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 articles
by authors

[Back to Top]