Skip to main content
Erschienen in: Annals of Finance 1/2020

28.08.2019 | Research Article

Asian options pricing in Hawkes-type jump-diffusion models

verfasst von: Riccardo Brignone, Carlo Sgarra

Erschienen in: Annals of Finance | Ausgabe 1/2020

Einloggen

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this paper we propose a method for pricing Asian options in market models with the risky asset dynamics driven by a Hawkes process with exponential kernel. For these processes the couple \( (\lambda (t), X(t) ) \) is affine, this property allows to extend the general methodology introduced by Hubalek et al. (Quant Finance 17:873–888, 2017) for Geometric Asian option pricing to jump-diffusion models with stochastic jump intensity. Although the system of ordinary differential equations providing the characteristic function of the related affine process cannot be solved in closed form, a COS-type algorithm allows to obtain the relevant quantities needed for options valuation. We describe, by means of graphical illustrations, the dependence of Asian options prices by the main parameters of the driving Hawkes process. Finally, by using Geometric Asian options values as control variates, we show that Arithmetic Asian options prices can be computed in a fast and efficient way by a standard Monte Carlo method.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Fußnoten
1
Similarly, posing \(u_3=u\), \(u_1 = u_2 =u_4 = 0\) one gets the cumulants of Y(T).
 
Literatur
Zurück zum Zitat Abate, J., Whitt, W.: The Fourier-series method for inverting transforms of probability distributions. Queueing Syst Theory Appl 10, 5–88 (1992)CrossRef Abate, J., Whitt, W.: The Fourier-series method for inverting transforms of probability distributions. Queueing Syst Theory Appl 10, 5–88 (1992)CrossRef
Zurück zum Zitat Ait-Sahalia, Y., Chaco-Diaz, S., Laeven, R.: Modelling financial contagion using mutually exciting jump processes. J Financ Econ 117, 585–606 (2015)CrossRef Ait-Sahalia, Y., Chaco-Diaz, S., Laeven, R.: Modelling financial contagion using mutually exciting jump processes. J Financ Econ 117, 585–606 (2015)CrossRef
Zurück zum Zitat Bacry, E., Delattre, S., Hoffman, M., Muzy, J.F.: Modelling microstructure noise by mutually exciting point processes. Quant Finance 13, 65–77 (2013)CrossRef Bacry, E., Delattre, S., Hoffman, M., Muzy, J.F.: Modelling microstructure noise by mutually exciting point processes. Quant Finance 13, 65–77 (2013)CrossRef
Zurück zum Zitat Barndorff-Nielsen, O.E., Shephard, N.: Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc (Ser B) 63, 167–241 (2001)CrossRef Barndorff-Nielsen, O.E., Shephard, N.: Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc (Ser B) 63, 167–241 (2001)CrossRef
Zurück zum Zitat Barndorff-Nielsen, O.E., Nicolato, E., Shephard, N.: Some recent developments in stochastic volatility modelling. Quant Finance 2, 11–23 (2002)CrossRef Barndorff-Nielsen, O.E., Nicolato, E., Shephard, N.: Some recent developments in stochastic volatility modelling. Quant Finance 2, 11–23 (2002)CrossRef
Zurück zum Zitat Bates, D.: Jumps and stochastic volatility: the exchange rate processes implicit in Deutsche Mark options. Rev Financ Stud 9, 69–107 (1996)CrossRef Bates, D.: Jumps and stochastic volatility: the exchange rate processes implicit in Deutsche Mark options. Rev Financ Stud 9, 69–107 (1996)CrossRef
Zurück zum Zitat Bates, D.: Post-’87 crash fears in the S&P 500 futures option market. J Econom 94, 181–238 (2000)CrossRef Bates, D.: Post-’87 crash fears in the S&P 500 futures option market. J Econom 94, 181–238 (2000)CrossRef
Zurück zum Zitat Bernis, G., Salhi, K., Scotti, S.: Sensitivity analysis for marked Hawkes processes: application to CLO pricing. Math Financ Econ 12, 541–559 (2018)CrossRef Bernis, G., Salhi, K., Scotti, S.: Sensitivity analysis for marked Hawkes processes: application to CLO pricing. Math Financ Econ 12, 541–559 (2018)CrossRef
Zurück zum Zitat Carr, P., Wu, L.: Time-changed Lévy processes and option pricing. J Finan Econ 71, 113–141 (2004)CrossRef Carr, P., Wu, L.: Time-changed Lévy processes and option pricing. J Finan Econ 71, 113–141 (2004)CrossRef
Zurück zum Zitat Carr, P., Geman, H., Madan, D., Yor, M.: Stochastic volatility for Lévy processes. Math Finance 13, 345–382 (2003)CrossRef Carr, P., Geman, H., Madan, D., Yor, M.: Stochastic volatility for Lévy processes. Math Finance 13, 345–382 (2003)CrossRef
Zurück zum Zitat Cont, R., Tankov, P.: Financial Modelling With Jump Processes. Boca Raton: Chapman & Hall/CRC (2004) Cont, R., Tankov, P.: Financial Modelling With Jump Processes. Boca Raton: Chapman & Hall/CRC (2004)
Zurück zum Zitat Dassios, A., Zhao, H.: Exact simulation of Hawkes process with exponentially decaying intensity. Electron Commun Probab 18, 1–13 (2013)CrossRef Dassios, A., Zhao, H.: Exact simulation of Hawkes process with exponentially decaying intensity. Electron Commun Probab 18, 1–13 (2013)CrossRef
Zurück zum Zitat Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann Appl Probab 13, 984–1053 (2003)CrossRef Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann Appl Probab 13, 984–1053 (2003)CrossRef
Zurück zum Zitat Dufresne, D.: The integral of geometric Brownian motion. Adv Appl Probab 33, 223–241 (2001)CrossRef Dufresne, D.: The integral of geometric Brownian motion. Adv Appl Probab 33, 223–241 (2001)CrossRef
Zurück zum Zitat Errais, E., Giesecke, K., Goldberg, L.R.: Affine point processes and portfolio credit risk. SIAM J Financ Math 1, 642–665 (2010)CrossRef Errais, E., Giesecke, K., Goldberg, L.R.: Affine point processes and portfolio credit risk. SIAM J Financ Math 1, 642–665 (2010)CrossRef
Zurück zum Zitat Fang, F., Oosterlee, C.W.: A novel pricing method for European options based on Fourier-Cosine series expansions. SIAM J Sci Comput 31, 826–848 (2008)CrossRef Fang, F., Oosterlee, C.W.: A novel pricing method for European options based on Fourier-Cosine series expansions. SIAM J Sci Comput 31, 826–848 (2008)CrossRef
Zurück zum Zitat Filimonov, V., Bicchetti, D., Maystre, M., Sornette, D.: Quantification of the high level of endogeneity and of structural regime shifts in commodity markets. J Int Money Finance 42, 174–192 (2014)CrossRef Filimonov, V., Bicchetti, D., Maystre, M., Sornette, D.: Quantification of the high level of endogeneity and of structural regime shifts in commodity markets. J Int Money Finance 42, 174–192 (2014)CrossRef
Zurück zum Zitat Fu, M.C., Madan, D.B., Wang, T.: Pricing continuous Asian options: a comparison of Monte Carlo and Laplace transform inversion methods. J Comput Finance 2, 49–74 (1999)CrossRef Fu, M.C., Madan, D.B., Wang, T.: Pricing continuous Asian options: a comparison of Monte Carlo and Laplace transform inversion methods. J Comput Finance 2, 49–74 (1999)CrossRef
Zurück zum Zitat Fulop, A., Li, J., Yu, J.: Self-exciting jumps, learning, and asset pricing implications. Rev Financ Stud 28, 876–912 (2015)CrossRef Fulop, A., Li, J., Yu, J.: Self-exciting jumps, learning, and asset pricing implications. Rev Financ Stud 28, 876–912 (2015)CrossRef
Zurück zum Zitat Fusai, G., Kyriakou, I.: General optimized lower and upper bounds for discrete and continuous arithmetic Asian options. Math Oper Res 41, 531–559 (2016)CrossRef Fusai, G., Kyriakou, I.: General optimized lower and upper bounds for discrete and continuous arithmetic Asian options. Math Oper Res 41, 531–559 (2016)CrossRef
Zurück zum Zitat Geman, H., Yor, M.: Bessel processes, Asian options, and perpetuities. Math Finance 3, 349–375 (1993)CrossRef Geman, H., Yor, M.: Bessel processes, Asian options, and perpetuities. Math Finance 3, 349–375 (1993)CrossRef
Zurück zum Zitat Glasserman, P.: Monte Carlo Methods in Financial Engineering. New York: Springer (2004) Glasserman, P.: Monte Carlo Methods in Financial Engineering. New York: Springer (2004)
Zurück zum Zitat Hainaut, D., Moraux, F.: Hedging of options in the presence of jump clustering. J Comput Finance 22, 1–35 (2018)CrossRef Hainaut, D., Moraux, F.: Hedging of options in the presence of jump clustering. J Comput Finance 22, 1–35 (2018)CrossRef
Zurück zum Zitat Hainaut, D., Moraux, F.: A switching self-exciting jump diffusion process for stock prices. Ann Finance 15(2), 267–306 (2019)CrossRef Hainaut, D., Moraux, F.: A switching self-exciting jump diffusion process for stock prices. Ann Finance 15(2), 267–306 (2019)CrossRef
Zurück zum Zitat Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6, 327–343 (1993)CrossRef Heston, S.L.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6, 327–343 (1993)CrossRef
Zurück zum Zitat Hubalek, F., Sgarra, C.: On the explicit evaluation of the geometric Asian options in stochastic volatility models with jumps. J Comput Appl Math 235, 3355–3365 (2011)CrossRef Hubalek, F., Sgarra, C.: On the explicit evaluation of the geometric Asian options in stochastic volatility models with jumps. J Comput Appl Math 235, 3355–3365 (2011)CrossRef
Zurück zum Zitat Hubalek, F., Keller-Ressel, M., Sgarra, C.: Geometric Asian option pricing in general affine stochastic volatility models with jumps. Quant Finance 17, 873–888 (2017)CrossRef Hubalek, F., Keller-Ressel, M., Sgarra, C.: Geometric Asian option pricing in general affine stochastic volatility models with jumps. Quant Finance 17, 873–888 (2017)CrossRef
Zurück zum Zitat Jiao, Y., Ma, C., Scotti, S.: Alpha-CIR model with branching processes in sovereign interest rate modeling. Finance Stoch 21, 789–813 (2017)CrossRef Jiao, Y., Ma, C., Scotti, S.: Alpha-CIR model with branching processes in sovereign interest rate modeling. Finance Stoch 21, 789–813 (2017)CrossRef
Zurück zum Zitat Jiao, Y., Ma, C., Scotti, S., Sgarra, C.: A branching process approach to power markets. Energy Econ pp. 79, 1–13 (2018) Jiao, Y., Ma, C., Scotti, S., Sgarra, C.: A branching process approach to power markets. Energy Econ pp. 79, 1–13 (2018)
Zurück zum Zitat Keller-Ressel, M.: Moment explosions and long-term behavior of affine stochastic volatility models. Math Finance 21, 73–98 (2011)CrossRef Keller-Ressel, M.: Moment explosions and long-term behavior of affine stochastic volatility models. Math Finance 21, 73–98 (2011)CrossRef
Zurück zum Zitat Kemna, A.G.Z., Vorst, A.C.F.: A pricing method for options based on average asset values. J Bank Finance 14, 113–129 (1990)CrossRef Kemna, A.G.Z., Vorst, A.C.F.: A pricing method for options based on average asset values. J Bank Finance 14, 113–129 (1990)CrossRef
Zurück zum Zitat Kiesel, R., Paraschiv, F.: Econometric analysis of \(15\)-minute intraday electricity prices. Energy Econ 64, 77–90 (2017)CrossRef Kiesel, R., Paraschiv, F.: Econometric analysis of \(15\)-minute intraday electricity prices. Energy Econ 64, 77–90 (2017)CrossRef
Zurück zum Zitat Merton, R.: Option pricing when underlying stock returns are discontinuous. J Financ Econ 3, 125–144 (1976)CrossRef Merton, R.: Option pricing when underlying stock returns are discontinuous. J Financ Econ 3, 125–144 (1976)CrossRef
Zurück zum Zitat Ogata, Y.: On Lewis simulation method for point processes. IEEE Trans Inf Theory 27, 23–31 (1981)CrossRef Ogata, Y.: On Lewis simulation method for point processes. IEEE Trans Inf Theory 27, 23–31 (1981)CrossRef
Zurück zum Zitat Rambaldi, M., Pennesi, P., Lillo, F.: Modeling foreign exchange market activity around macroeconomic news: Hawkes process approach. Phys Rev E 91, 012819 (2015)CrossRef Rambaldi, M., Pennesi, P., Lillo, F.: Modeling foreign exchange market activity around macroeconomic news: Hawkes process approach. Phys Rev E 91, 012819 (2015)CrossRef
Zurück zum Zitat Wilmott, P., Dewynne, J., Howison, S.: The Mathematics of Financial Derivatives. Cambridge: Cambridge University Press (1995) Wilmott, P., Dewynne, J., Howison, S.: The Mathematics of Financial Derivatives. Cambridge: Cambridge University Press (1995)
Metadaten
Titel
Asian options pricing in Hawkes-type jump-diffusion models
verfasst von
Riccardo Brignone
Carlo Sgarra
Publikationsdatum
28.08.2019
Verlag
Springer Berlin Heidelberg
Erschienen in
Annals of Finance / Ausgabe 1/2020
Print ISSN: 1614-2446
Elektronische ISSN: 1614-2454
DOI
https://doi.org/10.1007/s10436-019-00352-1

Weitere Artikel der Ausgabe 1/2020

Annals of Finance 1/2020 Zur Ausgabe