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An iterative algorithm for evaluating approximations to the optimal exercise boundary for a nonlinear Black-Scholes equation

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  • Daniel Sevcovic

Abstract

The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a function of the second derivative of the option price itself. A motivation for studying the nonlinear Black--Scholes equation with a nonlinear volatility arises from option pricing models taking into account e.g. nontrivial transaction costs, investor's preferences, feedback and illiquid markets effects and risk from a volatile (unprotected) portfolio. We present a new method how to transform the free boundary problem for the early exercise boundary position into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. We furthermore propose an iterative numerical scheme that can be used to find an approximation of the free boundary. We present results of numerical approximation of the early exercise boundary for various types of nonlinear Black--Scholes equations and we discuss dependence of the free boundary on various model parameters.

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File URL: http://arxiv.org/pdf/0710.5301
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Paper provided by arXiv.org in its series Papers with number 0710.5301.

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Date of creation: Oct 2007
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Handle: RePEc:arx:papers:0710.5301

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  1. Roll, Richard, 1977. "An analytic valuation formula for unprotected American call options on stocks with known dividends," Journal of Financial Economics, Elsevier, Elsevier, vol. 5(2), pages 251-258, November.
  2. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  3. Avellaneda Marco & ParaS Antonio, 1994. "Dynamic hedging portfolios for derivative securities in the presence of large transaction costs," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 1(2), pages 165-194.
  4. Rachel Kuske & Joseph Keller, 1998. "Optimal exercise boundary for an American put option," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 5(2), pages 107-116.
  5. Geske, Robert & Johnson, Herb E, 1984. " The American Put Option Valued Analytically," Journal of Finance, American Finance Association, American Finance Association, vol. 39(5), pages 1511-24, December.
  6. Leland, Hayne E, 1985. " Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, American Finance Association, vol. 40(5), pages 1283-1301, December.
  7. Rüdiger Frey & Alexander Stremme, 1997. "Market Volatility and Feedback Effects from Dynamic Hedging," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 7(4), pages 351-374.
  8. Halil Mete Soner & Guy Barles, 1998. "Option pricing with transaction costs and a nonlinear Black-Scholes equation," Finance and Stochastics, Springer, vol. 2(4), pages 369-397.
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Cited by:
  1. Tomas Bokes & Daniel Sevcovic, 2009. "Early exercise boundary for American type of floating strike Asian option and its numerical approximation," Papers 0912.1321, arXiv.org.

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