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January  2021, 17(1): 133-149. doi: 10.3934/jimo.2019103

Pricing power exchange options with hawkes jump diffusion processes

Puneet Pasricha and  Anubha Goel ,

Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, Delhi, 110016, India

* Corresponding author: Anubha Goel

Received  June 2018 Revised  March 2019 Published  January 2021 Early access  September 2019

In this article, we propose a jump diffusion framework to price the power exchange options. We model the price dynamics of assets using a Hawkes jump diffusion model with common factors to describe the correlated jump risk and clustering of asset price jumps. In the proposed model, the jumps, reflecting common systematic risk and idiosyncratic risk, are modeled by self-exciting Hawkes process with exponential decay. A pricing formula for valuation of power exchange option is obtained following the measure-change technique. Existing models in the literature are shown to be special cases of the proposed model. Finally, sensitivity analysis is given to illustrate the effect of jump risk and jump clustering on option prices. We observe that jump clustering significantly effects the option prices.

Keywords: Jump clustering, Hawkes process, power exchange options, jump diffusion, change of measure.
Mathematics Subject Classification: Primary: 60G51; Secondary: 60G55.
Citation: Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial and Management Optimization, 2021, 17 (1) : 133-149. doi: 10.3934/jimo.2019103
References:
[1]

L. Adamopoulos, Cluster models for earthquakes: Regional comparisons, J. of the Internat. Assoc. for Math. Geology, 8 (1976), 463-475.  doi: 10.1007/BF01028982.

[2]

Y. Aït-Sahalia, J. Cacho-Diaz and R. J. Laeven, Modeling financial contagion using mutually exciting jump processes, J. Financial Economics, 117 (2015), 585-606.  doi: 10.1016/j.jfineco.2015.03.002.

[3]

G. Bakshi, C. Cao and Z. Chen, Empirical performance of alternative option pricing models, The Journal of Finance, 52 (1997), 2003-2049.  doi: 10.1111/j.1540-6261.1997.tb02749.x.

[4]

D. S. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options, The Review of Financial Studies, 9 (1996), 69-107.  doi: 10.1093/rfs/9.1.69.

[5]

D. S. Bates, Post-'87 crash fears in the S & P 500 futures option market, J. Econometrics, 94 (2000), 181-238.  doi: 10.1016/S0304-4076(99)00021-4.

[6]

L. P. Blenman and S. P. Clark, Power exchange options, Finance Research Letters, 2 (2005), 97-106.  doi: 10.1016/j.frl.2005.01.003.

[7]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081.  doi: 10.1287/mnsc.1110.1393.

[8]

P. Carr and L. Wu, Time-changed lévy processes and option pricing, J. Financial Economics, 71 (2004), 113-141.  doi: 10.1016/0304-405X(79)90015-1.

[9]

A. Dassios and H. Zhao, A dynamic contagion process, Adv. in Appl. Probab., 43 (2011), 814-846.  doi: 10.1239/aap/1316792671.

[10]

B. Eraker, Do stock prices and volatility jump? Reconciling evidence from spot and option prices, J. Finance, 59 (2004), 1367-1403.  doi: 10.1111/j.1540-6261.2004.00666.x.

[11]

B. Eraker, M. Johannes and N. Polson, The impact of jumps in volatility and returns, J. Finance, 58 (2003), 1269-1300.  doi: 10.1111/1540-6261.00566.

[12]

E. Errais, K. Giesecke and L. R. Goldberg, Affine point processes and portfolio credit risk, SIAM J. Financial Math., 1 (2010), 642-665.  doi: 10.1137/090771272.

[13]

S. Fischer, Call option pricing when the exercise price is uncertain, and the valuation of index bonds, J. Finance, 33 (1978), 169-176.  doi: 10.1111/j.1540-6261.1978.tb03396.x.

[14]

A. G. Hawkes, Point spectra of some mutually exciting point processes, J. Roy. Statist. Soc. Ser. B, 33 (1971), 438-443.  doi: 10.1111/j.2517-6161.1971.tb01530.x.

[15]

A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58 (1971), 83-90.  doi: 10.1093/biomet/58.1.83.

[16]

A. G. Hawkes, Hawkes processes and their applications to finance: A review, Quant. Finance, 18 (2018), 193-198.  doi: 10.1080/14697688.2017.1403131.

[17]

S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.  doi: 10.1287/mnsc.48.8.1086.166.

[18]

W. Liu and S.-P. Zhu, Pricing variance swaps under the Hawkes jump-diffusion process, J. Futures Markets, 39 (2019). doi: 10.1002/fut.21997.

[19]

T. Lux and M. Marchesi, Volatility clustering in financial markets: A microsimulation of interacting agents, Int. J. Theor. Appl. Finance, 3 (2000), 675-702.  doi: 10.1142/S0219024900000826.

[20]

Y. Ma, K. Shrestha and W. Xu, Pricing vulnerable options with jump clustering, J. Futures Markets, 37 (2017), 1155-1178.  doi: 10.1002/fut.21843.

[21]

Y. Ma and W. Xu, Structural credit risk modelling with Hawkes jump diffusion processes, J. Comput. Appl. Math., 303 (2016), 69-80.  doi: 10.1016/j.cam.2016.02.032.

[22]

J. M. Maheu and T. H. McCurdy, News arrival, jump dynamics, and volatility components for individual stock returns, J. Finance, 59 (2004), 755-793.  doi: 10.1111/j.1540-6261.2004.00648.x.

[23]

B. Mandelbrot, The variation of certain speculative prices, J. Business, 36 (1963), 394-419.  doi: 10.1086/294632.

[24]

W. Margrabe, The value of an option to exchange one asset for another, J. Finance, 33 (1978), 177-186.  doi: 10.1111/j.1540-6261.1978.tb03397.x.

[25]

S. Meyer, J. Elias and M. Höhle, A space–time conditional intensity model for invasive meningococcal disease occurrence, Biometrics, 68 (2012), 607-616.  doi: 10.1111/j.1541-0420.2011.01684.x.

[26]

Y. Ogata, On Lewis' simulation method for point processes, IEEE Transactions on Information Theory, 27 (1981), 23-31.  doi: 10.1109/TIT.1981.1056305.

[27]

Y. Ogata, Statistical models for earthquake occurrences and residual analysis for point processes, J. Amer. Statistical Association, 83 (1988), 9-27.  doi: 10.1080/01621459.1988.10478560.

[28]

J. Pan, The jump-risk premia implicit in options: Evidence from an integrated time-series study, J. of Financial Economics, 63 (2002), 3-50.  doi: 10.1016/S0304-405X(01)00088-5.

[29]

P. Pasricha and A. Goel, Pricing vulnerable power exchange options in an intensity based framework, J. Comput. Appl. Math., 355 (2019), 106-115.  doi: 10.1016/j.cam.2019.01.019.

[30]

A. Reinhart, A review of self-exciting spatio-temporal point processes and their applications, Statist. Sci., 33 (2018), 330-333.  doi: 10.1214/18-STS654.

[31]

R. Tompkins, Power options: hedging nonlinear risks, J. Risk, 2 (2000), 29-45.  doi: 10.21314/JOR.2000.022.

[32]

X. Wang, Pricing power exchange options with correlated jump risk, Finance Research Letters, 19 (2016), 90-97.  doi: 10.1016/j.frl.2016.06.009.

[33]

X. Wang, S. Song and Y. Wang, The valuation of power exchange options with counterparty risk and jump risk, J. Futures Markets, 37 (2017), 499-521.  doi: 10.1002/fut.21803.

[34]

J. Yu, Empirical characteristic function estimation and its applications, Econometric Rev., 23 (2004), 93-123.  doi: 10.1081/ETC-120039605.

show all references

References:
[1]

L. Adamopoulos, Cluster models for earthquakes: Regional comparisons, J. of the Internat. Assoc. for Math. Geology, 8 (1976), 463-475.  doi: 10.1007/BF01028982.

[2]

Y. Aït-Sahalia, J. Cacho-Diaz and R. J. Laeven, Modeling financial contagion using mutually exciting jump processes, J. Financial Economics, 117 (2015), 585-606.  doi: 10.1016/j.jfineco.2015.03.002.

[3]

G. Bakshi, C. Cao and Z. Chen, Empirical performance of alternative option pricing models, The Journal of Finance, 52 (1997), 2003-2049.  doi: 10.1111/j.1540-6261.1997.tb02749.x.

[4]

D. S. Bates, Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options, The Review of Financial Studies, 9 (1996), 69-107.  doi: 10.1093/rfs/9.1.69.

[5]

D. S. Bates, Post-'87 crash fears in the S & P 500 futures option market, J. Econometrics, 94 (2000), 181-238.  doi: 10.1016/S0304-4076(99)00021-4.

[6]

L. P. Blenman and S. P. Clark, Power exchange options, Finance Research Letters, 2 (2005), 97-106.  doi: 10.1016/j.frl.2005.01.003.

[7]

N. Cai and S. G. Kou, Option pricing under a mixed-exponential jump diffusion model, Management Science, 57 (2011), 2067-2081.  doi: 10.1287/mnsc.1110.1393.

[8]

P. Carr and L. Wu, Time-changed lévy processes and option pricing, J. Financial Economics, 71 (2004), 113-141.  doi: 10.1016/0304-405X(79)90015-1.

[9]

A. Dassios and H. Zhao, A dynamic contagion process, Adv. in Appl. Probab., 43 (2011), 814-846.  doi: 10.1239/aap/1316792671.

[10]

B. Eraker, Do stock prices and volatility jump? Reconciling evidence from spot and option prices, J. Finance, 59 (2004), 1367-1403.  doi: 10.1111/j.1540-6261.2004.00666.x.

[11]

B. Eraker, M. Johannes and N. Polson, The impact of jumps in volatility and returns, J. Finance, 58 (2003), 1269-1300.  doi: 10.1111/1540-6261.00566.

[12]

E. Errais, K. Giesecke and L. R. Goldberg, Affine point processes and portfolio credit risk, SIAM J. Financial Math., 1 (2010), 642-665.  doi: 10.1137/090771272.

[13]

S. Fischer, Call option pricing when the exercise price is uncertain, and the valuation of index bonds, J. Finance, 33 (1978), 169-176.  doi: 10.1111/j.1540-6261.1978.tb03396.x.

[14]

A. G. Hawkes, Point spectra of some mutually exciting point processes, J. Roy. Statist. Soc. Ser. B, 33 (1971), 438-443.  doi: 10.1111/j.2517-6161.1971.tb01530.x.

[15]

A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika, 58 (1971), 83-90.  doi: 10.1093/biomet/58.1.83.

[16]

A. G. Hawkes, Hawkes processes and their applications to finance: A review, Quant. Finance, 18 (2018), 193-198.  doi: 10.1080/14697688.2017.1403131.

[17]

S. G. Kou, A jump-diffusion model for option pricing, Management Science, 48 (2002), 1086-1101.  doi: 10.1287/mnsc.48.8.1086.166.

[18]

W. Liu and S.-P. Zhu, Pricing variance swaps under the Hawkes jump-diffusion process, J. Futures Markets, 39 (2019). doi: 10.1002/fut.21997.

[19]

T. Lux and M. Marchesi, Volatility clustering in financial markets: A microsimulation of interacting agents, Int. J. Theor. Appl. Finance, 3 (2000), 675-702.  doi: 10.1142/S0219024900000826.

[20]

Y. Ma, K. Shrestha and W. Xu, Pricing vulnerable options with jump clustering, J. Futures Markets, 37 (2017), 1155-1178.  doi: 10.1002/fut.21843.

[21]

Y. Ma and W. Xu, Structural credit risk modelling with Hawkes jump diffusion processes, J. Comput. Appl. Math., 303 (2016), 69-80.  doi: 10.1016/j.cam.2016.02.032.

[22]

J. M. Maheu and T. H. McCurdy, News arrival, jump dynamics, and volatility components for individual stock returns, J. Finance, 59 (2004), 755-793.  doi: 10.1111/j.1540-6261.2004.00648.x.

[23]

B. Mandelbrot, The variation of certain speculative prices, J. Business, 36 (1963), 394-419.  doi: 10.1086/294632.

[24]

W. Margrabe, The value of an option to exchange one asset for another, J. Finance, 33 (1978), 177-186.  doi: 10.1111/j.1540-6261.1978.tb03397.x.

[25]

S. Meyer, J. Elias and M. Höhle, A space–time conditional intensity model for invasive meningococcal disease occurrence, Biometrics, 68 (2012), 607-616.  doi: 10.1111/j.1541-0420.2011.01684.x.

[26]

Y. Ogata, On Lewis' simulation method for point processes, IEEE Transactions on Information Theory, 27 (1981), 23-31.  doi: 10.1109/TIT.1981.1056305.

[27]

Y. Ogata, Statistical models for earthquake occurrences and residual analysis for point processes, J. Amer. Statistical Association, 83 (1988), 9-27.  doi: 10.1080/01621459.1988.10478560.

[28]

J. Pan, The jump-risk premia implicit in options: Evidence from an integrated time-series study, J. of Financial Economics, 63 (2002), 3-50.  doi: 10.1016/S0304-405X(01)00088-5.

[29]

P. Pasricha and A. Goel, Pricing vulnerable power exchange options in an intensity based framework, J. Comput. Appl. Math., 355 (2019), 106-115.  doi: 10.1016/j.cam.2019.01.019.

[30]

A. Reinhart, A review of self-exciting spatio-temporal point processes and their applications, Statist. Sci., 33 (2018), 330-333.  doi: 10.1214/18-STS654.

[31]

R. Tompkins, Power options: hedging nonlinear risks, J. Risk, 2 (2000), 29-45.  doi: 10.21314/JOR.2000.022.

[32]

X. Wang, Pricing power exchange options with correlated jump risk, Finance Research Letters, 19 (2016), 90-97.  doi: 10.1016/j.frl.2016.06.009.

[33]

X. Wang, S. Song and Y. Wang, The valuation of power exchange options with counterparty risk and jump risk, J. Futures Markets, 37 (2017), 499-521.  doi: 10.1002/fut.21803.

[34]

J. Yu, Empirical characteristic function estimation and its applications, Econometric Rev., 23 (2004), 93-123.  doi: 10.1081/ETC-120039605.

Figure 1.  Option prices against time to maturity
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Figure 2.  Option prices against correlation coefficient when T = 1.5
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Figure 3.  Option prices against the parameters of common Hawkes process
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Figure 4.  Option prices against the parameters of Hawkes process for Asset 1
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Figure 5.  Option prices against the parameters of Hawkes process for Asset 2
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Figure 6.  Comparison of option prices against the parameters of common Hawkes process and asset specific Hawkes processes
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Figure 7.  Comparison of option prices against the parameters of amplitudes of jump sizes
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Table 1.  Values of the Parameters in the Base Case
Parameters Values Parameters Values
$ S_1(0) $ 10 $ S_2(0) $ 10
$ \sigma_1 $ 0.2 $ \sigma_2 $ 0.2
$ a_1 $ 0 $ a_2 $ 0
$ b_1 $ 0.01 $ b_2 $ 0.01
$ \lambda_{1, 0} $ 1 $ \lambda_{2, 0} $ 1
$ \theta_{1} $ 1 $ \theta_2 $ 1
$ \delta_1 $ 2 $ \delta_2 $ 2
$ \lambda_{0} $ 1 $ \alpha_1 $ 1
$ \theta $ 1 $ \alpha_2 $ 1
$ \delta $ 2 $ \eta_1 $ 1
$ r $ 0.02 $ \eta_2 $ 1
Table Options

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Parameters Values Parameters Values
$ S_1(0) $ 10 $ S_2(0) $ 10
$ \sigma_1 $ 0.2 $ \sigma_2 $ 0.2
$ a_1 $ 0 $ a_2 $ 0
$ b_1 $ 0.01 $ b_2 $ 0.01
$ \lambda_{1, 0} $ 1 $ \lambda_{2, 0} $ 1
$ \theta_{1} $ 1 $ \theta_2 $ 1
$ \delta_1 $ 2 $ \delta_2 $ 2
$ \lambda_{0} $ 1 $ \alpha_1 $ 1
$ \theta $ 1 $ \alpha_2 $ 1
$ \delta $ 2 $ \eta_1 $ 1
$ r $ 0.02 $ \eta_2 $ 1
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