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.

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.

[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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