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The Risk-Neutral Measure and Option Pricing under Log-Stable Uncertainty

  • J. Huston McCulloch

The fact that the expected payoffs on assets and call options are infinite under most log-stable distributions led Paul Samuelson and Robert Merton to conjecture that assets and derivatives could not be reasonably priced under these distributions, despite their many other attractive features. This paper demonstrates that when the observed distribution of future prices is log-stable, the Risk Neutral Measure (RNM) under which asset and derivative prices may be computed as expectations is not itself log-stable in the problematic cases. Instead, the RNM is determined by the convolution of two densities, one negatively skewed stable, and the other an exponentially tilted positively skewed stable. The resulting RNM gives finite expected payoffs, and therefore demonstrates that these fears were in fact unfounded. Carr and Madan (1999) have shown how the Fast Fourier Transform (FFT) can be used to quickly evaluate options directly from the characteristic function of any RNM. The log-stable RNM characteristic function presented here therefore greatly facilitates the pricing of options on log-stable assets, by means of this new methodology, provided a Romberg adaptation of the FFT is employed. The full paper is at .

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Paper provided by Econometric Society in its series Econometric Society 2004 North American Winter Meetings with number 428.

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Date of creation: 11 Aug 2004
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Handle: RePEc:ecm:nawm04:428
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  1. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  2. Peter Carr & Liuren Wu, 2002. "The Finite Moment Log Stable Process and Option Pricing," Finance 0207012, EconWPA.
  3. Mehra, Rajnish & Prescott, Edward C., 1985. "The equity premium: A puzzle," Journal of Monetary Economics, Elsevier, vol. 15(2), pages 145-161, March.
  4. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  5. McCulloch, J. Huston, 1985. "Interest-risk sensitive deposit insurance premia : Stable ACH estimates," Journal of Banking & Finance, Elsevier, vol. 9(1), pages 137-156, March.
  6. Alvaro Cartea & Sam Howison, 2002. "Distinguished Limits of Levy-Stable Processes, and Applications to Option Pricing," OFRC Working Papers Series 2002mf04, Oxford Financial Research Centre.
  7. Smith, Clifford Jr., 1976. "Option pricing : A review," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 3-51.
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