• Previous Article
    An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts
  • JIMO Home
  • This Issue
  • Next Article
    Two penalized mixed–integer nonlinear programming approaches to tackle multicollinearity and outliers effects in linear regression models
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, 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]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[2]

Emre Esentürk, Juan Velazquez. Large time behavior of exchange-driven growth. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 747-775. doi: 10.3934/dcds.2020299

[3]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[4]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[5]

Harrison Bray. Ergodicity of Bowen–Margulis measure for the Benoist 3-manifolds. Journal of Modern Dynamics, 2020, 16: 305-329. doi: 10.3934/jmd.2020011

[6]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[7]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327

[8]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

[9]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[10]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[11]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[12]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020354

[13]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[14]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[15]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[16]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[17]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[18]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[19]

Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (241)
  • HTML views (478)
  • Cited by (0)

Other articles
by authors

[Back to Top]