CALCULATION OF EXOTIC OPTIONS IN INCOMPLETE MARKETS
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CALCULATION OF EXOTIC OPTIONS IN INCOMPLETE MARKETS
Annotation
PII
S042473880000540-3-1
Publication type
Article
Status
Published
Authors
Elena Shelemeh
Edition
Pages
78-92
Abstract

We consider incomplete market with discreet time and without transaction costs, consisting of a risky and a risk-free assets. Return on risky asset is supposed to take on each step one of three values, so that increase, decrease and constancy of risky asset’s price are possible. There is exotic option in the market. In the article exotic option is a contract, for which payoff and moment of execution depend on some event, such as risky asset’s price reaches some given value. To calculate such an option in the incomplete market minimax approach was adopted. The approach reflekts seller’s point of view and implies the following. Seller’s risk function is supposed to be exponential and to depend on the option’s initial price and on deficit of seller’s portfolio at the execution moment. Participants of the market do not know the distribution of risky asset’s price. The seller is prudent, i.e. the seller assumes that the distribution is the worst one (it maximazes seller’s expected risk) and seeks to minimize expected risk by portfolio management. For this problem we have derived a new recurrent equation describing evolution of minimax value for seller’s expected risk. A self-financing portfolio has been found, which minimizes maximum value of seller’s expected risk (optimal portfolio). It has been proved that a corresponding portfolio with consumption is a superhaging one with respect to any discrete measure and it’s capital does not exceed capital of any other superhaging portfolio. Thus it is impossible to improve obtained solution choosing another risk functions. By use of this resuts formulas for superhedging portfolios of binary and barrier options have been derived.

Keywords
exotic option, incomplete market, minimax approach, superhedging, binary option, barrier option
Date of publication
01.07.2017
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0
Views
91
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## References

Acciaio B., Beiglbock M., Penkner F., Schachermayer W. (2016). Model-Free Version of the Fundamental Theorem of Asset Pricing and the Super-Replication Theorem. Mathematical Finance 26, 2, 233–251.
Beiglbock M., Henry-Labordere P., Penkner F. (2013). Model-Independent Bounds for Option Prices – a Mass Transport Approach. Finance and Stochastics 17, 3, 477–501.
Billingsley P. (1977). Convergence of Probability Measures. Moscow: Nauka (in Russian).
Carr P., Nadtochiy S. (2011). Static Hedging under Time-Homogeneous Diffusions. SIAM Journal on Financial Mathematics 2, 1, 794–838.
Dyomin N.S., Andreeva U.V. (2011). Exotic Call Options with Limited Payments and Guaranteed Income in Black–Scholes Model. Control Sciences 1, 33–39.
Fahim A., Huang Y. (2016). Model-Independent Superhedging under Portfolio Constraints. Finance and Stochastics 20, 1, 51–81.
Föllmer H., Schied A. (2008). Stochastic Finance. An Introduction in Discrete Time. Moscow: MTsNMO (in Russian).
Gushchin A.A. (2013). On the upper Hedging Price for Non-Negative Payoffs. Contemporary Problems of Mathematics and Mechanics VIII, Mathematics, 3, 60–72 (in Russian).
Hull J.C. (2009). Options, Futures and other Derivatives. Upper Saddle River: Pearson Prentice Hall.
Khametov V.M., Shelemekh E.A. (2015). Superhedging of American Options on an Incomplete Market with Discrete Time and Finite Horizon. Automatika i Telemekhanika 9, 125–149 (in Russian).
Kudryavtsev O.E., Levendorskii S.Z. (2009). Fast and Accurate Pricing of Barrier Options under Levy Processes. Finance and Stochastics. 13, 4, 531–562.
Petrosyan L.A. (1998). Game Theory. Moscow: Vysshaya shkola (in Russian).
Schachermayer W. (2012). Optimisation and Utility Functions. Documenta Mathematica. Extra Volume ISMP, 455–460.
Shiryaev A.N. (1998). Essentials of Stochastic Finance. Vol. 2. Theory. Moscow: Fazis (in Russian).
Shiryaev A.N. (2004). Probability. Moscow: MTsNMO (in Russian).