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Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment

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  • Maxim Bichuch

Abstract

We price a contingent claim liability (claim for short) using a utility indifference argument. We consider an agent with exponential utility, who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of a proportional transaction cost ε>0 in two cases: with and without a claim. Using the heuristic computations of Whalley and Wilmott (Math. Finance 7:307–324, 1997 ), under suitable technical conditions, we provide a rigorous derivation of the asymptotic expansion of the value function in powers of $\varepsilon^{\frac{1}{3}}$ in both cases with and without a claim. Additionally, using the utility indifference method, we derive the price of the claim at the leading order of $\varepsilon^{\frac{2}{3}}$ . In both cases, we also obtain a “nearly optimal” strategy, whose expected utility asymptotically matches the leading terms of the value function. We also present an example of how this methodology can be used to price more exotic barrier-type contingent claims. Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Maxim Bichuch, 2014. "Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment," Finance and Stochastics, Springer, vol. 18(3), pages 651-694, July.
    • Handle: RePEc:spr:finsto:v:18:y:2014:i:3:p:651-694
    DOI: 10.1007/s00780-014-0233-z
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    References listed on IDEAS

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    1. Halil Mete Soner & Nizar Touzi, 2012. "Homogenization and Asymptotics for Small Transaction Costs," Swiss Finance Institute Research Paper Series 12-13, Swiss Finance Institute.
    2. Monoyios, Michael, 2004. "Option pricing with transaction costs using a Markov chain approximation," Journal of Economic Dynamics and Control, Elsevier, vol. 28(5), pages 889-913, February.
    3. Michael Roper & Marek Rutkowski, 2009. "On The Relationship Between The Call Price Surface And The Implied Volatility Surface Close To Expiry," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(04), pages 427-441.
    4. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    5. repec:dau:papers:123456789/6707 is not listed on IDEAS
    6. Yuri M. Kabanov & Günter Last, 2002. "Hedging under Transaction Costs in Currency Markets: a Continuous‐Time Model," Mathematical Finance, Wiley Blackwell, vol. 12(1), pages 63-70, January.
    7. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    8. (*), Thaleia Zariphopoulou & George M. Constantinides, 1999. "Bounds on prices of contingent claims in an intertemporal economy with proportional transaction costs and general preferences," Finance and Stochastics, Springer, vol. 3(3), pages 345-369.
    9. M. H. A. Davis & A. R. Norman, 1990. "Portfolio Selection with Transaction Costs," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 676-713, November.
    10. Dumas, Bernard & Luciano, Elisa, 1991. "An Exact Solution to a Dynamic Portfolio Choice Problem under Transactions Costs," Journal of Finance, American Finance Association, vol. 46(2), pages 577-595, June.
    11. Boyle, Phelim P & Vorst, Ton, 1992. "Option Replication in Discrete Time with Transaction Costs," Journal of Finance, American Finance Association, vol. 47(1), pages 271-293, March.
    12. repec:dau:papers:123456789/1533 is not listed on IDEAS
    13. George M. Constantinides & Thaleia Zariphopoulou, 2001. "Bounds on Derivative Prices in an Intertemporal Setting with Proportional Transaction Costs and Multiple Securities," Mathematical Finance, Wiley Blackwell, vol. 11(3), pages 331-346, July.
    14. Clewlow, Les & Hodges, Stewart, 1997. "Optimal delta-hedging under transactions costs," Journal of Economic Dynamics and Control, Elsevier, vol. 21(8-9), pages 1353-1376, June.
    15. Kumar Muthuraman & Sunil Kumar, 2006. "Multidimensional Portfolio Optimization With Proportional Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 301-335, April.
    16. A. E. Whalley & P. Wilmott, 1997. "An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 7(3), pages 307-324, July.
    17. Burdzy, Krzysztof & Kang, Weining & Ramanan, Kavita, 2009. "The Skorokhod problem in a time-dependent interval," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 428-452, February.
    18. Karel Janeček & Steven Shreve, 2004. "Asymptotic analysis for optimal investment and consumption with transaction costs," Finance and Stochastics, Springer, vol. 8(2), pages 181-206, May.
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    Cited by:

    1. Bouchard, Bruno & Muhle-Karbe, Johannes, 2022. "Simple bounds for utility maximization with small transaction costs," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 98-113.
    2. Lu, Xiaoping & Yan, Dong & Zhu, Song-Ping, 2022. "Optimal exercise of American puts with transaction costs under utility maximization," Applied Mathematics and Computation, Elsevier, vol. 415(C).

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    More about this item

    Keywords

    Transaction costs; Optimal control; Asymptotic analysis; Utility maximization; Option pricing; 91G80; 60H30; G13;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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