# American Institute of Mathematical Sciences

January  2014, 10(1): 207-218. doi: 10.3934/jimo.2014.10.207

## Variance-optimal hedging for target volatility options

 1 School of Mathematical Sciences, Nankai University, Tianjin 300071, China 2 School of Business, Nankai University, Tianjin 300071, China

Received  March 2012 Revised  March 2013 Published  October 2013

In this paper, we consider a variance-optimal hedge for target volatility options, under exponential Lévy dynamics. Since the payoff of target volatility options is related with realized volatility of some underlying asset, which is path-dependent, it is difficult to price this instrument. Here we will derive an explicit Föllmer-Schweizer decomposition of the contingent claim of target volatility options and then give the explicit expressions of hedging strategies in both discrete time and continuous time.
Citation: Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial and Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207
##### References:
 [1] P. Carr, R. Lee and L. Wu, Variance swaps on time-changed Lévy processes, Finance and Stochastics, 16 (2012), 335-355. doi: 10.1007/s00780-011-0157-9. [2] G. Di Graziano and L. Torricelli, Target Volatility Option Pricing, International Journal of Theoretical and Applied Finance, 15 (2012), 1250005-1-1250005-17. doi: 10.1142/S0219024911006474. [3] R. Elloitt, T. Siu and L. Chan, Pricing volatility swaps under Heston's stochastic volatility model with regime switching, Applied Mathematical Finance, 14 (2007), 41-62. doi: 10.1080/13504860600659222. [4] R. Engle, Risk and Volatility: Econometric models and financial practice, American Economic Review, 94 (2004), 405-420. doi: 10.1257/0002828041464597. [5] H. Föllmer and M. Schweizer, Hedging of contingent claims under incomplete information, in "Applied Stochastic Analysis" (eds. M. Davis and R. Elliott), Gordon and Breach, (1991), 389-414. [6] J. Fu, X. Wang and Y. Wang, Variance-optimal hedging for volatility swaps, preprint, (2012). [7] S. Howison, A. Rafailidis and H. Rasmussen, On the pricing and hedging of volatility derivatives, Applied Mathematical Finance, 11 (2004), 317-346. doi: 10.1080/1350486042000254024. [8] F. Hubalek, J. Kallsen and L. Karwczyk, Variance-optimal hedging for processes with stationary independent increments, Annals of Applied Probability, 16 (2006), 853-885. doi: 10.1214/105051606000000178. [9] P. Protter, "Stochastic Integration and Differential Equations," Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. [10] K. Sato, "Lévy Processes and Infinitely Divisible Distributions," Cambridge University Press, Cambridge, 2001. [11] D. Schweizer, Approximating random variables by stochastic integrals, Annals of Probability, 22 (1994), 1536-1575. doi: 10.1214/aop/1176988611. [12] D. Schweizer, Variance-optimal hedging in discrete time, Mathematics of Operations Research, 20 (1995), 1-32. doi: 10.1287/moor.20.1.1. [13] S. Song, X. Wang and Y. Wang, Pricing volatility derivatives with jump underlying and stochastic volatility, preprint, (2011). [14] S. Zhu and G. Lian, A closed-form exact solution for pricing variance swaps with stochastic volatility, Mathematical Finance, 21 (2011), 233-256. doi: 10.1111/j.1467-9965.2010.00436.x.

show all references

##### References:
 [1] P. Carr, R. Lee and L. Wu, Variance swaps on time-changed Lévy processes, Finance and Stochastics, 16 (2012), 335-355. doi: 10.1007/s00780-011-0157-9. [2] G. Di Graziano and L. Torricelli, Target Volatility Option Pricing, International Journal of Theoretical and Applied Finance, 15 (2012), 1250005-1-1250005-17. doi: 10.1142/S0219024911006474. [3] R. Elloitt, T. Siu and L. Chan, Pricing volatility swaps under Heston's stochastic volatility model with regime switching, Applied Mathematical Finance, 14 (2007), 41-62. doi: 10.1080/13504860600659222. [4] R. Engle, Risk and Volatility: Econometric models and financial practice, American Economic Review, 94 (2004), 405-420. doi: 10.1257/0002828041464597. [5] H. Föllmer and M. Schweizer, Hedging of contingent claims under incomplete information, in "Applied Stochastic Analysis" (eds. M. Davis and R. Elliott), Gordon and Breach, (1991), 389-414. [6] J. Fu, X. Wang and Y. Wang, Variance-optimal hedging for volatility swaps, preprint, (2012). [7] S. Howison, A. Rafailidis and H. Rasmussen, On the pricing and hedging of volatility derivatives, Applied Mathematical Finance, 11 (2004), 317-346. doi: 10.1080/1350486042000254024. [8] F. Hubalek, J. Kallsen and L. Karwczyk, Variance-optimal hedging for processes with stationary independent increments, Annals of Applied Probability, 16 (2006), 853-885. doi: 10.1214/105051606000000178. [9] P. Protter, "Stochastic Integration and Differential Equations," Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. [10] K. Sato, "Lévy Processes and Infinitely Divisible Distributions," Cambridge University Press, Cambridge, 2001. [11] D. Schweizer, Approximating random variables by stochastic integrals, Annals of Probability, 22 (1994), 1536-1575. doi: 10.1214/aop/1176988611. [12] D. Schweizer, Variance-optimal hedging in discrete time, Mathematics of Operations Research, 20 (1995), 1-32. doi: 10.1287/moor.20.1.1. [13] S. Song, X. Wang and Y. Wang, Pricing volatility derivatives with jump underlying and stochastic volatility, preprint, (2011). [14] S. Zhu and G. Lian, A closed-form exact solution for pricing variance swaps with stochastic volatility, Mathematical Finance, 21 (2011), 233-256. doi: 10.1111/j.1467-9965.2010.00436.x.
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