# Option Pricing with Fractional Stochastic Volatility and Discontinuous Payoff Function of Polynomial Growth

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## Abstract

We consider the pricing problem related to payoffs of polynomial growth that can have discontinuities of the 1st kind. The asset price dynamic is modeled within the Black-Scholes framework characterized by a stochastic volatility term driven by a fractional Ornstein-Uhlenbeck process. In order to solve the aforementioned problem, we consider three approaches. The first one consists in a suitable transformation of the initial value of the asset price, in order to eliminate possible discontinuities. Then we discretize both the Wiener process and the fractional Brownian motion and estimate the rate of convergence of the related discretized price to its real value whose closed-form analytical expression is usually difficult to obtain. The second approach consists in considering the conditional expectation with respect to the entire trajectory of the fractional Brownian motion (fBm). Here we derive a presentation for the option price which involves only an integral functional depending on the fBm trajectory, and then discretize the fBm and estimate the rate of convergence of the associated numerical scheme. In both cases the rate of convergence is the same and equals *n*^{−rH}, where *n* is the partition size, *H* is the Hurst index of the fBm, and *r* is the Hölder exponent of the volatility function. The third method consists in calculating the density of the integral functional depending on the trajectory of the fBm via Malliavin calculus and providing the option price in terms of the associated probability density.

## Keywords

Option pricing Stochastic volatility Black–Scholes model Wiener process Discontinuous payoff function Polynomial growth Rate of convergence Discretization Conditioning Malliavin calculus Stochastic derivative Skorokhod integral## Mathematics Subject Classification (2010)

91G20 91B24 91B25 91G60 60G22 60H07## Preview

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## References

- Alòs E (2012) A decomposition formula for option prices in the Heston model and applications to option pricing approximation. Finance Stoch 16(3):403–422MathSciNetCrossRefMATHGoogle Scholar
- Altmayer M, Neuenkirch A (2015) Multilevel MOnte Carlo quadrature of discontinuous payoffs in the generalized Heston model using Malliavin integration by parts. SIAM J Financial Math 6(1):22–52MathSciNetCrossRefMATHGoogle Scholar
- Barndorff-Nielsen OE, Shephard N (2001) Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J R Stat Soc Ser B Stat Methodol 63(2):167–241MathSciNetCrossRefMATHGoogle Scholar
- Barndorff-Nielsen OE, Shephard N (2002) Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J R Stat Soc Ser B Stat Methodol 64(2):253–280MathSciNetCrossRefMATHGoogle Scholar
- Bayer C, Friz P, Gatheral J (2016) Pricing under rough volatility. Quant Finan 16(6):887–904MathSciNetCrossRefGoogle Scholar
- Bergomi L, Guyon J (2012) Stochastic volatility’s orderly smiles. Risk 25(5):60–66Google Scholar
- Bertholon H, Monfort A, Pegoraro F (2007) Pricing and inference with mixtures of conditionally normal processes. Direction Générale des études et des relations internationales, Banque de France, pp 1–59Google Scholar
- Bollerslev T, Mikkelsen HO (1996) Modeling and pricing long memory in stock market volatility. J Econ 73(1):151–184CrossRefMATHGoogle Scholar
- Carrasco M, Chen X (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Economet Theor 18(1):17–39MathSciNetCrossRefMATHGoogle Scholar
- Cheridito P, Kawaguchi H, Maejima M (2003) ORnstein-Uhlenbeck processes. Electron J Probab 8(3):14 (electronic)MathSciNetMATHGoogle Scholar
- Chronopoulou A, Viens FG (2012) Estimation and pricing under long-memory stochastic volatility. Ann Finance 8(2-3):379–403MathSciNetCrossRefMATHGoogle Scholar
- Comte F, Coutin L, Renault E (2012) Affine fractional stochastic volatility models. Ann Finance 8(2-3):337–378MathSciNetCrossRefMATHGoogle Scholar
- Cont R, Tankov P (2004) Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca RatonMATHGoogle Scholar
- Fernique X (1975) Regularité des trajectoires des fonctions aléatoires gaussiennes. In: École d’Été de Probabilités de Saint-Flour, IV-1974, vol 480. Springer, Berlin, pp 1–96. Lecture Notes in MathGoogle Scholar
- Fouque J-P, Papanicolaou G, Sircar KR (2000) Derivatives in financial markets with stochastic volatility. Cambridge University Press, CambridgeMATHGoogle Scholar
- Funahashi H, Kijima M (2017) Does the Hurst index matter for option prices under fractional volatility. Ann Finance 13(1):55–74MathSciNetCrossRefMATHGoogle Scholar
- Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2):327–343CrossRefMATHGoogle Scholar
- Hull JC, White A (1987) The pricing of options on assets with stochastic volatilities. J Financ 42(2):281–300CrossRefMATHGoogle Scholar
- Kahl C (2008) Modelling and simulation of stochastic volatility in finance. Universal-PublishersGoogle Scholar
- Knight J, Satchell S (eds) (2011) Forecasting volatility in the financial markets, 3rd edn. Butterworth-Heinemann, OxfordGoogle Scholar
- Kuchuk-Iatsenko S, Mishura Y (2015) Pricing the E,uropean call option in the model with stochastic volatility driven by Ornstein-Uhlenbeck process. Exact formulas Mod Stoch Theory Appl 2(3):233–249CrossRefMATHGoogle Scholar
- Kyprianou AE, Schoutens W (eds) (2005) Exotic option pricing and advanced Lévy models. Wiley, ChichesterMATHGoogle Scholar
- Ledoux M (1996) Isoperimetry and gaussian analysis. In: Lectures on probability theory and statistics, pp 165–294. SpringerGoogle Scholar
- León JA, Nualart D (1998) Stochastic evolution equations with random generators. Ann Probab 26(1):149–186MathSciNetCrossRefMATHGoogle Scholar
- Nicolato E, Venardos E (2003) Option pricing in stochastic volatility models of the O,rnstein-Uhlenbeck type. Math Financ 13(4):445–466CrossRefMATHGoogle Scholar
- Norros I, Valkeila E, Virtamo J (1999) An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions. Bernoulli 5(4):571–587MathSciNetCrossRefMATHGoogle Scholar
- Nualart D (2006) The Malliavin calculus and related topics. Probability and its Applications (New York), 2nd edn. Springer, BerlinMATHGoogle Scholar
- Palm FC (1996) 7 GARCH models of volatility. Handbook of statistics 14:209–240MathSciNetCrossRefGoogle Scholar
- Pospíšil J, Sobotka T (2016) Market calibration under a long memory stochastic volatility model. Appl Math Finance 23(5):323–343MathSciNetCrossRefGoogle Scholar
- Schobel R, Zhu J (1999) Stochastic volatility with an Ornstein–Uhlenbeck process: an extension. Ann Finance 3(1):23–46MATHGoogle Scholar
- Schweizer M (1995) On the minimal martingale measure and the Föllmer-Schweizer decomposition. Stochastic Anal Appl 13(5):573–599MathSciNetCrossRefMATHGoogle Scholar
- Shephard N (1996) Statistical aspects of ARCH and stochastic volatility. Monographs on Statistics and Applied Probability 65:1–68Google Scholar
- Talagrand M (1994) Sharper bounds for Gaussian and empirical processes. Ann Probab 22(1):28–76MathSciNetCrossRefMATHGoogle Scholar