# American Institute of Mathematical Sciences

October  2014, 10(4): 1209-1224. doi: 10.3934/jimo.2014.10.1209

## Hedging strategies for discretely monitored Asian options under Lévy processes

 1 School of Mathematical Sciences, Nankai University, Tianjin 300071 2 School of Business, Nankai University, Tianjin 300071

Received  May 2013 Revised  December 2013 Published  February 2014

In this work, we consider a variance-optimal hedging strategy for discretely sampled geometric Asian options, under exponential Lévy dynamics. Since it is difficult to hedge these instruments perfectly, here we choose to maximize a quadratic utility function and give the expressions of hedging strategies explicitly, based on the derived Föllmer-Schweizer decomposition of the contingent claim of geometric Asian options monitored at discrete times. The expression of its corresponding error is also given.
Citation: 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
##### References:
 [1] J. Angus, A note on pricing Asian derivatives with continuous geometric averaging, Journal of Futures Markets, 19 (1999), 845-858. doi: 10.1002/(SICI)1096-9934(199910)19:7<845::AID-FUT6>3.3.CO;2-4. [2] E. Bayraktar and H. Xing, Pricing Asian options for jump diffusion, Mathematical Finance, 21 (2011), 117-143. doi: 10.1111/j.1467-9965.2010.00426.x. [3] N. Cai and S. G. Kou, Pricing Asian options under a hyper-exponential jump diffusion model, Operations Research, 60 (2012), 64-77. doi: 10.1287/opre.1110.1006. [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] P. Foschi, S. Pagliarani and A. Pascucci, Approximations for Asian options in local volatility models, Journal of Computational and Applied Mathematics, 237 (2013), 442-459. doi: 10.1016/j.cam.2012.06.015. [7] G. Fusai and A. Meucci, Pricing discretely monitored Asian options under Lévy processes, Journal of Banking and Finance, 32 (2008), 2076-2088. doi: 10.1016/j.jbankfin.2007.12.027. [8] S. Hodges and A. Neuberger, Optimal replication of contingent claims under transactions costs, Review of Forward Markets, 8 (1989), 222-239. [9] 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. [10] F. Hubalek and C. Sgarra, On the explicit evaluation of the geometric Asian options in stochastic volatility models with jumps, Journal of Computational and Applied Mathematics, 235 (2011), 3355-3365. doi: 10.1016/j.cam.2011.01.049. [11] B. Kim and I. S. Wee, Pricing of geometric Asian options under Heston's stochastic volatility model,, Quantitative Finance., ().  doi: 10.1080/14697688.2011.596844. [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] X. Wang and Y. Wang, Variance-optimal hedging for target volatility options, Journal of Industrial and Management Optimization, 10 (2014), 207-218. doi: 10.3934/jimo.2014.10.207.

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##### References:
 [1] J. Angus, A note on pricing Asian derivatives with continuous geometric averaging, Journal of Futures Markets, 19 (1999), 845-858. doi: 10.1002/(SICI)1096-9934(199910)19:7<845::AID-FUT6>3.3.CO;2-4. [2] E. Bayraktar and H. Xing, Pricing Asian options for jump diffusion, Mathematical Finance, 21 (2011), 117-143. doi: 10.1111/j.1467-9965.2010.00426.x. [3] N. Cai and S. G. Kou, Pricing Asian options under a hyper-exponential jump diffusion model, Operations Research, 60 (2012), 64-77. doi: 10.1287/opre.1110.1006. [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] P. Foschi, S. Pagliarani and A. Pascucci, Approximations for Asian options in local volatility models, Journal of Computational and Applied Mathematics, 237 (2013), 442-459. doi: 10.1016/j.cam.2012.06.015. [7] G. Fusai and A. Meucci, Pricing discretely monitored Asian options under Lévy processes, Journal of Banking and Finance, 32 (2008), 2076-2088. doi: 10.1016/j.jbankfin.2007.12.027. [8] S. Hodges and A. Neuberger, Optimal replication of contingent claims under transactions costs, Review of Forward Markets, 8 (1989), 222-239. [9] 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. [10] F. Hubalek and C. Sgarra, On the explicit evaluation of the geometric Asian options in stochastic volatility models with jumps, Journal of Computational and Applied Mathematics, 235 (2011), 3355-3365. doi: 10.1016/j.cam.2011.01.049. [11] B. Kim and I. S. Wee, Pricing of geometric Asian options under Heston's stochastic volatility model,, Quantitative Finance., ().  doi: 10.1080/14697688.2011.596844. [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] X. Wang and Y. Wang, Variance-optimal hedging for target volatility options, Journal of Industrial and Management Optimization, 10 (2014), 207-218. doi: 10.3934/jimo.2014.10.207.
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