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

Hedging strategies for discretely monitored Asian options under Lévy processes

Xingchun Wang 1, and  Yongjin Wang 2,

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.
Keywords: Föllmer-Schweizer decomposition, Kunita-Watanabe decomposition, Asian options, variance-optimal hedging., Lévy processes.
Mathematics Subject Classification: Primary: 60G51, 91B70; Secondary: 65K1.
Citation: 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
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

R. Engle, Risk and Volatility: Econometric models and financial practice, American Economic Review, 94 (2004), 405-420. doi: 10.1257/0002828041464597.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Hodges and A. Neuberger, Optimal replication of contingent claims under transactions costs, Review of Forward Markets, 8 (1989), 222-239. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

D. Schweizer, Variance-optimal hedging in discrete time, Mathematics of Operations Research, 20 (1995), 1-32. doi: 10.1287/moor.20.1.1.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

R. Engle, Risk and Volatility: Econometric models and financial practice, American Economic Review, 94 (2004), 405-420. doi: 10.1257/0002828041464597.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

S. Hodges and A. Neuberger, Optimal replication of contingent claims under transactions costs, Review of Forward Markets, 8 (1989), 222-239. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

D. Schweizer, Variance-optimal hedging in discrete time, Mathematics of Operations Research, 20 (1995), 1-32. doi: 10.1287/moor.20.1.1.  Google Scholar

[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.  Google Scholar

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