α-Quantile Option in a Jump-Diffusion Economy

  • Laura Ballotta
Part of the Applied Optimization book series (APOP, volume 70)

Abstract

In this note, we extend the analysis of the behaviour of the α-quantile option to the case of a contract’s underlying security driven by a Lévy process. To this aim, a simulation procedure based on the order statistics is implemented. The results produced are also used to study the connections between the occurring of a jump in the market and option prices. In particular, we show that, no matter the risk-neutral valuation framework chosen, the occurring of a jump affects the tails of the distribution of the functional which defines the option payoff. Since options pay a premium for the probability mass existing in the tails of such a distribution, this fact might be seen as a first key to interpret the observed biases.

Keywords

Lookback option α-quantile option Lévy processes Lévy-Khintchine formula incomplete markets order stastistic. 

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References

  1. [1]
    Akahori, J. (1995): Some formulae for a new type of path-dependent option, The Annals of Applied Probability, 5, 383–388.CrossRefGoogle Scholar
  2. [2]
    Ballotta, L. (2001): Lévy processes, option valuation and pricing of the a-quantile option, PhD Thesis - Università degli studi di Bergamo.Google Scholar
  3. [3]
    Ballotta, L. and Kyprianou, A. E. (2000): A note on the a-quantile option, Actuarial Research Paper N° 128, Department of Actuarial Science and Statistics, City University London.Google Scholar
  4. [4]
    Dassios, A. (1995): The distribution of the quantile of Brownian motion with drift and the pricing of related path-dependent options, The Annals of Applied Probability, 5, 389–398.CrossRefGoogle Scholar
  5. [5]
    Dassios, A. (1996): Sample quantiles of stochastic processes with stationary and independent increments, The Annals of Applied Probability, 6, 1041–1043.CrossRefGoogle Scholar
  6. [6]
    Föllmer, H. and Schweizer, M. (1991): Hedging of contingent claims under incomplete information, Applied Stochastic Analysis, 389–414.Google Scholar
  7. [7]
    Gerber, H. U. and E. S. W. Shiu (1994): Option pricing by Esscher transforms (with discussion), Transactions of the Society of Actuaries, 46, 99140; discussion: 141–191.Google Scholar
  8. [8]
    Heynen, R. and H. Kat (1994): Selective memory, RISK, Vol. 7, N° 11, 73–76.Google Scholar
  9. [9]
    Merton, R. C. (1976): Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 125–144.Google Scholar
  10. [10]
    Miura, R. (1992): A note on lookback options based on order statistics, Hitotsubashi Journal of Commerce and Management, 27, 15–28.Google Scholar
  11. [11]
    Nahum, E. (1998): On the distribution of the supremum of the sum of a Brownian motion with drift and a marked point process, and the pricing of lookback options, Technical Report N° 516, Department of Statistics, University of California, Berkeley.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Laura Ballotta
    • 1
  1. 1.Department of Actuarial Science and Statistics School of MathematicsCity University LondonUK

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