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.
 [1] Xingchun Wang, Yongjin Wang. Hedging strategies for discretely monitored Asian options under Lévy processes. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1209-1224. doi: 10.3934/jimo.2014.10.1209 [2] Edson Pindza, Francis Youbi, Eben Maré, Matt Davison. Barycentric spectral domain decomposition methods for valuing a class of infinite activity Lévy models. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 625-643. doi: 10.3934/dcdss.2019040 [3] Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli. Pricing realized variance options using integrated stochastic variance options in the Heston stochastic volatility model. Conference Publications, 2007, 2007 (Special) : 354-363. doi: 10.3934/proc.2007.2007.354 [4] Yang Yang, Kaiyong Wang, Jiajun Liu, Zhimin Zhang. Asymptotics for a bidimensional risk model with two geometric Lévy price processes. Journal of Industrial and Management Optimization, 2019, 15 (2) : 481-505. doi: 10.3934/jimo.2018053 [5] Mingshang Hu, Shige Peng. G-Lévy processes under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 1-22. doi: 10.3934/puqr.2021001 [6] A. Settati, A. Lahrouz, Mohamed El Fatini, A. El Haitami, M. El Jarroudi, M. Erriani. A Markovian switching diffusion for an SIS model incorporating Lévy processes. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022072 [7] Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764 [8] Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521 [9] Hao Chang, Jiaao Li, Hui Zhao. Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1393-1423. doi: 10.3934/jimo.2021025 [10] Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial and Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241 [11] Walter Farkas, Ludovic Mathys. Geometric step options and Lévy models: Duality, PIDEs, and semi-analytical pricing. Frontiers of Mathematical Finance, 2022, 1 (1) : 1-51. doi: 10.3934/fmf.2021001 [12] Ian Knowles, Ajay Mahato. The inverse volatility problem for American options. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3473-3489. doi: 10.3934/dcdss.2020235 [13] Siwei Yu, Jianwei Ma, Stanley Osher. Geometric mode decomposition. Inverse Problems and Imaging, 2018, 12 (4) : 831-852. doi: 10.3934/ipi.2018035 [14] Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549 [15] Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial and Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001 [16] Duo Wang, Zheng-Fen Jin, Youlin Shang. A penalty decomposition method for nuclear norm minimization with l1 norm fidelity term. Evolution Equations and Control Theory, 2019, 8 (4) : 695-708. doi: 10.3934/eect.2019034 [17] Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282 [18] Wei Zhong, Yongxia Zhao, Ping Chen. Equilibrium periodic dividend strategies with non-exponential discounting for spectrally positive Lévy processes. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2639-2667. doi: 10.3934/jimo.2020087 [19] Wenyuan Wang, Ran Xu. General drawdown based dividend control with fixed transaction costs for spectrally negative Lévy risk processes. Journal of Industrial and Management Optimization, 2022, 18 (2) : 795-823. doi: 10.3934/jimo.2020179 [20] Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435

2020 Impact Factor: 1.801

Metrics

• PDF downloads (335)
• HTML views (0)
• Cited by (10)

Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]