Sensitivity analysis of the optimal exercise boundary of the American put option

Nasir Rehman, Sultan Hussain and Wasim Ul-Haq

Abstract

We consider the American put problem in a general one-dimensional diffusion model. The risk-free interest rate is constant, and volatility is assumed to be a function of time and stock price. We use the well-known parabolic obstacle problem and establish the continuity estimate of the optional exercise boundaries of the American put option with respect to the local volatilities, which may be considered as a generalization of the Achdou results [1].

  • [1]

    Achdou Y., An inverse problem for a parabolic variational inequality arising in volatility calibration with American options, SIAM J. Control Optim. 43 (2005), no. 5, 1583–1615.

  • [2]

    Anderson S., A simple approach to the pricing of Bermudan swaptions in the multifactor LIBOR market model, J. Comput. Finance 3 (2000), 5–32.

  • [3]

    Ekström E., Properties of American option prices, Stochastic Process. Appl. 114 (2004), no. 2, 265–278.

  • [4]

    El-Karoui N., Jeanblance-Picqué M. and Shreve S. E., Robustness of the Black and Scholes formula, Math. Finance 8 (1998), no. 2, 93–126.

  • [5]

    Hobson D. G., Volatility misspecification, option pricing and superreplication via coupling, Ann. Appl. Probab. 8 (1998), no. 1, 193–205.

  • [6]

    Jacka S., Optimal stopping and the American put, Math. Finance 1 (1991), no. 2, 1–14.

  • [7]

    Karatzas I. and Shreve S. E., Methods of Mathematical Finance, Appl. Math. 39, Springer, Berlin, 1998.

  • [8]

    Kim I. J., The analytic valuation of the American options, Rev. Fin. Stud. 3 (1990), no. 4, 547–572.

  • [9]

    Lamberton D. and Villeneuve S., Critical price near maturity for an American option on a dividend-paying stock, Ann. Appl. Probab. 13 (2003), no. 2, 800–815.

  • [10]

    Pham H., Optimal stopping, free boundary, and American option in a jump-diffusion model, Appl. Math. Optim. 35 (1997), no. 2, 145–164.

  • [11]

    Rehman N. and Shashiashvili M., The American foreign exchange option in time-dependent one-dimensional diffusion model for exchange rate, Appl. Math. Optim. 59 (2009), no. 3, 329–363.

  • [12]

    Shreve S. E., Stochastic Calculus for Finance. II: Continuous-Time Models, Springer Finance, Springer, New York, 2004.

  • [13]

    Villeneuve S., Exercise regions of American options on several assets, Finance Stoch. 3 (1999), no. 3, 295–322.

  • [14]

    Wilmott P., Dewynne J. and Howison S., Option Pricing: Mathematical Models and Computation, Oxford Financial Press, Oxford, 1993.

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The Georgian Mathematical Journal was founded by the Georgian National Academy of Sciences and A. Razmadze Mathematical Institute, and is jointly produced with De Gruyter.

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