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

Variance-optimal hedging for target volatility options

Xingchun Wang 1, and  Yongjin Wang 2,

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.
Keywords: Galtchouk-Kunita-Watanabe decomposition., target volatility options, Föllmer-Schweizer decomposition, Lévy processes, Variance-optimal hedging.
Mathematics Subject Classification: Primary: 60G51, 91B70; Secondary: 65K1.
Citation: Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

R. Engle, Risk and Volatility: Econometric models and financial practice, American Economic Review, 94 (2004), 405-420. doi: 10.1257/0002828041464597.  Google Scholar

[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.  Google Scholar

[6]

J. Fu, X. Wang and Y. Wang, Variance-optimal hedging for volatility swaps, preprint, (2012). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. Sato, "Lévy Processes and Infinitely Divisible Distributions," Cambridge University Press, Cambridge, 2001. Google Scholar

[11]

D. Schweizer, Approximating random variables by stochastic integrals, Annals of Probability, 22 (1994), 1536-1575. doi: 10.1214/aop/1176988611.  Google Scholar

[12]

D. Schweizer, Variance-optimal hedging in discrete time, Mathematics of Operations Research, 20 (1995), 1-32. doi: 10.1287/moor.20.1.1.  Google Scholar

[13]

S. Song, X. Wang and Y. Wang, Pricing volatility derivatives with jump underlying and stochastic volatility, preprint, (2011). Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

R. Engle, Risk and Volatility: Econometric models and financial practice, American Economic Review, 94 (2004), 405-420. doi: 10.1257/0002828041464597.  Google Scholar

[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.  Google Scholar

[6]

J. Fu, X. Wang and Y. Wang, Variance-optimal hedging for volatility swaps, preprint, (2012). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. Sato, "Lévy Processes and Infinitely Divisible Distributions," Cambridge University Press, Cambridge, 2001. Google Scholar

[11]

D. Schweizer, Approximating random variables by stochastic integrals, Annals of Probability, 22 (1994), 1536-1575. doi: 10.1214/aop/1176988611.  Google Scholar

[12]

D. Schweizer, Variance-optimal hedging in discrete time, Mathematics of Operations Research, 20 (1995), 1-32. doi: 10.1287/moor.20.1.1.  Google Scholar

[13]

S. Song, X. Wang and Y. Wang, Pricing volatility derivatives with jump underlying and stochastic volatility, preprint, (2011). Google Scholar

[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.  Google Scholar

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