Abstract
Modelling financial and insurance time series with Lévy processes or with exponential Lévy processes is a relevant actual practice and an active area of research. It allows qualitatively and quantitatively good adaptation to the empirical statistical properties of asset returns. Due to model incompleteness it is a problem of considerable interest to determine the dependence of option prices in these models on the choice of pricing measures and to establish nontrivial price bounds. In this paper we review and extend ordering results of stochastic and convex type for this class of models. We also extend the ordering results to processes with independent increments (PII) and present several examples and applications as to α-stable processes, NIG-processes, GH-distributions, and others. Criteria are given for the Lévy measures which imply corresponding comparison results for European type options in (exponential) Lévy models.
Similar content being viewed by others
References
Barndorff-Nielsen O (1998) Processes of normal inverse Gaussian type. Financ Stoch 2:41–68
Barndorff-Nielsen OE (1977) Exponentially decreasing distributions for the logarithm of particle size. Proc R Soc Lond Ser A 353:401–419
Bellamy N, Jeanblanc MA (2000) Incompleteness of markets driven by a mixed diffusion. Financ Stoch 4:209–222
Bergenthum J (2005) Comparison of semimartingales and Lévy processes with applications to finanical mathematics. PhD thesis, University of Freiburg
Bergenthum J, Rüschendorf L (2006) Comparison of option prices in semimartingale models. Financ Stoch 10(2):222–249
Bergenthum J, Rüschendorf L (2007) Comparison of semimartingales and Lévy processes. Ann Probab 35:228–254
Carr P, Geman H, Madan DB, Yor M (2003) Stochastic volatility of Lévy processes. Math Financ 13:345–382
Cont R, Tankov P (2004) Financial modelling with jump processes. Chapman and Hall/CRC Financial Mathematics Series, London
Delbaen F, Haezendonck J (1989) A martingale approach to premium calculation principles in an arbitrage free market. Insur Math Econ 8:269–277
Eberlein E, Keller U (1995) Hyperbolic distributions in finance. Bernoulli 1(3):281–299
El Karoui N, Jeanblanc-Picqué M, Shreve SE (1998) Robustness of the black and scholes formula. Math Financ 8(2):93–126
Embrechts P (2000) Actuarial versus financial pricing of insurance. Risk Financ 1(4):17–26
Goll T, Rüschendorf L (2001) Minimax and minimal distance martingale measures and their relationship to portfolio optimization. Financ Stoch 5:557–581
Gushchin AA, Mordecki E (2002) Bounds for option prices in the semimartingale market models. In: Proceedings of the Steklov Mathematical Institute, vol 237, pp 73–113
Henderson V, Hobson D (2003) Coupling and option price comparisons in a jump diffusion model. Stoch Stoch Rep 75(3):79–101
Hobson D (1998) Volatility misspecification, option pricing and superreplication via coupling. Ann Appl Probab 8(1):193–205
Jacod J, Shiryaev AN (2003) Limit theorems for stochastic processes. Springer, Heidelberg
Karlin S, Novikoff A (1963) Generalized convex inequalities. Pac J Math 13:1251–1279
Madan D, Seneta E (1990) The vg model for share market returns. J Bus 63:511–524
Mandelbrot B (1960) The Pareto–Lévy law and the distribution of income. Int Econ Rev 1:79–106
Merton RC (1976) Option pricing when underlying stock returns are discontinuous. J Financ Econ 3:125–144
Møller T (2002) On valuation and risk management at the interface of insurance and finance. Br Actuar J 8(IV):787–827
Møller T (2004) Stochastic orders in dynamic reinsurance markets. Financ Stoch 8(4):479–499
Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, London
Norberg R (1993) Prediction of outstanding liabilities in non-life insurance. ASTIN Bull 23(1):95–115
Rachev S, Mittnik S (2000) Stable Paretian models in finance. Wiley, London
Sato K-I (1999) Lévy processes and infinitely divisible distributions. Cambridge Studies in Advanced Mathematics, vol 68. Cambridge University Press, London
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bergenthum, J., Rüschendorf, L. Convex ordering criteria for Lévy processes. ADAC 1, 143–173 (2007). https://doi.org/10.1007/s11634-007-0008-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11634-007-0008-x