Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T20:53:06.741Z Has data issue: false hasContentIssue false
  • English
  • Français

LOCALLY RISK-MINIMIZING HEDGING FOR EUROPEAN CONTINGENT CLAIMS WRITTEN ON NON-TRADABLE ASSETS WITH COMMON JUMP RISK

Published online by Cambridge University Press:  01 March 2021

Xiaonan Su ,
Yu Xing ,
Wei Wang Open the ORCID record for Wei Wang [Opens in a new window]  and
Wensheng Wang
Xiaonan Su
Affiliation:
School of Statistics and Mathematics, Nanjing Audit University, Nanjing, China
Yu Xing
Affiliation:
School of Finance, Nanjing Audit University, Nanjing, China
Wei Wang
Affiliation:
School of Mathematics and Statistics, Ningbo University, Ningbo, China E-mail: [email protected]
Wensheng Wang
Affiliation:
School of Economics, Hangzhou Dianzi University, Hangzhou, China
Get access Rights & Permissions [Opens in a new window]

Abstract

This article investigates the optimal hedging problem of the European contingent claims written on non-tradable assets. We assume that the risky assets satisfy jump diffusion models with a common jump process which reflects the correlated jump risk. The non-tradable asset and jump risk lead to an incomplete financial market. Hence, the cross-hedging method will be used to reduce the potential risk of the contingent claims seller. First, we obtain an explicit closed-form solution for the locally risk-minimizing hedging strategies of the European contingent claims by using the Föllmer–Schweizer decomposition. Then, we consider the hedging for a European call option as a special case. The value of the European call option under the minimal martingale measure is derived by the Fourier transform method. Next, some semi-closed solution formulae of the locally risk-minimizing hedging strategies for the European call option are obtained. Finally, some numerical examples are provided to illustrate the sensitivities of the optimal hedging strategies. By comparing the optimal hedging strategies when the underlying asset is a non-tradable asset or a tradable asset, we find that the liquidity risk has a significant impact on the optimal hedging strategies.

Keywords

common jump riskhedging strategylocal risk minimizationnon-tradable assets
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 3 , July 2022 , pp. 581 - 605
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bajo, E., Barbi, M., & Romagnoli, S. (2014). Optimal corporate hedging using options with basis and production risk. North American Journal of Economics and Finance 30: 56–71.CrossRefGoogle Scholar
Bakshi, G. & Madan, D. (2000). Spanning and derivative-security valuation. Journal of Financial Economics 55(2): 205–238.CrossRefGoogle Scholar
Bates, D. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options. Review of Financial Studies 9: 69–107.CrossRefGoogle Scholar
Biagini, F. & Cretarola, A. (2010). Local risk minimization for defaultable markets. Mathematical Finance 19(4): 669–689.CrossRefGoogle Scholar
Carr, P. & Madan, D. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance 2(4): 61–73.CrossRefGoogle Scholar
Colwell, D., El-Hassan, N., & Kwon, O.K. (2007). Hedging diffusion processes by local risk minimization with applications to index tracking. Journal of Economic Dynamics and Control 31: 2135–2151.CrossRefGoogle Scholar
Davis, M. (2006). Optimal hedging with basis risk. In Y. Kabanov, R. Liptser, & J. Stoyanov (eds), From stochastic calculus to mathematical finance. Berlin, Heidelberg: Springer, pp. 169–187.CrossRefGoogle Scholar
Duffie, D., Pan, J., & Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6): 1343–1376.CrossRefGoogle Scholar
Elliott, R.J., Siu, T.K., Chan, L.G., & Lau, J.W. (2007). Pricing options under a generalized Markov-modulated jump-diffusion model. Stochastic Analysis and Applications 25(4): 821–843.CrossRefGoogle Scholar
Föllmer, H. & Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In M.H.A. Davis & R.J. Elliott (eds), Applied stochastic analysis. Stochastics Monographs, vol. 5. London/New York: Gordon and Breach, pp. 389–414.Google Scholar
Föllmer, H. & Sondermann, D. (1986). Hedging of non-redundant contingent claims. In W. Hildenbrand & A. Mas-Colell (eds), Contributions to mathematical economics. Amsterdam: North-Holland, pp. 205–223.Google Scholar
Fu, J. & Yang, H. (2012). Equilibrium approach of asset pricing under Lévy process. European Journal of Operational Research 223(3): 701–708.CrossRefGoogle Scholar
Henriksen, L.F.B. & Møller, T. (2015). Local risk-minimization with longevity bonds. Applied Stochastic Models in Business and Industry 31(2): 241–263.CrossRefGoogle Scholar
Heston, S. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6: 327–343.CrossRefGoogle Scholar
Kou, S.G. (2002). A jump diffusion model for option pricing. Management Science 48(8): 1086–1101.CrossRefGoogle Scholar
Lee, K. & Protter, P. (2008). Hedging claims with feed back jumps in the price process. Communications on Stochastic Analysis 2(1): 125–143.CrossRefGoogle Scholar
Ma, Y., Pan, D., Shrestha, K., & Xu, W. (2020). Pricing and hedging foreign equity options under Hawkes jump-diffusion processes. Physica A: Statistical Mechanics and Its Applications 537: 122645.CrossRefGoogle Scholar
Merton, R.C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics 3: 125–144.CrossRefGoogle Scholar
Møller, T. (1998). Risk minimizing hedging strategies for unit-linked life insurance contracts. Astin Bulletin 28(1): 17–47.CrossRefGoogle Scholar
Okhrati, R., Balbás, A., & Garrido, J. (2014). Hedging of defaultable claims in a structural model using a locally risk-minimizing approach. Stochastic Processes and Their Applications 124: 2869–2891.CrossRefGoogle Scholar
Pansera, J. (2012). Discrete-time local risk minimization of payment processes and applications to equity-linked life-insurance contracts. Insurance: Mathematics and Economics 50(1): 1–11.Google Scholar
Qian, L., Jin, Z., Wang, W., & Chen, L. (2018). Pricing dynamic fund protections for a hyperexponential jump diffusion process. Communications in Statistics – Theory and Methods 47(1): 210–221.CrossRefGoogle Scholar
Schweizer, M. (1988). Hedging of options in a general semimartingale model. Ph.D. thesis, ETH, Zurich, Switzerland.Google Scholar
Schweizer, M. (2001). A guided tour through quadratic hedging approaches. In E. Jouini, M. Museiela, & J. Cvitanic (eds), Option Pricing Interest Rates, and Risk Management. Cambridge: Cambridge University Press, pp. 538–574.CrossRefGoogle Scholar
Shen, Y. & Zeng, Y. (2015). Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process. Insurance: Mathematics and Economics 62: 118–137.Google Scholar
Shen, Y., Zhang, X., & Siu, T.K. (2014). Mean-variance portfolio selection under a constant elasticity of variance model. Operations Research Letters 42(5): 337–342.CrossRefGoogle Scholar
Su, X., Wang, W., & Kyo-Shin, H. (2012). Risk-minimizing option pricing under a Markov-modulated jump-diffusion model with stochastic volatility. Statistics and Probability Letters 82(10): 1777–1785.CrossRefGoogle Scholar
Vandaele, N. & Vanmaele, M. (2008). A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a L$\acute {e}$vy process financial market. Insurance: Mathematics and Economics 42: 1128–1137.Google Scholar
Wang, X. (2016). Pricing power exchange options with correlated jump risk. Finance Research Letters 19: 90–97.CrossRefGoogle Scholar
Wang, W., Zhuo, J., Qian, L., & Su, X. (2016). Local risk minimization for vulnerable European contingent claims on nontradable assets under regime switching models. Stochastic Analysis and Applications 34(4): 662–678.CrossRefGoogle Scholar
Xing, Y., Xu, Y., & Niu, H. (2020). Equilibrium valuation of currency options under a discontinuous model with co-jumps. Probability in the Engineering and Informational Sciences. doi:10.1017/S0269964819000500Google Scholar
Xue, X., Zhang, J., & Weng, C. (2019). Mean-variance hedging with basis risk. Applied Stochastic Models in Business and Industry 35(3): 704–716.CrossRefGoogle Scholar
Yu, X., Wan, Z., Tu, X., & Li, Y. (2020). The optimal multi-period hedging model of currency futures and options with exponential utility. Journal of Computational and Applied Mathematics 366: 112412.CrossRefGoogle Scholar
Zhang, J., Tan, K.S., & Weng, C. (2017). Optimal hedging with basis risk under mean variance criterion. Insurance: Mathematics and Economics 75: 1–15.Google Scholar