Annals of Applied Probability

Convex pricing by a generalized entropy penalty

Johannes Leitner

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In an incomplete Brownian-motion market setting, we propose a convex monotonic pricing functional for nonattainable bounded contingent claims which is compatible with prices for attainable claims. The pricing functional is defined as the convex conjugate of a generalized entropy penalty functional and an interpretation in terms of tracking with instantaneously vanishing risk can be given.

Article information

Ann. Appl. Probab., Volume 18, Number 2 (2008), 620-631.

First available in Project Euclid: 20 March 2008

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Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control 91B28 58E17: Pareto optimality, etc., applications to economics [See also 90C29]

Hedging with vanishing risk generalized entropy quadratic BSDE


Leitner, Johannes. Convex pricing by a generalized entropy penalty. Ann. Appl. Probab. 18 (2008), no. 2, 620--631. doi:10.1214/07-AAP466.

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  • Artzner, P., Delbaen, F., Eber J.-M. and Heath, D. (1999). Coherent measures of risk. Math. Finance 9 203–228.
  • Barrieu, P. and El Karoui, N. (2004). Optimal derivatives design under dynamic risk measures. In Math. Finance (G. Yin and Q. Zhang, eds.) 13–25. Amer. Math. Soc., Providence, RI.
  • Barrieu, P. and El Karoui, N. (2005). Inf-convolution of risk measures and optimal risk transfer. Finance Stoch. 9 269–298.
  • Barrieu, P. and El Karoui, N. (2007). Pricing, hedging and optimally designing derivatives via minimization of risk measures. Volume on Indifference Pricing (R. Carmona ed.). Princeton Univ. Press. To appear.
  • Briand, P. and Hu, Y. (2006). BSDE with quadratic growth and unbounded terminal value. Probab. Theory Related Fields 136 604–618.
  • Choulli, T. and Stricker, C. (2005). Minimal entropy-Hellinger martingale measure in incomplete markets. Math. Finance 15 465–490.
  • Csiszár, I. (1975). I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 146–158.
  • Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. North-Holland, Amsterdam.
  • Delbaen, F. (2001). Coherent Risk Measures. Scuola Normale Superiore, Classe di Scienze, Pisa.
  • Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M. and Stricker, C. (2002). Exponential hedging and entropic penalties. Math. Finance 12 99–123.
  • Ekeland, I. and Témam, R. (1999). Convex Analysis and Variational Problems. SIAM, Philadelphia.
  • El Karoui, N. and Mazliak, L., eds. (1997). Backward Stochastic Differential Equations. Longman, Harlow.
  • El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • Föllmer, H. and Schied, A. (2002). Stochastic Finance. de Gruyter, Berlin.
  • Föllmer, H. and Schweizer, M. (1990). Hedging of continuous claims under incomplete information. In Applied Stochastic Analysis (M. Davis and R. Elliot, eds.) 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.
  • Hu, Y., Imkeller, P. and Müller, M. (2005). Partial equilibrium and market completion. Internat. J. Theoret. Appl. Finance 8 483–508.
  • Kazamaki, N. (1994). Continuous Exponential Martingales and BMO. Springer, Berlin.
  • Klöppel, S. and Schweizer, M. (2007). Dynamic indifference valuation via convex risk measures. Math. Finance 17 599–627.
  • Kobylanski, M. (1997). Résultats d’existence et d’unicité pour des équations différentielles stochastiques rétrogrades avec des générateurs à croissance quadratique. C. R. Acad. Sci. Par. Sér. I Math. 324 81–86.
  • Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558–602.
  • Lazrak, A. and Quenez, M. C. (2003). A generalized stochastic differential utility. Math. Oper. Res. 28 154–180.
  • Lepeltier, J.-P. and San Martin, J. (1998). Existence for BSDE with superlinear-quadratic coefficients. Stochastics Stochastics Rep. 63 227–240.
  • Leitner, J. (2006). Pricing and hedging with globally and instantaneously vanishing risk. Submitted.
  • Ma, J. and Yong, J. (1999). Forward–Backward Stochastic Differential Equations and Their Applications. Springer, Berlin.
  • Mania, M. and Schweizer, M. (2005). Dynamic exponential utility indifference valuation. Ann. Appl. Probab. 13 2113–2143.
  • Molchanov, I. (2005). Theory of Random Sets. Springer, London.
  • Peng, S. (2004). Nonlinear expectations, nonlinear evaluations and risk measures Stochastic Methods in Finance. Lecture Notes in Math. 1856 165–253. Springer, Berlin.
  • Rouge, R. and El Karoui, N. (2000). Pricing via utility maximization and entropy. Math. Finance 10 259–276.