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doi: 10.3934/jimo.2022024
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Pricing path-dependent options under the Hawkes jump diffusion process

School of International Trade and Economics, University of International Business and Economics, Beijing 100029, China

Received  August 2021 Revised  January 2022 Early access February 2022

Fund Project: This study was supported by the National Natural Science Foundation of China (No. 11701084)

In this paper, we investigate the pricing of a path-dependent option with default risk under the Hawkes jump diffusion process. For each asset, its dynamics are driven by a Hawkes jump diffusion process, and their diffusive components, Hawkes jumps as well as jump amplitudes are all correlated. In the proposed pricing framework, we obtain the prices of fader options with/without default risk in closed form. Finally, we present numerical examples to illustrate the prices of fader options with default risk.

Citation: Xingchun Wang. Pricing path-dependent options under the Hawkes jump diffusion process. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022024
References:
[1]

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

[2]

O. Brockhaus, A. Ferraris, C. Gallus, D. Long, R. Martin and M. Overhaus, Modelling and Hedging Equity Derivatives, Risk Books, London, 1999.

[3]

M. Cao and J. Wei, Vulnerable options, risky corporate bond, and credit spread, Journal of Futures Markets, 21 (2001), 301-327.  doi: 10.1002/1096-9934(200104)21:4<301::AID-FUT1>3.0.CO;2-K.

[4]

M. EscobarM. MahlstedtS. Panz and R. Zagst, Vulnerable exotic derivatives, Journal of Derivatives, 24 (2017), 84-102.  doi: 10.3905/jod.2017.24.3.084.

[5]

J. D. Fonseca and R. Zaatour, Hawkes process: Fast calibration, application to trade clustering, and diffusive limit, Journal of Futures Markets, 34 (2014), 548-579.  doi: 10.2139/ssrn.2294112.

[6]

S. Griebsch and U. Wystup, On the valuation of fader and discrete barrier options in Heston's stochastic volatility model, Quant. Finance, 11 (2011), 693-709.  doi: 10.1080/14697688.2010.503375.

[7]

J. Hakala and U. Wystup, Foreign Exchange Risk, Risk Books, London, 2002.

[8]

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

[9]

A. 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.

[10]

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

[11]

A. Hawkes, Hawkes jump-diffusions and finance: a brief history and review, European Journal of Finance, 2020. doi: 10.1080/1351847X.2020.1755712.

[12]

C. HuiC. Lo and H. Lee, Pricing vulnerable Black-Scholes options with dynamic default barriers, Journal of Derivatives, 10 (2003), 62-69.  doi: 10.3905/jod.2003.319206.

[13]

J. JeonJ. Yoon and M. Kang, Valuing vulnerable geometric Asian options, Computers and Mathematics with Applications, 71 (2016), 676-691.  doi: 10.1016/j.camwa.2015.12.038.

[14]

J. JeonJ. Yoon and M. Kang, Pricing vulnerable path-dependent options using integral transforms, J. Comput. Appl. Math., 313 (2017), 259-272.  doi: 10.1016/j.cam.2016.09.024.

[15]

H. Johnson and R. Stulz, The pricing of options with default risk, Journal of Finance, 42 (1987), 267-280.  doi: 10.1111/j.1540-6261.1987.tb02567.x.

[16]

P. Klein, Pricing Black-Scholes options with correlated credit risk, Journal of Banking and Finance, 20 (1996), 1211-1229.  doi: 10.1016/0378-4266(95)00052-6.

[17]

P. Klein and M. Inglis, Pricing vulnerable European options when the option's payoff can increase the risk of financial distress, Journal of Banking and Finance, 25 (2001), 993-1012.  doi: 10.1016/S0378-4266(00)00109-6.

[18]

G. Liang and X. Ren, The credit risk and pricing of OTC options, Asia-Pacific Financial Markets, 14 (2007), 45-68.  doi: 10.1007/s10690-007-9053-x.

[19]

W. Liu and S. Zhu, Pricing variance swaps under the Hawkes jump-diffusion process, Journal of Futures Markets, 39 (2019), 635-655.  doi: 10.1002/fut.21997.

[20]

Y. MaD. Pan and T. Wang, Exchange options under clustered jump dynamics, Quant. Finance, 20 (2020), 949-967.  doi: 10.1080/14697688.2019.1704045.

[21]

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

[22]

R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2.

[23]

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

[24]

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.

[25]

P. Pasricha and A. Goel, Pricing power exchange options with Hawkes jump diffusion processes, J. Ind. Manag. Optim., 17 (2021), 133-149.  doi: 10.3934/jimo.2019103.

[26]

P. Pasricha, X. Lu and S. Zhu, A note on the calculation of default probabilities in "Structural credit risk modeling with Hawkes jump-diffusion processes", J. Comput. Appl. Math., 381 (2021), Paper No. 113037, 8 pp. doi: 10.1016/j.cam.2020.113037.

[27]

N. Shephard, From characteristic function to a distribution function: A simple framework for theory, Econometric Theory, 7 (1991), 519-529.  doi: 10.1017/S0266466600004746.

[28]

L. TianG. WangX. Wang and Y. Wang, Pricing vulnerable options with correlated credit risk under jump-diffusion processes, Journal of Futures Markets, 34 (2014), 957-979.  doi: 10.1002/fut.21629.

[29]

C. Tsao and C. Liu, Asian options with credit risks: Pricing and sensitivity analysis, Emerging Markets Finance and Trade, 48 (2012), 96-115.  doi: 10.2753/REE1540-496X4805S306.

[30]

G. WangX. Wang and K. Zhou, Pricing vulnerable options with stochastic volatility, Phys. A, 485 (2017), 91-103.  doi: 10.1016/j.physa.2017.04.146.

[31]

H. Wang, J. Zhang and K. Zhou, On pricing of vulnerable barrier options and vulnerable double barrier options, Finance Research Letters, 44 (2022). doi: 10.1016/j.frl.2021.102100.

[32]

X. Wang, Differences in the prices of vulnerable options with different counterparties, Journal of Futures Markets, 37 (2017), 148-163.  doi: 10.1002/fut.21789.

[33]

X. Wang, Analytical valuation of Asian options with counterparty risk under stochastic volatility models, Journal of Futures Markets, 40 (2020), 410-429.  doi: 10.1002/fut.22064.

[34]

X. Wang, Valuing fade-in options with default risk in Heston-Nandi GARCH models, Review of Derivatives Research, 2021. doi: 10.1007/s11147-021-09179-3.

[35]

X. Wang, Pricing vulnerable options with jump risk and liquidity risk, Review of Derivatives Research, 24 (2021), 243-260.  doi: 10.1007/s11147-021-09177-5.

[36]

X. Wang, Valuation of options on the maximum of two prices with default risk under GARCH models, North American Journal of Economics and Finance, 57 (2021), 101422.  doi: 10.1016/j.najef.2021.101422.

[37]

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

[38]

U. Wystup, FX Options and Structured Products, Wiley, New York, 2006. doi: 10.1002/9781118673355.

[39]

W. XuW. XuH. Li and W. Xiao, A jump-diffusion approach to modelling vulnerable option pricing, Finance Research Letters, 9 (2012), 48-56.  doi: 10.1016/j.frl.2011.07.001.

[40]

S. YangM. Lee and J. Kim, Pricing vulnerable options under a stochastic volatility model, Appl. Math. Lett., 34 (2014), 7-12.  doi: 10.1016/j.aml.2014.03.007.

show all references

References:
[1]

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

[2]

O. Brockhaus, A. Ferraris, C. Gallus, D. Long, R. Martin and M. Overhaus, Modelling and Hedging Equity Derivatives, Risk Books, London, 1999.

[3]

M. Cao and J. Wei, Vulnerable options, risky corporate bond, and credit spread, Journal of Futures Markets, 21 (2001), 301-327.  doi: 10.1002/1096-9934(200104)21:4<301::AID-FUT1>3.0.CO;2-K.

[4]

M. EscobarM. MahlstedtS. Panz and R. Zagst, Vulnerable exotic derivatives, Journal of Derivatives, 24 (2017), 84-102.  doi: 10.3905/jod.2017.24.3.084.

[5]

J. D. Fonseca and R. Zaatour, Hawkes process: Fast calibration, application to trade clustering, and diffusive limit, Journal of Futures Markets, 34 (2014), 548-579.  doi: 10.2139/ssrn.2294112.

[6]

S. Griebsch and U. Wystup, On the valuation of fader and discrete barrier options in Heston's stochastic volatility model, Quant. Finance, 11 (2011), 693-709.  doi: 10.1080/14697688.2010.503375.

[7]

J. Hakala and U. Wystup, Foreign Exchange Risk, Risk Books, London, 2002.

[8]

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

[9]

A. 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.

[10]

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

[11]

A. Hawkes, Hawkes jump-diffusions and finance: a brief history and review, European Journal of Finance, 2020. doi: 10.1080/1351847X.2020.1755712.

[12]

C. HuiC. Lo and H. Lee, Pricing vulnerable Black-Scholes options with dynamic default barriers, Journal of Derivatives, 10 (2003), 62-69.  doi: 10.3905/jod.2003.319206.

[13]

J. JeonJ. Yoon and M. Kang, Valuing vulnerable geometric Asian options, Computers and Mathematics with Applications, 71 (2016), 676-691.  doi: 10.1016/j.camwa.2015.12.038.

[14]

J. JeonJ. Yoon and M. Kang, Pricing vulnerable path-dependent options using integral transforms, J. Comput. Appl. Math., 313 (2017), 259-272.  doi: 10.1016/j.cam.2016.09.024.

[15]

H. Johnson and R. Stulz, The pricing of options with default risk, Journal of Finance, 42 (1987), 267-280.  doi: 10.1111/j.1540-6261.1987.tb02567.x.

[16]

P. Klein, Pricing Black-Scholes options with correlated credit risk, Journal of Banking and Finance, 20 (1996), 1211-1229.  doi: 10.1016/0378-4266(95)00052-6.

[17]

P. Klein and M. Inglis, Pricing vulnerable European options when the option's payoff can increase the risk of financial distress, Journal of Banking and Finance, 25 (2001), 993-1012.  doi: 10.1016/S0378-4266(00)00109-6.

[18]

G. Liang and X. Ren, The credit risk and pricing of OTC options, Asia-Pacific Financial Markets, 14 (2007), 45-68.  doi: 10.1007/s10690-007-9053-x.

[19]

W. Liu and S. Zhu, Pricing variance swaps under the Hawkes jump-diffusion process, Journal of Futures Markets, 39 (2019), 635-655.  doi: 10.1002/fut.21997.

[20]

Y. MaD. Pan and T. Wang, Exchange options under clustered jump dynamics, Quant. Finance, 20 (2020), 949-967.  doi: 10.1080/14697688.2019.1704045.

[21]

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

[22]

R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2.

[23]

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

[24]

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.

[25]

P. Pasricha and A. Goel, Pricing power exchange options with Hawkes jump diffusion processes, J. Ind. Manag. Optim., 17 (2021), 133-149.  doi: 10.3934/jimo.2019103.

[26]

P. Pasricha, X. Lu and S. Zhu, A note on the calculation of default probabilities in "Structural credit risk modeling with Hawkes jump-diffusion processes", J. Comput. Appl. Math., 381 (2021), Paper No. 113037, 8 pp. doi: 10.1016/j.cam.2020.113037.

[27]

N. Shephard, From characteristic function to a distribution function: A simple framework for theory, Econometric Theory, 7 (1991), 519-529.  doi: 10.1017/S0266466600004746.

[28]

L. TianG. WangX. Wang and Y. Wang, Pricing vulnerable options with correlated credit risk under jump-diffusion processes, Journal of Futures Markets, 34 (2014), 957-979.  doi: 10.1002/fut.21629.

[29]

C. Tsao and C. Liu, Asian options with credit risks: Pricing and sensitivity analysis, Emerging Markets Finance and Trade, 48 (2012), 96-115.  doi: 10.2753/REE1540-496X4805S306.

[30]

G. WangX. Wang and K. Zhou, Pricing vulnerable options with stochastic volatility, Phys. A, 485 (2017), 91-103.  doi: 10.1016/j.physa.2017.04.146.

[31]

H. Wang, J. Zhang and K. Zhou, On pricing of vulnerable barrier options and vulnerable double barrier options, Finance Research Letters, 44 (2022). doi: 10.1016/j.frl.2021.102100.

[32]

X. Wang, Differences in the prices of vulnerable options with different counterparties, Journal of Futures Markets, 37 (2017), 148-163.  doi: 10.1002/fut.21789.

[33]

X. Wang, Analytical valuation of Asian options with counterparty risk under stochastic volatility models, Journal of Futures Markets, 40 (2020), 410-429.  doi: 10.1002/fut.22064.

[34]

X. Wang, Valuing fade-in options with default risk in Heston-Nandi GARCH models, Review of Derivatives Research, 2021. doi: 10.1007/s11147-021-09179-3.

[35]

X. Wang, Pricing vulnerable options with jump risk and liquidity risk, Review of Derivatives Research, 24 (2021), 243-260.  doi: 10.1007/s11147-021-09177-5.

[36]

X. Wang, Valuation of options on the maximum of two prices with default risk under GARCH models, North American Journal of Economics and Finance, 57 (2021), 101422.  doi: 10.1016/j.najef.2021.101422.

[37]

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

[38]

U. Wystup, FX Options and Structured Products, Wiley, New York, 2006. doi: 10.1002/9781118673355.

[39]

W. XuW. XuH. Li and W. Xiao, A jump-diffusion approach to modelling vulnerable option pricing, Finance Research Letters, 9 (2012), 48-56.  doi: 10.1016/j.frl.2011.07.001.

[40]

S. YangM. Lee and J. Kim, Pricing vulnerable options under a stochastic volatility model, Appl. Math. Lett., 34 (2014), 7-12.  doi: 10.1016/j.aml.2014.03.007.

Figure 1.  Fader option prices against strike prices. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively
Figure 2.  Fader option prices against initial levels of the intensity. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively
Figure 3.  Fader option prices against the values of $ \theta $. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively
Figure 4.  Fader option prices against the values of $ \delta $. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively
Figure 5.  Fader option prices against initial values of issuer's assets. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively
Figure 6.  Fader option prices against the levels of issuer's debt. The dotted, solid, dashed and dot-dashed lines correspond to fader option prices with default risk and $ \alpha = 0.40 $, fader option prices with default risk and $ \alpha = 0.50 $, fader option prices with default risk and $ \alpha = 0.60 $, and fader option prices without default risk, respectively
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