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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Becherer, D.(2003): Rational hedging and valuation of integrated risks under constant absolute risk aversion. Insurance: Mathematics and Economics 33, 1–28.
Bellini, F. and Frittelli, M.(2002): On the existence of minimax martingale measures. Mathematical Finance 12, 1–21.
Belomestny, D. and Reiß, M.(2006): Spectral calibration of exponential Lévy models. Finance & Stochastics 10, 449–474.
Björk, T.(2004): Arbitrage Theory in Continuous Time.2nd edition. Oxford University Press, Oxford.
Carr, P. and Madan, D.(1999): Option valuation using the fast Fourier transform. The Journal of Computational Finance 2, 61–73.
Černý, A. and Kallsen, J. (2007): On the structure of general mean-variance hedging strategies. The Annals of Probability to appear.
Cont, R.(2006): Model uncertainty and its impact on the pricing of derivative instruments. Mathematical Finance 16, 519–547.
Cont, R. and Tankov, P.(2004): Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton.
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.
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.
Delbaen, F. and Schachermayer, W.(1994): A general version of the fundamental theorem of asset pricing. Mathematische Annalen 300, 463–520.
Delbaen, F. and Schachermayer, W.(1998): The fundamental theorem of asset pricing for unbounded stochastic processes. Mathematische Annalen 312, 215–250.
Delbaen, F. and Schachermayer, W.(2006): The Mathematics of Arbitrage. Springer, Berlin.
Duffie, D. (2001): Dynamic Asset Pricing Theory. 3rd edition. Princeton University Press, Princeton.
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.
Föllmer, H. and Schied, A. (2004): Stochastic Finance: An Introduction in Discrete Time. 2nd edition. Walter de Gruyter, Berlin.
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.
Frittelli, M.(2000): Introduction to a theory of value coherent with the no-arbitrage principle. Finance & Stochastics 4, 275–297.
Goll, T. and Kallsen, J.(2000): Optimal portfolios for logarithmic utility. Stochastic Processes and their Applications 89, 31–48.
Goll, T. and Kallsen, J.(2003): A complete explicit solution to the log-optimal portfolio problem. The Annals of Applied Probability, 13, 774–799.
Gourieroux, C., Laurent, J. and Pham, H.(1998): Mean-variance hedging and numéraire. Mathematical Finance 8, 179–200.
Harrison, M. and Kreps, D.(1979): Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20, 381–408.
Harrison, M. and Pliska, S.(1981): Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11, 215–260.
Henderson, V.(2002): Valuation of claims on non-traded assets using utility maximization. Mathematical Finance 12, 351–371.
Jacod, J. and Protter, P. (2006): Risk neutral compatibility with option prices. Preprint, 2006.
Kallsen, J.(2000): Optimal portfolios for exponential Lévy processes. Mathematical Methods of Operations Research 51, 357–374.
Kallsen, J.(2002): Derivative pricing based on local utility maximization. Finance & Stochastics 6, 115–140.
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.
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.
Karatzas, I. and Kou, S.(1996): On the pricing of contingent claims under constraints. The Annals of Applied Probability 6, 321–369.
Karatzas, I. and Shreve, S.(1998): Methods of Mathematical Finance. Springer, Berlin.
Kramkov, D.(1996): Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probability Theory and Related Fields 105, 459–479.
Kramkov, D. and Sîrbu, M. (2006): Asymptotic analysis of utility based hedging strategies for a small number of contingent claims. Preprint, 2006.
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.
Mania, M. and Schweizer, M.(2005): Dynamic exponential utility indifference valuation. The Annals of Applied Probability 15, 2113–2143.
Raible, S. (2000): Lévy Processes in Finance: Theory, Numerics, and Empirical Facts. PhD thesis, University of Freiburg.
Rheinländer, T. and Schweizer, M.(1997): On L 2-projections on a space of stochastic integrals. The Annals of Probability 25, 1810–1831.
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.
Schweizer, M.(1994): Approximating random variables by stochastic integrals. The Annals of Probability 22, 1536–1575.
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.
Stricker, C.(1990): Arbitrage et lois de martingale. Annales de l'Institut Henri Poincaré 26, 451–460.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kallsen, J. (2009). Option Pricing. In: Mikosch, T., Kreiß, JP., Davis, R., Andersen, T. (eds) Handbook of Financial Time Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71297-8_26
Download citation
DOI: https://doi.org/10.1007/978-3-540-71297-8_26
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71296-1
Online ISBN: 978-3-540-71297-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)