On Market Completions Approach to Option Pricing
https://doi.org/10.26794/2308-944X-2021-9-3-77-93
Abstract
Option pricing is one of the most important problems of contemporary quantitative finance. It can be solved in complete markets with non-arbitrage option price being uniquely determined via averaging with respect to a unique risk-neutral measure. In incomplete markets, an adequate option pricing is achieved by determining an interval of non-arbitrage option prices as a region of negotiation between seller and buyer of the option. End points of this interval characterise the minimum and maximum average of discounted pay-off function over the set of equivalent risk-neutral measures. By estimating these end points, one constructs super hedging strategies providing a risk-management in such contracts. The current paper analyses an interesting approach to this pricing problem, which consists of introducing the necessary amount of auxiliary assets such that the market becomes complete with option price uniquely determined. One can estimate the interval of non-arbitrage prices by taking minimal and maximal price values from various numbers calculated with the help of different completions. It is a dual characterisation of option prices in incomplete markets, and it is described here in detail for the multivariate diffusion market model. Besides that, the paper discusses how this method can be exploited in optimal investment and partial hedging problems.
Keywords
About the Authors
I. VasilevCanada
Ilia Vasilev — PhD Candidate, MSc Moscow Institute of Physics and Technology (State University), BSc Moscow Institute of Physics and Technology (State University). Edmonton, AB. WoS ResearcherID AAR‑3773-2021
A. Melnikov
Canada
Alexander Melnikov — Professor, DSc from Steklov Mathematical Institute, Russia, PhD from Steklov Mathematical Institute, Russia, MSc from Moscow State University, Russia. Edmonton, AB. Web of Science ResearcherID P‑4620-2017
References
1. Bajeux-Besnainou, I., & Portait, R. (1997). The numeraire portfolio: a new perspective on financial theory. The European Journal of Finance, 3(4), 291–309. https://EconPapers.repec.org/RePEc:taf:eurjfi:v:3:y:1997:i:4:p:291–309
2. Capinski, M. (2014). Hedging Conditional Value at Risk with Options. European Journal of Operational Research. https://doi.org/10.1016/j.ejor.2014.11.011
3. Cong, J., Tan, K. S., & Weng, C. (2014). Conditional value-at-risk-based optimal partial hedging. The Journal of Risk, 16(3), 49–83.
4. Corcuera, J. M., Nualart, D., & Schoutens, W. (2005). Completion of a Lévy market by power-jump assets. Finance and Stochastics, 9(1), 109.
5. Dhaene, J., Kukush, A., & Linders, D. (2013). The Multivariate Black-Scholes Market: Conditions for Completeness and No-Arbitrage. Theory of Probability and Mathematical Statistics, 88, 1–14. https://doi.org/10.2139/ssrn.2186830
6. Eyraud-Loisel, A. (2019). How Does Asymmetric Information Create Market Incompleteness? Methodology and Computing in Applied Probability, 21(2). https://doi.org/10.1007/s11009–018–9672-x
7. Föllmer, H., & Leukert, P. (1999). Quantile hedging. Finance and Stochastics, 3(3), 251–273. https://EconPapers.repec.org/RePEc:spr:finsto:v:3:y:1999:i:3:p:251–273
8. Föllmer, H., & Leukert, P. (2000). Efficient hedging: Cost versus shortfall risk. Finance and Stochastics, 4, 117–146.
9. Follmer, H., & Schweizer, M. (1991). Hedging of Contingent Claims Under Incomplete Information. (389–414). In M. H. A. Davis & R. J. Elliott (eds.), Applied Stochastic Analysis, Stochastics Monographs, Vol. 5, Gordon and Breach, London/New York.
10. Godin, F. (2015). Minimising CVaR in global dynamic hedging with transaction costs. Quantitative Finance, 16, 1–15. https://doi.org/10.1080/14697688.2015.1054865
11. Guilan, W. (1999). Pricing and hedging of American contingent claims in incomplete markets. Acta Mathematicae Applicatae Sinica, 15, 144–152. https://doi.org/10.1007/BF02720489
12. Hu, Y., Imkeller, P., & Müller, M. (2005). Partial Equilibrium and Market Completion. International Journal of Theoretical and Applied Finance (IJTAF), 08, 483–508. https://doi.org/10.1142/S0219024905003098
13. Karatzas, I., Lehoczky, J. P., Shreve, S. E., & Xu, G.-L. (1991). Martingale and Duality Methods for Utility Maximization in an Incomplete Market. SIAM J.Control Optim., 29(3), 702–730. https://doi.org/10.1137/0329039
14. Karatzas, I., & Shreve, S. (2000). Methods of Mathematical Finance. Journal of the American Statistical Association, 95. https://doi.org/10.2307/2669426
15. Kobylanski, M. (2000). Backward Stochastic Differential Equations and Partial Differential Equations with Quadratic Growth. The Annals of Probability, 28(2), 558–602.
16. Li, J., & Xu, M. (2013). Optimal Dynamic Portfolio with Mean-CVaR Criterion. Risks, ISSN 2227–9091, MDPI, Basel, Vol. 1, Iss. 3, pp. 119–147, http://dx.doi.org/10.3390/risks1030119
17. Melnikov, A. (1999). Financial Markets: Stochastic Analysis and the Pricing of Derivative Securities. American Mathematical Society.
18. Melnikov, A., & Smirnov, I. (2012). Dynamic hedging of conditional value-at-risk. Insurance: Mathematics and Economics, 51. https://doi.org/10.1016/j.insmatheco.2012.03.011
19. Melnikov, A., Volkov, S., & Nechaev, M. (2001). Mathematics of Financial Obligations.
20. Melnikov, A. V., & Feoktistov, K. M. (2001). Вопросы безарбитражности и полноты дискретных рынков и расчеты платежных обязательств [Arbitration-Free and Completeness Issues for Discrete Markets and Calculations of Payment Obligations]. Obozreniye Prikladnoy i Promyshlennoy Matematiki, 8(1), 28–40. (In Russian)
21. Miyahara, Y. (1995). Canonical Martingale Measures of Incomplete Assets Markets. Probability Theory and Mathematical Statistics: Proceedings of the Seventh Japan-Russia Symposium.
22. Spivak, G., & Cvitanic, J. (1999). Maximising the probability of a perfect hedge. The Annals of Applied Probability, 9(4), 1303–1328
23. Touchette, H. (n. d.). Legendre-Fenchel transforms in a nutshell. Retrieved from https://www.ise.ncsu.edu/fuzzyneural/wp-content/uploads/sites/9/2019/01/or706-LF-transform-1.pdf
24. Zhang, A. (2007). A secret to create a complete market from an incomplete market. Applied Mathematics and Computation, 191, 253–262. https://doi.org/10.1016/j.amc.2007.02.086
Review
For citations:
Vasilev I., Melnikov A. On Market Completions Approach to Option Pricing. Review of Business and Economics Studies. 2021;9(3):77-93. https://doi.org/10.26794/2308-944X-2021-9-3-77-93