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On the martingale property in the rough bergomi model. (English) Zbl 07068657
Summary: We consider a class of fractional stochastic volatility models (including the so-called rough Bergomi model), where the volatility is a superlinear function of a fractional Gaussian process. We show that the stock price is a true martingale if and only if the correlation \(\rho \) between the driving Brownian motions of the stock and the volatility is nonpositive. We also show that for each \(\rho <0\) and \(m> \frac{1} {{1-\rho ^{2}}}\), the \(m\)-th moment of the stock price is infinite at each positive time.

MSC:
60G44 Martingales with continuous parameter
60G22 Fractional processes, including fractional Brownian motion
91G20 Derivative securities (option pricing, hedging, etc.)
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References:
[1] Elisa Alòs, Jorge A. León, and Josep Vives, On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility, Finance Stoch. 11 (2007), no. 4, 571–589. · Zbl 1145.91020
[2] Christian Bayer, Peter Friz, and Jim Gatheral, Pricing under rough volatility, Quant. Finance 16 (2016), no. 6, 887–904.
[3] Stefan Blei and Hans-Jürgen Engelbert, On exponential local martingales associated with strong Markov continuous local martingales, Stochastic Process. Appl. 119 (2009), no. 9, 2859–2880. · Zbl 1192.60068
[4] Vladimir I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998. · Zbl 0913.60035
[5] Michelle Boué and Paul Dupuis, A variational representation for certain functionals of Brownian motion, Ann. Probab. 26 (1998), no. 4, 1641–1659. · Zbl 0936.60059
[6] H. Brunner and Z. W. Yang, Blow-up behavior of Hammerstein-type Volterra integral equations, J. Integral Equations Appl. 24 (2012), no. 4, 487–512. · Zbl 1261.45003
[7] Alexander Cherny, Brownian moving averages have conditional full support, Ann. Appl. Probab. 18 (2008), no. 5, 1825–1830. · Zbl 1151.91490
[8] P. K. Friz, P. Gassiat, and P. Pigato, Precise asymptotics: robust stochastic volatility models, ArXiv e-prints (2018).
[9] Masaaki Fukasawa, Asymptotic analysis for stochastic volatility: martingale expansion, Finance Stoch. 15 (2011), no. 4, 635–654. · Zbl 1303.91177
[10] A. M. Garsia, E. Rodemich, and H. Rumsey, Jr., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J. 20 (1970/1971), 565–578. · Zbl 0252.60020
[11] Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum, Volatility is rough, Quant. Finance 18 (2018), no. 6, 933–949. · Zbl 1400.91590
[12] Stefan Gerhold, Christoph Gerstenecker, and Arpad Pinter, Moment explosions in the rough heston model, Arxiv E-prints (2018).
[13] G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. · Zbl 0695.45002
[14] Archil Gulisashvili, Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions, ArXiv e-prints (2019), arXiv:1808.00421v7.
[15] B. Jourdain, Loss of martingality in asset price models with lognormal stochastic volatility, Preprint CERMICS 2004-267 (2004).
[16] P.-L. Lions and M. Musiela, Correlations and bounds for stochastic volatility models, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 1, 1–16. · Zbl 1108.62110
[17] Philip Protter, A mathematical theory of financial bubbles, Paris-Princeton Lectures on Mathematical Finance 2013, Lecture Notes in Math., vol. 2081, Springer, Cham, 2013, pp. 1–108. · Zbl 1277.91134
[18] Johannes Ruf, The martingale property in the context of stochastic differential equations, Electron. Commun. Probab. 20 (2015), No. 34, 10. · Zbl 1321.60085
[19] Carlos A. Sin, Complications with stochastic volatility models, Adv. in Appl. Probab. 30 (1998), no. 1, 256–268. · Zbl 0907.90026
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