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Structure-preserving equivalent martingale measures for ℋ-SII models

Part of:Stochastic processesMathematical economics

Published online by Cambridge University Press:  28 March 2018

David Criens
David Criens*
Affiliation:
Technical University of Munich
*
* Postal address: Technical University of Munich, Parkring 11-13, 85748 Garching b. München, Germany. Email address: david.criens@tum.de
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Abstract

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In this paper we relate the set of structure-preserving equivalent martingale measures ℳsp for financial models driven by semimartingales with conditionally independent increments to a set of measurable and integrable functions 𝒴. More precisely, we prove that ℳsp ≠ ∅ if and only if 𝒴 ≠ ∅, and connect the sets ℳsp and 𝒴 to the semimartingale characteristics of the driving process. As examples we consider integrated Lévy models with independent stochastic factors and time-changed Lévy models and derive mild conditions for ℳsp ≠ ∅.

Keywords

Equivalent martingale measurestochastic volatility modelconditionally independent increments

MSC classification

Primary:60G44: Martingales with continuous parameter60G48: Generalizations of martingales60G51: Processes with independent increments; L'evy processes91B70: Stochastic models
Secondary:60G22: Fractional processes, including fractional Brownian motion

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 1 , March 2018, pp. 1 - 14
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. B 63, 167–241. Google Scholar
[2]Bauer, H. (2002). Wahrscheinlichkeitstheorie, 5th edn. De Gruyter, Berlin. Google Scholar
[3]Bichteler, K. (2002). Stochastic Integration with Jumps. Cambridge University Press. Google Scholar
[4]Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. J. Political Economy 81, 637–654. Google Scholar
[5]Carr, P., Geman, H., Madan, D. B. and Yor, M. (2003). Stochastic volatility for Lévy processes. Math. Finance 13, 345–382. Google Scholar
[6]Eberlein, E. and Jacod, J. (1997). On the range of options prices. Finance Stoch. 1, 131–140. Google Scholar
[7]Grigelionis, B. (1975). Characterization of stochastic processes with conditionally independent increments. Lithuanian Math. J. 15, 562–567. Google Scholar
[8]He, S. W., Wang, J. G. and Yan, J. A. (1992). Semimartingale Theory and Stochastic Calculus. Science Press, Beijing. Google Scholar
[9]Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financial Studies 6, 327–343. CrossRefGoogle Scholar
[10]Hubalek, F. and Sgarra, C. (2006). Esscher transforms and the minimal entropy martingale measure for exponential Lévy models. Quant. Finance 6, 125–145. Google Scholar
[11]Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin. CrossRefGoogle Scholar
[12]Kabanov, J. M., Lipcer, R. Š. and Širjaev, A. N. (1979). Absolute continuity and singularity of locally absolutely continuous probability distributions. I. Math. USSR-Sbornik 35, 631–680. Google Scholar
[13]Kabanov, J. M., Lipcer, R. Š. and Širjaev, A. N. (1981). On the representation of integral-valued random measures and local martingales by means of random measures with deterministic compensators. Math. USSR-Sbornik 39, 267–280. Google Scholar
[14]Kallsen, J. (2004). σ-localization and σ-martingales. Theory Prob. Appl. 48, 152–163. Google Scholar
[15]Kallsen, J. and Muhle-Karbe, J. (2010). Exponentially affine martingales, affine measure changes and exponential moments of affine processes. Stoch. Process. Appl. 120, 163–181. Google Scholar
[16]Kallsen, J. and Muhle-Karbe, J. (2010). Utility maximization in models with conditionally independent increments. Ann. Appl. Prob. 20, 2162–2177. CrossRefGoogle Scholar
[17]Kallsen, J. and Shiryaev, A. N. (2002). The cumulant process and Esscher's change of measure. Finance Stoch. 6, 397–428. Google Scholar
[18]Kassberger, S. and Liebmann, T. (2011). Minimal q-entropy martingale measures for exponential time-changed Lévy processes. Finance Stoch. 15, 117–140. CrossRefGoogle Scholar
[19]Liptser, R. S. and Shiryaev, A. N. (1989). Theory of Martingales. Kluwer, Dordrecht. Google Scholar
[20]Nicolato, E. and Venardos, E. (2003). Option pricing in stochastic volatility models of the Ornstein-Uhlenbeck type. Math. Finance 13, 445–466. Google Scholar
[21]Selivanov, A. V. (2005). On the martingale measures in exponential Lévy models. Theory Prob. Appl. 49, 261–274. Google Scholar
[22]Stein, E. M. and Stein, J. C. (1991). Stock price distributions with stochastic volatility: an analytic approach. Rev. Financial Studies 4, 727–752. Google Scholar
[23]Stroock, D. W. (2011). Probability Theory: An Analytic View, 2nd edn. Cambridge University Press. Google Scholar