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

Pricing power exchange options with hawkes jump diffusion processes

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

* Corresponding author: Anubha Goel

Received  June 2018 Revised  March 2019 Published  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.

Citation: Puneet Pasricha, Anubha Goel. Pricing power exchange options with hawkes jump diffusion processes. Journal of Industrial & 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.  Google Scholar

[2]

Y. Aït-SahaliaJ. 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.  Google Scholar

[3]

G. BakshiC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

S. MeyerJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

X. WangS. 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.  Google Scholar

[34]

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

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.  Google Scholar

[2]

Y. Aït-SahaliaJ. 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.  Google Scholar

[3]

G. BakshiC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

S. MeyerJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

X. WangS. 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.  Google Scholar

[34]

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

Figure 1.  Option prices against time to maturity
Figure 2.  Option prices against correlation coefficient when T = 1.5
Figure 3.  Option prices against the parameters of common Hawkes process
Figure 4.  Option prices against the parameters of Hawkes process for Asset 1
Figure 5.  Option prices against the parameters of Hawkes process for Asset 2
Figure 6.  Comparison of option prices against the parameters of common Hawkes process and asset specific Hawkes processes
Figure 7.  Comparison of option prices against the parameters of amplitudes of jump sizes
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
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
[1]

Wan-Hua He, Chufang Wu, Jia-Wen Gu, Wai-Ki Ching, Chi-Wing Wong. Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021057

[2]

Wei Wang, Yang Shen, Linyi Qian, Zhixin Yang. Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021072

[3]

Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021039

[4]

Jun Moon. Linear-quadratic mean-field type stackelberg differential games for stochastic jump-diffusion systems. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021026

[5]

Zhenbing Gong, Yanping Chen, Wenyu Tao. Jump and variational inequalities for averaging operators with variable kernels. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021045

[6]

Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021, 3 (1) : 49-66. doi: 10.3934/fods.2021005

[7]

Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207

[8]

Jan Rychtář, Dewey T. Taylor. Moran process and Wright-Fisher process favor low variability. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3491-3504. doi: 10.3934/dcdsb.2020242

[9]

Lei Lei, Wenli Ren, Cuiling Fan. The differential spectrum of a class of power functions over finite fields. Advances in Mathematics of Communications, 2021, 15 (3) : 525-537. doi: 10.3934/amc.2020080

[10]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281

[11]

Alexander Tolstonogov. BV solutions of a convex sweeping process with a composed perturbation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021012

[12]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404

[13]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040

[14]

Ruchika Sehgal, Aparna Mehra. Worst-case analysis of Gini mean difference safety measure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1613-1637. doi: 10.3934/jimo.2020037

[15]

José Antonio Carrillo, Martin Parisot, Zuzanna Szymańska. Mathematical modelling of collagen fibres rearrangement during the tendon healing process. Kinetic & Related Models, 2021, 14 (2) : 283-301. doi: 10.3934/krm.2021005

[16]

Muberra Allahverdi, Harun Aydilek, Asiye Aydilek, Ali Allahverdi. A better dominance relation and heuristics for Two-Machine No-Wait Flowshops with Maximum Lateness Performance Measure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1973-1991. doi: 10.3934/jimo.2020054

[17]

Yunjuan Jin, Aifang Qu, Hairong Yuan. Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021048

[18]

Xianbang Chen, Yang Liu, Bin Li. Adjustable robust optimization in enabling optimal day-ahead economic dispatch of CCHP-MG considering uncertainties of wind-solar power and electric vehicle. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1639-1661. doi: 10.3934/jimo.2020038

[19]

Sumon Sarkar, Bibhas C. Giri. Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021048

[20]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (376)
  • HTML views (536)
  • Cited by (1)

Other articles
by authors

[Back to Top]