Mathematical Finance pp 537-593 | Cite as

# Arbitrage-Based Valuation and Hedging of Derivatives

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## Abstract

The valuation of derivative securities constitutes one of the main issues in modern Mathematical Finance. Economic theory has considered the genesis of asset prices for a long time.

## References

- 29.T. Björk,
*Arbitrage Theory in Continuous Time*, 3rd edn. (Oxford Univ. Press, Oxford, 2009)zbMATHGoogle Scholar - 40.D. Breeden, R. Litzenberger, Prices of state-contingent claims implicit in option prices. J. Bus., 621–651 (1978)CrossRefGoogle Scholar
- 47.P. Carr, R. Lee, Robust replication of volatility derivatives, in
*PRMIA Award for Best Paper in Derivatives, MFA 2008 Annual Meeting*, 2009Google Scholar - 48.P. Carr, D. Madan, Option valuation using the fast Fourier transform. J. Comput. Finance
**2**(4), 61–73 (1999)CrossRefGoogle Scholar - 53.P. Carr, R. Lee, L. Wu, Variance swaps on time-changed Lévy processes. Finance Stochast.
**16**(2), 335–355 (2012)zbMATHCrossRefGoogle Scholar - 59.R. Cont, Model uncertainty and its impact on the pricing of derivative instruments. Math. Finance
**16**(3), 519–547 (2006)MathSciNetzbMATHCrossRefGoogle Scholar - 60.R. Cont, P. Tankov,
*Financial Modelling with Jump Processes*(Chapman & Hall/CRC, Boca Raton, 2004)Google Scholar - 69.M. Davis, Complete-market models of stochastic volatility. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
**460**(2041), 11–26 (2004)MathSciNetzbMATHCrossRefGoogle Scholar - 70.F. Delbaen, W. Schachermayer, A general version of the fundamental theorem of asset pricing. Math. Ann.
**300**(3), 463–520 (1994)MathSciNetzbMATHCrossRefGoogle Scholar - 71.F. Delbaen, W. Schachermayer, The Banach space of workable contingent claims in arbitrage theory. Ann. Inst. H. Poincaré Probab. Statist.
**33**(1), 113–144 (1997)MathSciNetzbMATHCrossRefGoogle Scholar - 72.F. Delbaen, W. Schachermayer, The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann.
**312**(2), 215–250 (1998)MathSciNetzbMATHCrossRefGoogle Scholar - 73.F. Delbaen, W. Schachermayer,
*The Mathematics of Arbitrage*(Springer, Berlin, 2006)zbMATHGoogle Scholar - 81.E. Eberlein, J. Jacod, On the range of options prices. Finance Stochast.
**1**(2), 131–140 (1997)zbMATHCrossRefGoogle Scholar - 92.E. Eberlein, K. Glau, A. Papapantoleon, Analysis of Fourier transform valuation formulas and applications. Appl. Math. Finance
**17**(3), 211–240 (2010)MathSciNetzbMATHCrossRefGoogle Scholar - 93.E. Eberlein, K. Glau, A. Papapantoleon, Analyticity of the Wiener-Hopf factors and valuation of exotic options in Lévy models, in
*Advanced Mathematical Methods for Finance*(Springer, Heidelberg, 2011), pp. 223–245zbMATHGoogle Scholar - 97.N. El Karoui, M. Jeanblanc-Picqué, S. Shreve, Robustness of the Black and Scholes formula. Math. Finance
**8**(2), 93–126 (1998)MathSciNetzbMATHCrossRefGoogle Scholar - 101.F. Fang, C. Oosterlee, A novel pricing method for European options based on Fourier-cosine series expansions. SIAM J. Sci. Comput.
**31**(2), 826–848 (2008)MathSciNetzbMATHCrossRefGoogle Scholar - 120.R. Frey, C. Sin, Bounds on European option prices under stochastic volatility. Math. Finance
**9**(2), 97–116 (1999)MathSciNetzbMATHCrossRefGoogle Scholar - 122.H. Geman, N. El Karoui, J. Rochet, Changes of numéraire, changes of probability measure and option pricing. J. Appl. Probab.
**32**(2), 443–458 (1995)MathSciNetzbMATHCrossRefGoogle Scholar - 123.P. Glasserman,
*Monte Carlo Methods in Financial Engineering*(Springer, New York, 2004)zbMATHGoogle Scholar - 133.M. Harrison, S. Pliska, Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl.
**11**(3), 215–260 (1981)MathSciNetzbMATHCrossRefGoogle Scholar - 134.M. Harrison, S. Pliska, A stochastic calculus model of continuous trading: complete markets. Stoch. Process. Appl.
**15**(3), 313–316 (1983)MathSciNetzbMATHCrossRefGoogle Scholar - 143.N. Hilber, O. Reichmann, C. Schwab, and C. Winter.
*Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing*(Springer, Heidelberg, 2013)zbMATHCrossRefGoogle Scholar - 147.F. Hubalek, J. Kallsen, Variance-optimal hedging and Markowitz-efficient portfolios for multivariate processes with stationary independent increments with and without constraints (2004). preprintGoogle Scholar
- 148.F. Hubalek, J. Kallsen, L. Krawczyk, Variance-optimal hedging for processes with stationary independent increments. Ann. Appl. Probab.
**16**(2), 853–885 (2006)MathSciNetzbMATHCrossRefGoogle Scholar - 150.S. Jacka, A martingale representation result and an application to incomplete financial markets. Math. Finance
**2**(4), 239–250 (1992)zbMATHCrossRefGoogle Scholar - 171.J. Kallsen, C. Kühn, Convertible bonds: financial derivatives of game type.
*Exotic Option Pricing and Advanced Lévy Models*(Wiley, Chichester, 2005), pp. 277–291Google Scholar - 196.D. Kramkov, Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Relat. Fields
**105**(4), 459–479 (1996)MathSciNetzbMATHCrossRefGoogle Scholar - 204.D. Lamberton, B. Lapeyre,
*Stochastic Calculus Applied to Finance*(Chapman & Hall, London, 1996)zbMATHGoogle Scholar - 219.R. Merton, Theory of rational option pricing. Bell J. Econom. Manag. Sci.
**4**, 141–183 (1973)MathSciNetzbMATHCrossRefGoogle Scholar - 240.S. Raible,
*Lévy Processes in Finance: Theory, Numerics, and Empirical Facts*. PhD thesis, University of Freiburg, 2000Google Scholar - 264.W. Schoutens, E. Simons, J. Tistaert, A perfect calibration! Now what? Wilmott Magazine
**2004**, 66–78 (2004)CrossRefGoogle Scholar - 265.K. Schürger, Laplace transforms and suprema of stochastic processes.
*Advances in Finance and Stochastics*(Springer, Berlin, 2002), pp. 285–294zbMATHCrossRefGoogle Scholar - 272.R. Seydel,
*Tools for Computational Finance*, 4th edn. (Springer, Berlin, 2009)zbMATHGoogle Scholar - 279.S. Shreve,
*Stochastic Calculus for Finance II: Continuous-Time Models*(Springer, New York, 2004)zbMATHCrossRefGoogle Scholar - 281.C. Stricker, Arbitrage et lois de martingale. Ann. Inst. H. Poincaré Probab. Stat.
**26**(3), 451–460 (1990)MathSciNetzbMATHGoogle Scholar - 285.M. Taqqu, W. Willinger, The analysis of finite security markets using martingales. Adv. Appl. Probab.
**19**(1), 1–25 (1987)MathSciNetzbMATHCrossRefGoogle Scholar - 293.J. Xia, J. Yan, A new look at some basic concepts in arbitrage pricing theory. Sci. China Ser. A
**46**(6), 764–774 (2003)MathSciNetzbMATHCrossRefGoogle Scholar - 294.J. Yan, A new look at the fundamental theorem of asset pricing. J. Korean Math. Soc.
**35**(3), 659–673 (1998). International Conference on Probability Theory and Its Applications (Taejon, 1998)Google Scholar

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