## The Annals of Applied Probability

### The minimal entropy martingale measure for general Barndorff-Nielsen/Shephard models

#### Abstract

We determine the minimal entropy martingale measure for a general class of stochastic volatility models where both price process and volatility process contain jump terms which are correlated. This generalizes previous studies which have treated either the geometric Lévy case or continuous price processes with an orthogonal volatility process. We proceed by linking the entropy measure to a certain semi-linear integro-PDE for which we prove the existence of a classical solution.

#### Article information

Source
Ann. Appl. Probab. Volume 16, Number 3 (2006), 1319-1351.

Dates
First available: 2 October 2006

http://projecteuclid.org/euclid.aoap/1159804983

Digital Object Identifier
doi:10.1214/105051606000000240

Mathematical Reviews number (MathSciNet)
MR2260065

Zentralblatt MATH identifier
1154.28305

#### Citation

Rheinländer, Thorsten; Steiger, Gallus. The minimal entropy martingale measure for general Barndorff-Nielsen/Shephard models. The Annals of Applied Probability 16 (2006), no. 3, 1319--1351. doi:10.1214/105051606000000240. http://projecteuclid.org/euclid.aoap/1159804983.

#### References

• Barndorff-Nielsen, O. E. and Shephard, N. (2001). Modelling by Lévy processes for financial econometrics. In Lévy Processes (T. Mikosch et al., eds.) 283--318. Birkhäuser, Boston.
• Becherer, D. (2001). Rational hedging and valuation with utility-based preferences. Ph.D. thesis, Technical Univ. Berlin. Available at http://e-docs.tu_berlin.de/diss/2001/ becherer_dirk.htm.
• Becherer, D. (2004). Utility indifference hedging and valuation via reaction--diffusion systems. Proc. Roy. Soc. Ser. A 460 27--51.
• Bellini, F. and Frittelli, M. (2002). On the existence of minimax martingale measures. Math. Finance 12 1--21.
• Benth, F. E., Karlsen, K. H. and Reikvam, K. (2003). Merton's portfolio optimization problem in a Black & Scholes market with non-Gaussian stochastic volatility of Ornstein--Uhlenbeck type. Math. Finance 13 215--244.
• Benth, F. E. and Meyer-Brandis, T. (2005). The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps. Finance and Stochastics 9 563--575.
• Chan, T. (1999). Pricing contingent claims on stocks driven by Lévy processes. Ann. Appl. Probab. 9 502--528.
• Choulli, T. and Stricker, C. (2005). Minimal entropy-Hellinger martingale measure in incomplete markets. Math. Finance 15 465--490.
• Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M. and Stricker, Ch. (2002). Exponential hedging and entropic penalties. Math. Finance 12 99--123.
• Dellacherie, C. and Meyer, P.-A. (1980). Probabilités et Potentiel, 2nd ed. Hermann, Paris.
• Esche, F. and Schweizer, M. (2005). Minimal entropy preserves the Lévy property: How and why. Stochastic Process. Appl. 115 299--327.
• Föllmer, H. and Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In Applied Stochastics Monographs (M. H. A. Davis and R. J. Elliott, eds.) 5 389--414. Gordon and Breach, New York.
• Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10 39--52.
• Fujiwara, T. and Miyahara, Y. (2003). The minimal entropy martingale measures for geometric Lévy processes. Finance and Stochastics 7 509--531.
• Grandits, P. (1999). On martingale measures for stochastic processes with independent increments. Theory Probab. Appl. 44 39--50.
• Grandits, P. and Rheinländer, T. (2002). On the minimal entropy martingale measure. Ann. Probab. 30 1003--1038.
• He, S., Wang, J. and Yan, J. (1992). Semimartingale Theory and Stochastic Calculus. Science Press and CRC Press, Beijing.
• Hobson, D. (2004). Stochastic volatility models, correlation and the $q$-optimal measure. Math. Finance 14 537--556.
• Jacod, J. and Shiryaev, A. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Springer, Berlin.
• Kallsen, J. (2001). Utility-based derivative pricing in incomplete markets. In Mathematical Finance---Bachelier Congress 2000 (H. Geman et al., eds.) 313--338. Springer, Berlin.
• Lepingle, D. and Mémin, J. (1978). Sur l'intégrabilité uniforme des martingales exponentielles. Z. Wahrsch. Verw. Gebiete 42 175--203.
• Mandelbrot, B. (1963). The variation of certain speculative prices. J. Business 36 392--417.
• Miyahara, Y. (2001). Geometric Lévy Processes and MEMM pricing model and related estimation problems. Financial Engineering and the Japanese Markets 8 45--60.
• Nicolato, E. and Venardos, E. (2003). Option pricing in stochastic volatility models of the Ornstein--Uhlenbeck type. Math. Finance 13 445--466.
• Protter, P. (2004). Stochastic Integration and Differential Equations---A New Approach, 2nd ed. Springer, Berlin.
• Rheinländer, T. (2005). An entropy approach to the Stein and Stein model with correlation. Finance and Stochastics 9 399--413.
• Steiger, G. (2005). The optimal martingale measure for investors with exponential utility function. Ph.D. thesis, ETH Zürich. Available http://www.math.ethz.ch/~gsteiger.
• Zeidler, E. (1986). Nonlinear Functional Analysis and its Applications I. Fixed-Point Theorems. Springer, New York.