The Annals of Applied Probability

Recovering a time-homogeneous stock price process from perpetual option prices

Erik Ekström and David Hobson

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Abstract

It is well known how to determine the price of perpetual American options if the underlying stock price is a time-homogeneous diffusion. In the present paper we consider the inverse problem, that is, given prices of perpetual American options for different strikes, we show how to construct a time-homogeneous stock price model which reproduces the given option prices.

Article information

Source
Ann. Appl. Probab. Volume 21, Number 3 (2011), 1102-1135.

Dates
First available in Project Euclid: 2 June 2011

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1307020393

Digital Object Identifier
doi:10.1214/10-AAP720

Mathematical Reviews number (MathSciNet)
MR2830614

Zentralblatt MATH identifier
1228.91068

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 91G20: Derivative securities
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Keywords
American options generalized diffusions exact calibration of volatility inverse problems

Citation

Ekström, Erik; Hobson, David. Recovering a time-homogeneous stock price process from perpetual option prices. Ann. Appl. Probab. 21 (2011), no. 3, 1102--1135. doi:10.1214/10-AAP720. https://projecteuclid.org/euclid.aoap/1307020393


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References

  • [1] Alfonsi, A. and Jourdain, B. (2009). Exact volatility calibration based on a Dupire-type call-put duality for perpetual American options. NoDEA Nonlinear Differential Equations Appl. 16 523–554.
  • [2] Amir, M. (1991). Sticky Brownian motion as the strong limit of a sequence of random walks. Stochastic Process. Appl. 39 221–237.
  • [3] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd ed. Probability and Its Applications. Birkhäuser, Basel.
  • [4] Dupire, B. (1994). Pricing with a smile. Risk 7 18–20.
  • [5] Heath, D., Jarrow, R. and Morton, A. (1992). Bond pricing and the term structure of interest rates. Econometrica 60 77–106.
  • [6] Hobson, D. (1998). Robust hedging of the lookback option. Finance Stoch. 2 329–347.
  • [7] Itô, K. and McKean, H. P., Jr. (1965). Diffusion Processes and Their Sample Paths. Grundlehren der Mathematischen Wissenschaften 125. Academic Press, New York.
  • [8] Knight, F. B. (1981). Characterization of the Levy measures of inverse local times of gap diffusion. In Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981). Progr. Prob. Statist. 1 53–78. Birkhäuser, Boston, MA.
  • [9] Kotani, S. and Watanabe, S. (1982). Kreĭn’s spectral theory of strings and generalized diffusion processes. In Functional Analysis in Markov Processes (Katata/Kyoto, 1981). Lecture Notes in Math. 923 235–259. Springer, Berlin.
  • [10] Madan, D. B. and Yor, M. (2002). Making Markov martingales meet marginals: With explicit constructions. Bernoulli 8 509–536.
  • [11] Monroe, I. (1978). Processes that can be embedded in Brownian motion. Ann. Probab. 6 42–56.
  • [12] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [13] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2: Itô calculus. Cambridge Univ. Press, Cambridge.
  • [14] Schweizer, M. and Wissel, J. (2008). Arbitrage-free market models for option prices: The multi-strike case. Finance Stoch. 12 469–505.
  • [15] Warren, J. (1997). Branching processes, the Ray–Knight theorem, and sticky Brownian motion. In Séminaire de Probabilités, XXXI. Lecture Notes in Math. 1655 1–15. Springer, Berlin.