Option Pricing

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Abstract

This chapter reviews basic concepts of derivative pricing in financial mathematics.We distinguish market prices and individual values of a potential seller. We focus mainly on arbitrage theory. In addition, two hedgingbased valuation approaches are discussed. The first relies on quadratic hedging whereas the second involves a first-order approximation to utility indifference prices.

Keywords

Asset Price Option Price Contingent Claim Incomplete Market Arbitrage Opportunity 
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References

  1. Becherer, D.(2003): Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insurance: Mathematics and Economics 33, 1–28.MATHCrossRefMathSciNetGoogle Scholar
  2. Bellini, F. and Frittelli, M.(2002): On the existence of minimax martingale measures. Mathematical Finance 12, 1–21.MATHCrossRefMathSciNetGoogle Scholar
  3. Belomestny, D. and Reiß, M.(2006): Spectral calibration of exponential Lévy models. Finance & Stochastics 10, 449–474.MATHCrossRefGoogle Scholar
  4. Björk, T.(2004): Arbitrage Theory in Continuous Time.2nd edition. Oxford University Press, Oxford.MATHGoogle Scholar
  5. Carr, P. and Madan, D.(1999): Option valuation using the fast Fourier transform. The Journal of Computational Finance 2, 61–73.Google Scholar
  6. Černý, A. and Kallsen, J. (2007): On the structure of general mean-variance hedging strategies. The Annals of Probability to appear.Google Scholar
  7. Cont, R.(2006): Model uncertainty and its impact on the pricing of derivative instruments. Mathematical Finance 16, 519–547.MATHCrossRefMathSciNetGoogle Scholar
  8. Cont, R. and Tankov, P.(2004): Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton.MATHGoogle Scholar
  9. Dalang, R., Morton, A. and Willinger, W.(1990): Equivalent martingale measures and no-arbitrage in stochastic security market models. Stochastics and Stochastics Reports 29, 185–202.MATHMathSciNetGoogle Scholar
  10. Davis, M.(1997): Option pricing in incomplete markets. In: Dempster, M. and Pliska, S. (Eds.): Mathematics of Derivative Securities, 216–226. Cambridge University Press, Cambridge.Google Scholar
  11. Delbaen, F. and Schachermayer, W.(1994): A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300, 463–520.MATHCrossRefMathSciNetGoogle Scholar
  12. Delbaen, F. and Schachermayer, W.(1998): The fundamental theorem of asset pricing for unbounded stochastic processes. Mathematische Annalen 312, 215–250.MATHCrossRefMathSciNetGoogle Scholar
  13. Delbaen, F. and Schachermayer, W.(2006): The Mathematics of Arbitrage. Springer, Berlin.MATHGoogle Scholar
  14. Duffie, D. (2001): Dynamic Asset Pricing Theory. 3rd edition. Princeton University Press, Princeton.MATHGoogle Scholar
  15. El Karoui, N. and Quenez, M.(1995): Dynamic programming and pricing of contingent claims in an incomplete market. SIAM Journal on Control and Optimization 33, 29–66.MATHCrossRefMathSciNetGoogle Scholar
  16. Föllmer, H. and Schied, A. (2004): Stochastic Finance: An Introduction in Discrete Time. 2nd edition. Walter de Gruyter, Berlin.MATHGoogle Scholar
  17. Föllmer, H. and Sondermann, D.(1986): Hedging of nonredundant contingent claims. In: Hildenbrand, W. and Mas-Colell, A. (Eds.): Contributions to Mathematical Economics, 205–223. North-Holland, Amsterdam.Google Scholar
  18. Frittelli, M.(2000): Introduction to a theory of value coherent with the no-arbitrage principle. Finance & Stochastics 4, 275–297.MATHCrossRefMathSciNetGoogle Scholar
  19. Goll, T. and Kallsen, J.(2000): Optimal portfolios for logarithmic utility. Stochastic Processes and their Applications 89, 31–48.MATHCrossRefMathSciNetGoogle Scholar
  20. Goll, T. and Kallsen, J.(2003): A complete explicit solution to the log-optimal portfolio problem. The Annals of Applied Probability, 13, 774–799.MATHCrossRefMathSciNetGoogle Scholar
  21. Gourieroux, C., Laurent, J. and Pham, H.(1998): Mean-variance hedging and numéraire. Mathematical Finance 8, 179–200.MATHCrossRefMathSciNetGoogle Scholar
  22. Harrison, M. and Kreps, D.(1979): Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20, 381–408.MATHCrossRefMathSciNetGoogle Scholar
  23. Harrison, M. and Pliska, S.(1981): Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11, 215–260.MATHCrossRefMathSciNetGoogle Scholar
  24. Henderson, V.(2002): Valuation of claims on non-traded assets using utility maximization. Mathematical Finance 12, 351–371.MATHCrossRefMathSciNetGoogle Scholar
  25. Jacod, J. and Protter, P. (2006): Risk neutral compatibility with option prices. Preprint, 2006. Google Scholar
  26. Kallsen, J.(2000): Optimal portfolios for exponential Lévy processes. Mathematical Methods of Operations Research 51, 357–374.CrossRefMathSciNetGoogle Scholar
  27. Kallsen, J.(2002): Derivative pricing based on local utility maximization. Finance & Stochastics 6, 115–140.MATHCrossRefMathSciNetGoogle Scholar
  28. Kallsen, J.(2002): Utility-based derivative pricing in incomplete markets. In: Geman, H., Madan, D., Pliska, S. and Vorst, T. (Eds.): Mathematical Finance – Bachelier Congress 2000, 313–338. Berlin, Springer.Google Scholar
  29. Kallsen, J. and Kühn, C.(2005): Convertible bonds: Financial derivatives of game type. In: Kyprianou, A., Schoutens, W. and Wilmott, P. (Eds.): Exotic Option Pricing and Advanced Lévy Models, 277–291. Wiley, New York.Google Scholar
  30. Karatzas, I. and Kou, S.(1996): On the pricing of contingent claims under constraints. The Annals of Applied Probability 6, 321–369.MATHCrossRefMathSciNetGoogle Scholar
  31. Karatzas, I. and Shreve, S.(1998): Methods of Mathematical Finance. Springer, Berlin.MATHGoogle Scholar
  32. Kramkov, D.(1996): Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probability Theory and Related Fields 105, 459–479.MATHMathSciNetCrossRefGoogle Scholar
  33. Kramkov, D. and Sîrbu, M. (2006): Asymptotic analysis of utility based hedging strategies for a small number of contingent claims. Preprint, 2006. Google Scholar
  34. Kramkov, D. and Sîrbu, M.(2006): The sensitivity analysis of utility based prices and the risk-tolerance wealth processes. The Annals of Applied Probability 16, 2140–2194.MATHCrossRefMathSciNetGoogle Scholar
  35. Mania, M. and Schweizer, M.(2005): Dynamic exponential utility indifference valuation. The Annals of Applied Probability 15, 2113–2143.MATHCrossRefMathSciNetGoogle Scholar
  36. Raible, S. (2000): Lévy Processes in Finance: Theory, Numerics, and Empirical Facts. PhD thesis, University of Freiburg. Google Scholar
  37. Rheinländer, T. and Schweizer, M.(1997): On L 2-projections on a space of stochastic integrals. The Annals of Probability 25, 1810–1831.MATHCrossRefMathSciNetGoogle Scholar
  38. Schoutens, W., Simons, E. and Tistaert, J.(2005): Model risk for exotic and moment derivatives. In: Kyprianou, A., Schoutens, W. and Wilmott, P. (Eds.): Exotic Option Pricing and Advanced Lévy Models, 67–97. Wiley, New York.Google Scholar
  39. Schweizer, M.(1994): Approximating random variables by stochastic integrals. The Annals of Probability 22, 1536–1575.MATHCrossRefMathSciNetGoogle Scholar
  40. Schweizer, M.(2001): A guided tour through quadratic hedging approaches. In: Jouini, E., Cvitanic, J. and Musiela, M. (Eds.): Option Pricing, Interest Rates and Risk Management, 538–574. Cambridge University Press, Cambridge.Google Scholar
  41. Stricker, C.(1990): Arbitrage et lois de martingale. Annales de l'Institut Henri Poincaré 26, 451–460.MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Christian-Albrechts-Universität zu KielKielGermany

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