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RISK-NEUTRAL MEASURES AND PRICING FOR A PURE JUMP PRICE PROCESS

A STOCHASTIC CONTROL APPROACH

Published online by Cambridge University Press:  21 December 2009

Anna Gerardi  and
Paola Tardelli
Anna Gerardi
Affiliation:
Department of Electrical and Information Engineering, University of L’ Aquila, L’ Aquila, Italy E-mail: [email protected]; [email protected]
Paola Tardelli
Affiliation:
Department of Electrical and Information Engineering, University of L’ Aquila, L’ Aquila, Italy E-mail: [email protected]; [email protected]
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Abstract

This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process, modeled again by a marked point process. Taking into account the presence of catastrophic events, the possibility of common jump times between the risky asset price process and the arrivals process is allowed. By setting and solving a suitable control problem, the characterization of the minimal entropy martingale measure is obtained. In a particular case, a pricing problem is also discussed.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 1 , January 2010 , pp. 47 - 76
Copyright
Copyright © Cambridge University Press 2009

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References

1.Ansel, J.P. & Stricker, C. (1993). Unicité et existence de la loi minimale. Séminaire de Probabilités XXVII. Lecture Notes in Mathematics Vol. 1557, Berlin: Springer, 22–29.Google Scholar
2.Bellini, F. & Frittelli, M. (2002). On the existence of minimax martingale measures. Mathematical Finance 12(1): 1–21.CrossRefGoogle Scholar
3.Biagini, S. & Frittelli, M. (2005). Utility maximization in incomplete markets for unbounded processes. Finance and Stochastics 9(4): 493–517.CrossRefGoogle Scholar
4.Brémaud, P. (1981). Point processes and queues. Springer Series in Statistics. New York: Springer-Verlag.CrossRefGoogle Scholar
5.Ceci, C. (2006). Risk minimizing hedging for a partially observed high frequency data model. Stochastics 78(1): 13–31.CrossRefGoogle Scholar
6.Ceci, C. & Gerardi, A. (2009). Pricing for geometric marked point processes under partial information: Entropy approach. International Journal of Theoretical and Applied Finance 12: 179–207.CrossRefGoogle Scholar
7.Centanni, S. & Minozzo, M. (2006). Estimation and filtering by reversible jump MCMC for a doubly stochastic Poisson model for ultra-high-frequency financial data. Statistical Modelling 6(2): 97–118.CrossRefGoogle Scholar
8.Centanni, S. & Minozzo, M. (2006). A Monte Carlo approach to filtering for a class of marked doubly stochastic Poisson processes. Journal of the American Statistical Association 101(476): 1582–1597.CrossRefGoogle Scholar
9.Delbaen, F., Grandits, P., Rheinlander, T., Samperi, D., Schweizer, M. & Stricker, C. (2002). Exponential hedging and entropic penalties. Mathematical Finance 12(2): 99–123.CrossRefGoogle Scholar
10.Doléans-Dade, C. (1970). Quelques applications de la formule de changement de variables pour le semimartingales. Zeitschriftfür Wahrscheinlichkeitstheorie und Verwunde Gebiete 16: 181–194.CrossRefGoogle Scholar
11.Ethier, S.N. & Kurtz, T.G. (1986). Markov processes: Characterization and convergence. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.CrossRefGoogle Scholar
12.Follmer, H. & Schweizer, M. (1991). Hedging of contingent claims under incomplete information. Stochastic Monographs, 5. New York: Gordon and Breach, 389–414.Google Scholar
13.Frey, R. (2000). Risk minimization with incomplete information in a model for high frequency data. INFORMS Applied Probability Conference (Ulm, 1999). Mathematical Finance 10(2): 215–225.CrossRefGoogle Scholar
14.Frey, R. & Runggaldier, W.J. (2001). A nonlinear filtering approach to volatility estimation with a view towards high frequency data. Information modeling in finance (Evry, 2000). International Journal of Theoretical and Applied Finance 4(2): 199–210.CrossRefGoogle Scholar
15.Frey, R. & Runggaldier, W.J. (2009). Credit risk and incomplete information: A nonlinear filtering approach. To appear in Finance and Stochastics. Preprint. Available from www.math.unipd.it/~runggal/wolfganglp.html.Google Scholar
16.Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete market. Mathematical Finance 10(1): 39–52.CrossRefGoogle Scholar
17.Fujiwara, T. & Miyahara, Y. (2003). The minimal entropy martingale measures for geometric Lévy processes. Finance and Stochastics 7(4): 509–531.CrossRefGoogle Scholar
18.Gerardi, A. & Tardelli, P. (2006). Filtering on a partially observed ultra-high-frequency data model. Acta Applicandae Mathematical 91(2) 193–205.CrossRefGoogle Scholar
19.Gombani, A. & Jaschke, S., & Runggaldier, W.J. (2007). Consistent price systems for subfiltrations. ESAIM Probability and Statistics 11: 35–39.CrossRefGoogle Scholar
20.Grandits, P. & Rheinlander, T. (2002). On the minimal entropy martingale measures. Annals of Probability 30(3): 1003–1038.CrossRefGoogle Scholar
21.Jacod, J. (1975). Multivariate point processes: Predictable projection, Radon–Nikodym derivatives, representation of martingales. Zeitschrift für Wahrscheinlichkeitstheorie und Verwunde Gebiete 31: 235–253.CrossRefGoogle Scholar
22.Kliemann, W., Koch, G. & Marchetti, F. (1990). On the unnormalized solution of the filtering problem with counting process observations. IEEE Transactions on Information Theory 36(6): 1415–1425.CrossRefGoogle Scholar
23.Mania, M. & Schweizer, M. (2005). Dynamic exponential utility indifference valuation. Annals of Applied Probability 15(3): 2113–2143.Google Scholar
24.Prigent, J.L. (2001). Option pricing with a general marked point process. Mathematics of Operations Research 26(1): 50–66.CrossRefGoogle Scholar
25.Ridberg, T.H. & Shephard, N. (2000). A modelling framework for the prices and times of trades made on the New York Stock Exchange. In, Fitzgerald, W.J., Smith, R.L., Walden, A.T. & Young, P.C., (eds) Nonlinear and nonstationary signal processing, Cambridge: Cambridge University Press. 217–246.Google Scholar
26.Rogers, L.C.G. & Zane, O. (1998). Designing models for high frequency data, Preprint, University of Bath.Google Scholar
27.Schweizer, M. (1995). On the minimal martingale measure and the Follmer–Schweizer decomposition. Stochastic Analysis and Applications 13(5): 573–599.CrossRefGoogle Scholar
28.Schweizer, M. (2001). A guided tour through quadratic hedging approaches. In Option pricing, interest rates and risk management. Cambridge: Cambridge University Press. 538–574,CrossRefGoogle Scholar
29.Zariphopoulou, T. (2001). A solution approach to valuation with unhedgeable risks. Finance and Stochastics 5(1): 61–82.CrossRefGoogle Scholar