
-
Previous Article
A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem
- JIMO Home
- This Issue
-
Next Article
Biobjective optimization over the efficient set of multiobjective integer programming problem
Pricing power exchange options with hawkes jump diffusion processes
Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, Delhi, 110016, India |
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.
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. |







Parameters | Values | Parameters | Values |
10 | 10 | ||
0.2 | 0.2 | ||
0 | 0 | ||
0.01 | 0.01 | ||
1 | 1 | ||
1 | 1 | ||
2 | 2 | ||
1 | 1 | ||
1 | 1 | ||
2 | 1 | ||
0.02 | 1 |
Parameters | Values | Parameters | Values |
10 | 10 | ||
0.2 | 0.2 | ||
0 | 0 | ||
0.01 | 0.01 | ||
1 | 1 | ||
1 | 1 | ||
2 | 2 | ||
1 | 1 | ||
1 | 1 | ||
2 | 1 | ||
0.02 | 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] |
Yuanshi Wang. Asymmetric diffusion in a two-patch mutualism system characterizing exchange of resource for resource. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 963-985. doi: 10.3934/dcdsb.2020149 |
[3] |
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 |
[4] |
Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477 |
[5] |
Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 |
[6] |
Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 |
[7] |
Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021018 |
[8] |
Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020404 |
[9] |
Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050 |
[10] |
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 |
[11] |
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 |
[12] |
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 |
[13] |
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 |
[14] |
Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020172 |
[15] |
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 |
[16] |
Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041 |
[17] |
Sugata Gangopadhyay, Constanza Riera, Pantelimon Stănică. Gowers $ U_2 $ norm as a measure of nonlinearity for Boolean functions and their generalizations. Advances in Mathematics of Communications, 2021, 15 (2) : 241-256. doi: 10.3934/amc.2020056 |
[18] |
Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169 |
[19] |
Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3427-3450. doi: 10.3934/dcds.2020029 |
[20] |
Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021003 |
2019 Impact Factor: 1.366
Tools
Metrics
Other articles
by authors
[Back to Top]