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Change of measure in a Heston–Hawkes stochastic volatility model

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  24 July 2024

David R. Baños ,
Salvador Ortiz-Latorre  and
Oriol Zamora Font Open the ORCID record for Oriol Zamora Font [Opens in a new window]
David R. Baños*
Affiliation:
University of Oslo
Salvador Ortiz-Latorre*
Affiliation:
University of Oslo
Oriol Zamora Font*
Affiliation:
University of Oslo
*
*Postal address: Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway.
*Postal address: Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway.
*Postal address: Department of Mathematics, University of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway.
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Abstract

We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology, and other fields. We prove a general result on the existence of a family of equivalent (local) martingale measures. We apply this result to a particular example where the sizes of the jumps are exponentially distributed. Finally, a practical application to efficient computation of exposures is discussed.

Keywords

Stochastic volatilitychange of measurerisk-neutral measureexistence of equivalent martingale measurenon-Markovian modelHawkes processvolatility with jumps

MSC classification

Primary: 60G55: Point processes 60H10: Stochastic ordinary differential equations
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 30
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Alfonsi, A. (2015). Affine Diffusions and Related Processes: Simulation, Theory and Applications. Springer, Cham.Google Scholar
Alòs, E., León, J. A. and Vives, J. (2007). On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch. 11, 571–589.CrossRefGoogle Scholar
Andersen, T. G., Benzoni, L. and Lund, J. (2002). An empirical investigation of continuous-time equity return models. J. Finance 57, 1239–1284.CrossRefGoogle Scholar
Bacry, E., Mastromatteo, I. and Muzy, J.-F. (2015). Hawkes processes in finance. Market Microstruct. Liq. 01, article no. 1550005.CrossRefGoogle Scholar
Bates, D. S. (1996). Jumps and stochastic volatility: exchange rate processes implicit in Deutschemark options. Rev. Financial Studies 9, 69–107.CrossRefGoogle Scholar
Bates, D. S. (2000). Post-’87 crash fears in the S&P 500 futures option market. J. Econometrics 94, 181–238.CrossRefGoogle Scholar
Bernis, G., Brignone, R., Scotti, S. and Sgarra, C. (2021). A gamma Ornstein–Uhlenbeck model driven by a Hawkes process. Math. Financial Econom. 15, 747–773.CrossRefGoogle ScholarPubMed
Bernis, G., Garcin, M., Scotti, S. and Sgarra, C. (2023). Interest rates term structure models driven by Hawkes processes. SIAM J. Financial Math. 14, 1062–1079.CrossRefGoogle Scholar
Bibby, B. and Sørensen, M. (1996). A hyperbolic diffusion model for stock prices. Finance Stoch. 1, 25–41.CrossRefGoogle Scholar
Broadie, M. and Glasserman, P. (2004). A stochastic mesh method for pricing high-dimensional American options. J. Comput. Finance 7, 35–72.CrossRefGoogle Scholar
Callegaro, G., Mazzoran, A. and Sgarra, C. (2022). A self-exciting modeling framework for forward prices in power markets. Appl. Stoch. Models Business Industry 38, 27–48.CrossRefGoogle Scholar
Ceci, C. and Gerardi, A. (2006). A model for high frequency data under partial information: a filtering approach. Internat. J. Theoret. Appl. Finance 9, 555–576.CrossRefGoogle Scholar
Cesari, G. et al. (2009). Modelling, Pricing, and Hedging Counterparty Credit Exposure. Springer, Berlin.CrossRefGoogle Scholar
Chernov, M. and Ghysels, E. (2000). A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation. J. Financial Econom. 56, 407–458.CrossRefGoogle Scholar
Comte, F. and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8, 291–323.CrossRefGoogle Scholar
Cont, R. (2007). Volatility clustering in financial markets: empirical facts and agent-based models. In Long Memory in Economics, Springer, Berlin, pp. 289–309.CrossRefGoogle Scholar
Cont, R. and Kokholm, T. (2013). A consistent pricing model for index options and volatility derivatives. Math. Finance 23, 248–274.CrossRefGoogle Scholar
Delbaen, F. and Schachermayer, W. (1994). A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520.CrossRefGoogle Scholar
Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 984–1053.CrossRefGoogle Scholar
Duffie, D., Pan, J. and Singleton, K. (2000). Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376.CrossRefGoogle Scholar
El Euch, O., Fukasawa, M. and Rosenbaum, M. (2018). The microstructural foundations of leverage effect and rough volatility. Finance Stoch. 22, 241–280.CrossRefGoogle Scholar
El Euch, O. and Rosenbaum, M. (2018). Perfect hedging in rough Heston models. Ann. Appl. Prob. 28, 3813–3856.CrossRefGoogle Scholar
El Euch, O. and Rosenbaum, M. (2019). The characteristic function of rough Heston models. Math. Finance 29, 3–38.CrossRefGoogle Scholar
Eraker, B., Johannes, M. and Polson, N. (2003). The impact of jumps in volatility and returns. J. Finance 58, 1269–1300.CrossRefGoogle Scholar
Filimonov, V., Bicchetti, D., Maystre, N. and Sornette, D. (2014). Quantification of the high level of endogeneity and of structural regime shifts in commodity markets. J. Internat. Money Finance 42, 174–192.CrossRefGoogle Scholar
Gatheral, J., Jaisson, T. and Rosenbaum, M. (2018). Volatility is rough. Quant. Finance 18, 933–949.CrossRefGoogle Scholar
Glasserman, P. and Kim, K.-K. (2011). Gamma expansion of the Heston stochastic volatility model. Finance Stoch. 15, 267–296.CrossRefGoogle Scholar
Gonzato, L. and Sgarra, C. (2021). Self-exciting jumps in the oil market: Bayesian estimation and dynamic hedging. Energy Econom. 99, article no. 105279.CrossRefGoogle Scholar
Grasselli, M. (2017). The 4/2 stochastic volatility model: a unified approach for the Heston and the 3/2 model. Math. Finance 27, 1013–1034.CrossRefGoogle Scholar
Hartman, P. (1982). Ordinary Differential Equations. Birkhäuser, Boston.Google Scholar
Jaber, E. A., Illand, C. and Li, S. (2022). Joint SPX-VIX calibration with Gaussian polynomial volatility models: deep pricing with quantization hints. Preprint. Available at https://arxiv.org/abs/2212.08297.Google Scholar
Jaber, E. A., Illand, C. and Li, S. (2023). The quintic Ornstein–Uhlenbeck volatility model that jointly calibrates SPX VIX smiles. Preprint. Available at https://arxiv.org/abs/2212.10917.Google Scholar
Jain, S. and Oosterlee, C. W. (2012). Pricing high-dimensional Bermudan options using the stochastic grid method. Internat. J. Comput. Math. 89, 1186–1211.CrossRefGoogle Scholar
Jaisson, T. (2015). Market impact as anticipation of the order flow imbalance. Quant. Finance 15, 1123–1135.CrossRefGoogle Scholar
Jaisson, T. and Rosenbaum, M. (2015). Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Prob. 25, 600–631.CrossRefGoogle Scholar
Jaisson, T. and Rosenbaum, M. (2016). Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes. Ann. Appl. Prob. 26, 2860–2882.CrossRefGoogle Scholar
Jiao, Y., Ma, C., Scotti, S. and Zhou, C. (2021). The alpha-Heston stochastic volatility model. Math. Finance 31, 943–978.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
Keller-Ressel, M. (2011). Moment explosions and long-term behavior of affine stochastic volatility models. Math. Finance 21, 73–98.CrossRefGoogle Scholar
Keller-Ressel, M. and Mayerhofer, E. (2015). Exponential moments of affine processes. Ann. Appl. Prob. 25, 714–752.CrossRefGoogle Scholar
Kokholm, T. and Stisen, M. (2015). Joint pricing of VIX and SPX options with stochastic volatility and jump models. J. Risk Finance 16, 27–48.CrossRefGoogle Scholar
Longstaff, F. A. and Schwartz, E. S. (2015). Valuing American options by simulation: a simple least-squares approach. Rev. Financial Studies 14, 113–147.CrossRefGoogle Scholar
Pacati, C., Pompa, G. and Renò, R. (2018). Smiling twice: the Heston++ model. J. Banking Finance 96, 185–206.CrossRefGoogle Scholar
Pan, J. (2002). The jump-risk premia implicit in options: evidence from an integrated time-series study. J. Financial Econom. 63, 3–50.CrossRefGoogle Scholar
Protter, P. E. (2005). Stochastic integration and differential equations. Springer, Berlin.CrossRefGoogle Scholar
Recchioni, M. C., Iori, G., Tedeschi, G. and Ouellette, M. S. (2021). The complete Gaussian kernel in the multi-factor Heston model: option pricing and implied volatility applications. Europ. J. Operat. Res. 293, 336–360.CrossRefGoogle Scholar
Rydberg, T. H. (1997). A note on the existence of unique equivalent martingale measures in a Markovian setting. Finance Stoch. 1, 251–257.CrossRefGoogle Scholar
Rydberg, T. H. (1999). Generalized hyperbolic diffusion processes with applications in finance. Math. Finance 9, 183–201.CrossRefGoogle Scholar
Rømer, S. E. (2022). Empirical analysis of rough and classical stochastic volatility models to the SPX and VIX markets. Quant. Finance 22, 1805–1838.CrossRefGoogle Scholar
Stein, H. J. (2013). Joining risks and rewards. Available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2368905.Google Scholar
Stein, H. J. (2016). Fixing risk neutral risk measures. Internat. J. Theoret. Appl. Finance 19, article no. 1650021.CrossRefGoogle Scholar
Wong, B. and Heyde, C. C. (2006). On changes of measure in stochastic volatility models. J. Appl. Math. Stoch. Anal. 2006, article no. 18130.CrossRefGoogle Scholar