20, 381–408.

King, A. J. (2002). Duality and martingales: a stochastic programming perspective on contingent claims. Math. Program., Ser. B, 91, 543–562.

Lindberg, P. (2012). Optimal partial hedging of an American option: shifting the focus to the expiration date. Math. Method. Oper. Res., 75(3), 221–243.

McGarvey, G. (2007). Sequence A135361. The On-Line Encyclopedia of Integer Sequences. URL: http://oeis.org/A135361.

Merton, R.C. (1973). Theory of rational option pricing. Bell J. Econ., 4(1), 141–183.

Novikov, A.A. (1999). Hedging of options with a given probability. Theory Probab. Appl., 43(1), 135–143.

Perez-Hernandez, L. (2007). On the existence of an efficient hedge for an American contingent claim within a discrete time market. Quant. Financ., 7(5), 547–551.

Pinar, M.C. (2011). Buyer’s quantile hedge portfolios in discrete-time trading. Quant. Financ., 13(5), 729–738.

Pliska, S.R. (1997). Introduction to Mathematical Finance: Discrete Time Models. Blackwell Publishers: Malden, MA.

Rockafellar, R.T. and S. Uryasev (2000). Optimization of conditional value-at-risk. J. Risk, 2(3), 21–41.

Rockafellar, R.T. and S. Uryasev (2002). Conditional value-at-risk for general loss distributions. J. Bank. Financ., 26, 1443–1471.

Sarykalin, S., G. Serraino and S. Uryasev (2008). VaR vs. CVaR in risk management and optimization. In: Tutorials in Operations Research: State-of-the-art Decision-making Tools in the Information-Intensive Age (Chen, Z.-L. and S. Raghavan, eds), Chap. 13, pp. 270–294. INFORMS: Hanover, MD.

Shiryaev, A.N. (1999). Essentials of Stochastic Finance: Facts, Models, Theory. Advanced Series on Statistical Science & Applied Probability, Vol. 3. World Scientific: Singapore.

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Published

2022-05-01

How to Cite

I. Soloviev, A. (2022). Minimax Estimation of Value-at-Risk under Hedging of an American Contingent Claim in a Discrete Financial Market. Contributions to Game Theory and Management, 9. Retrieved from https://gametheory.spbu.ru/article/view/13357

Issue

Vol. 9 (2016)

Section

Articles

License

Articles of "Contributions to Game Theory and Management" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.