On Asymptotic Behaviors of Exponential Hedging in the Basis-Risk Model
Author(s)
Kazuhiro Takino
ABSTRACT
In this article, we consider the exponential hedging and the mean-variance hedging in the basis-risk model. We construct hedging strategies for multiple units of claim and calculate hedging errors. We then observe how the hedge error risk increases when the investor raises trading volumes of the claim. Under our definition of the hedge error risk amount, the risk increases in a linear way, according to the claim volume for the mean-variance hedging. As to the exponential hedging, it does not, i.e., nonlinear increment. The hedging error for the exponential hedging, however, tends to have the same properties to the mean-variance hedging when either risk-averse parameter or claim volume goes to zero. We numerically demonstrate this fact. Our numerical demonstration with the results of the previous researches verifies that the indifference price converges to the mean-variance hedging cost when the claim volume goes to zero under the basis-risk model.
In this article, we consider the exponential hedging and the mean-variance hedging in the basis-risk model. We construct hedging strategies for multiple units of claim and calculate hedging errors. We then observe how the hedge error risk increases when the investor raises trading volumes of the claim. Under our definition of the hedge error risk amount, the risk increases in a linear way, according to the claim volume for the mean-variance hedging. As to the exponential hedging, it does not, i.e., nonlinear increment. The hedging error for the exponential hedging, however, tends to have the same properties to the mean-variance hedging when either risk-averse parameter or claim volume goes to zero. We numerically demonstrate this fact. Our numerical demonstration with the results of the previous researches verifies that the indifference price converges to the mean-variance hedging cost when the claim volume goes to zero under the basis-risk model.
Cite this paper
Takino, K. (2015) On Asymptotic Behaviors of Exponential Hedging in the Basis-Risk Model. Journal of Mathematical Finance, 5, 212-231. doi: 10.4236/jmf.2015.52020.
Takino, K. (2015) On Asymptotic Behaviors of Exponential Hedging in the Basis-Risk Model. Journal of Mathematical Finance, 5, 212-231. doi: 10.4236/jmf.2015.52020.
References
[1] Davis, M.H.A. (2006) Optimal Hedging with Basis Risk. In: Kabanov, Y., Liptser, R. and Stoyanov, J., Eds., From Stochastic Calculus to Mathematical Finance, Springer-Verlag, Berlin, 169-187.
http://dx.doi.org/10.1007/978-3-540-30788-4_8 [2] Delbaen, F., Grandits, P., Rheinlander, T., Samperi, M., Schweizer, M. and Stricker, C. (2002) Exponential Hedging and Entropic Penalties. Mathematical Finance, 12, 99-123.
http://dx.doi.org/10.1111/1467-9965.02001 [3] Duffie, D. and Richardson, H.R. (1991) Mean-Variance Hedging in Continuous Time. The Annals of Applied Probability, 1, 1-15.
http://dx.doi.org/10.1214/aoap/1177005978 [4] Frittelli, M. (2000) The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets. Mathematical Finance, 10, 39-52.
http://dx.doi.org/10.1111/1467-9965.00079 [5] Heath, P., Platen, E. and Schweizer, M. (2001) A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets. Mathematical Finance, 11, 385-413.
http://dx.doi.org/10.1111/1467-9965.00122 [6] Henderson, V. (2002) Valuation of Claims on Nontraded Assets Using Utility Maximization. Mathematical Finance, 12, 351-373.
http://dx.doi.org/10.1111/j.1467-9965.2002.tb00129.x [7] Hodges, S.D. and Neuberger, A. (1989) Optimal Replication of Contingent Claims under Transaction Costs. Review of Futures Markets, 8, 222-239. [8] Ilhan, A., Jonsson, M. and Sircar, R. (2004) Portfolio Optimization with Derivatives and Indifference Pricing. In: Carmona, R., Ed., Indifference Pricing—Theory and Applications, Princeton University Press, Princeton, 183-210. [9] Mania, M. and Shweizer, M. (2005) Dynamic Exponential Utility Indifference Valuation. The Annals of Applied Probability, 15, 2113-2143.
http://dx.doi.org/10.1214/105051605000000395 [10] Monoyios, M. (2004) Performance of Utility-Based Strategies for Hedging Basis Risk. Quantitative Finance, 4, 245-255.
http://dx.doi.org/10.1088/1469-7688/4/3/001 [11] Monoyios, M. (2008) Optimal Hedging and Parameter Uncertainty. IMA Journal of Management Mathematics, 18, 331-351.
http://dx.doi.org/10.1093/imaman/dpm022 [12] Musiela, M. and Zariphopoulou, T. (2004) An Example of Indifference Prices under Exponential Preferences. Finance and Stochastics, 8, 229-239.
http://dx.doi.org/10.1007/s00780-003-0112-5 [13] Owari, K. (2010) Robust Exponential Hedging and Indifference Valuation. International Journal of Theoretical and Applied Finance, 13, 1075-1101.
http://dx.doi.org/10.1142/S0219024910006121 [14] Pham, H. (2009) Continuous-time Stochastic Control and Optimization with Financial Applications. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-540-89500-8 [15] Schweizer, M. (1992) Mean-Variance Hedging for General Claims. The Annals of Applied Probability, 2, 171-179.
http://dx.doi.org/10.1214/aoap/1177005776 [16] Schweizer, M. (1996) Approximation Pricing and the Variance-Optimal Martingale Measure. Annals of Probability, 24, 206-236.
http://dx.doi.org/10.1214/aop/1042644714 [17] Schweizer, M. (2001) A Guided Tour through Quadratic Hedging Approaches. In: Jouni, E., Cvitanic, J. and Musiela, M., Eds., Advances in Mathematical Finance, Cambridge University Press, Cambridge, 538-574.
[1] Davis, M.H.A. (2006) Optimal Hedging with Basis Risk. In: Kabanov, Y., Liptser, R. and Stoyanov, J., Eds., From Stochastic Calculus to Mathematical Finance, Springer-Verlag, Berlin, 169-187.
http://dx.doi.org/10.1007/978-3-540-30788-4_8 [2] Delbaen, F., Grandits, P., Rheinlander, T., Samperi, M., Schweizer, M. and Stricker, C. (2002) Exponential Hedging and Entropic Penalties. Mathematical Finance, 12, 99-123.
http://dx.doi.org/10.1111/1467-9965.02001 [3] Duffie, D. and Richardson, H.R. (1991) Mean-Variance Hedging in Continuous Time. The Annals of Applied Probability, 1, 1-15.
http://dx.doi.org/10.1214/aoap/1177005978 [4] Frittelli, M. (2000) The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets. Mathematical Finance, 10, 39-52.
http://dx.doi.org/10.1111/1467-9965.00079 [5] Heath, P., Platen, E. and Schweizer, M. (2001) A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets. Mathematical Finance, 11, 385-413.
http://dx.doi.org/10.1111/1467-9965.00122 [6] Henderson, V. (2002) Valuation of Claims on Nontraded Assets Using Utility Maximization. Mathematical Finance, 12, 351-373.
http://dx.doi.org/10.1111/j.1467-9965.2002.tb00129.x [7] Hodges, S.D. and Neuberger, A. (1989) Optimal Replication of Contingent Claims under Transaction Costs. Review of Futures Markets, 8, 222-239. [8] Ilhan, A., Jonsson, M. and Sircar, R. (2004) Portfolio Optimization with Derivatives and Indifference Pricing. In: Carmona, R., Ed., Indifference Pricing—Theory and Applications, Princeton University Press, Princeton, 183-210. [9] Mania, M. and Shweizer, M. (2005) Dynamic Exponential Utility Indifference Valuation. The Annals of Applied Probability, 15, 2113-2143.
http://dx.doi.org/10.1214/105051605000000395 [10] Monoyios, M. (2004) Performance of Utility-Based Strategies for Hedging Basis Risk. Quantitative Finance, 4, 245-255.
http://dx.doi.org/10.1088/1469-7688/4/3/001 [11] Monoyios, M. (2008) Optimal Hedging and Parameter Uncertainty. IMA Journal of Management Mathematics, 18, 331-351.
http://dx.doi.org/10.1093/imaman/dpm022 [12] Musiela, M. and Zariphopoulou, T. (2004) An Example of Indifference Prices under Exponential Preferences. Finance and Stochastics, 8, 229-239.
http://dx.doi.org/10.1007/s00780-003-0112-5 [13] Owari, K. (2010) Robust Exponential Hedging and Indifference Valuation. International Journal of Theoretical and Applied Finance, 13, 1075-1101.
http://dx.doi.org/10.1142/S0219024910006121 [14] Pham, H. (2009) Continuous-time Stochastic Control and Optimization with Financial Applications. Springer-Verlag, Berlin.
http://dx.doi.org/10.1007/978-3-540-89500-8 [15] Schweizer, M. (1992) Mean-Variance Hedging for General Claims. The Annals of Applied Probability, 2, 171-179.
http://dx.doi.org/10.1214/aoap/1177005776 [16] Schweizer, M. (1996) Approximation Pricing and the Variance-Optimal Martingale Measure. Annals of Probability, 24, 206-236.
http://dx.doi.org/10.1214/aop/1042644714 [17] Schweizer, M. (2001) A Guided Tour through Quadratic Hedging Approaches. In: Jouni, E., Cvitanic, J. and Musiela, M., Eds., Advances in Mathematical Finance, Cambridge University Press, Cambridge, 538-574.