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 & 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. 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. 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. doi: 10.1080/13504860600659222.

[4]

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

[5]

H. Föllmer and M. Schweizer, Hedging of contingent claims under incomplete information,, in, (1991), 389.

[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. 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. doi: 10.1214/105051606000000178.

[9]

P. Protter, "Stochastic Integration and Differential Equations,", Second edition. Applications of Mathematics (New York), (2004).

[10]

K. Sato, "Lévy Processes and Infinitely Divisible Distributions,", Cambridge University Press, (2001).

[11]

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

[12]

D. Schweizer, Variance-optimal hedging in discrete time,, Mathematics of Operations Research, 20 (1995), 1. 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. 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. 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. 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. doi: 10.1080/13504860600659222.

[4]

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

[5]

H. Föllmer and M. Schweizer, Hedging of contingent claims under incomplete information,, in, (1991), 389.

[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. 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. doi: 10.1214/105051606000000178.

[9]

P. Protter, "Stochastic Integration and Differential Equations,", Second edition. Applications of Mathematics (New York), (2004).

[10]

K. Sato, "Lévy Processes and Infinitely Divisible Distributions,", Cambridge University Press, (2001).

[11]

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

[12]

D. Schweizer, Variance-optimal hedging in discrete time,, Mathematics of Operations Research, 20 (1995), 1. 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. 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 & 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 & 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 & Management Optimization, 2019, 15 (2) : 481-505. doi: 10.3934/jimo.2018053

[5]

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 & Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521

[6]

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

[7]

Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial & Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241

[8]

Yongxia Zhao, Rongming Wang, Chuancun Yin. Optimal dividends and capital injections for a spectrally positive Lévy process. Journal of Industrial & Management Optimization, 2017, 13 (1) : 1-21. doi: 10.3934/jimo.2016001

[9]

Siwei Yu, Jianwei Ma, Stanley Osher. Geometric mode decomposition. Inverse Problems & Imaging, 2018, 12 (4) : 831-852. doi: 10.3934/ipi.2018035

[10]

Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549

[11]

Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282

[12]

Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435

[13]

Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 2014, 1 (2) : 391-421. doi: 10.3934/jcd.2014.1.391

[14]

Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu, J. Nathan Kutz. Compressed sensing and dynamic mode decomposition. Journal of Computational Dynamics, 2015, 2 (2) : 165-191. doi: 10.3934/jcd.2015002

[15]

Fritz Colonius, Paulo Régis C. Ruffino. Nonlinear Iwasawa decomposition of control flows. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 339-354. doi: 10.3934/dcds.2007.18.339

[16]

Thiago Ferraiol, Mauro Patrão, Lucas Seco. Jordan decomposition and dynamics on flag manifolds. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 923-947. doi: 10.3934/dcds.2010.26.923

[17]

Mauro Patrão, Luiz A. B. San Martin. Morse decomposition of semiflows on fiber bundles. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 561-587. doi: 10.3934/dcds.2007.17.561

[18]

Manman Li, George Yin. Optimal threshold strategies with capital injections in a spectrally negative Lévy risk model. Journal of Industrial & Management Optimization, 2019, 15 (2) : 517-535. doi: 10.3934/jimo.2018055

[19]

Jacinto Marabel Romo. A closed-form solution for outperformance options with stochastic correlation and stochastic volatility. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1185-1209. doi: 10.3934/jimo.2015.11.1185

[20]

Jingzhen Liu, Ka Fai Cedric Yiu, Alain Bensoussan. The optimal mean variance problem with inflation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 185-203. doi: 10.3934/dcdsb.2016.21.185

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]