Affine fractional stochastic volatility models

Abstract

By fractional integration of a square root volatility process, we propose in this paper a long memory extension of the Heston (Rev Financ Stud 6:327-343, 1993) option pricing model. Long memory in the volatility process allows us to explain some option pricing puzzles as steep volatility smiles in long term options and co-movements between implied and realized volatility. Moreover, we take advantage of the analytical tractability of affine diffusion models to clearly disentangle long term components and short term variations in the term structure of volatility smiles. In addition, we provide a recursive algorithm of discretization of fractional integrals in order to be able to implement a method of moments based estimation procedure from the high frequency observation of realized volatilities.

Realizing Smiles: Options Pricing with Realized Volatility

“…This feature has considerable option pricing implications as documented by Bollerslev and Mikkelsen (1996) and Comte, Coutin, and Renault (2003). In addition, Adrian and Rosenberg (2008) show that a multicomponents volatility model substantially improves the cross-sectional pricing of volatility risk.…”

“…Two exceptions are Comte, Coutin, and Renault (2003), who employ a fractional stochastic volatility model, and Carr and Wu (2003), who apply alpha-stable processes to slow down the central limit theorem and obtain negative skewness and excess kurtosis for long-maturity options.…”

Black-Scholes in a CEV Random Environment: A New Approach to Smile Modelling

“…Their study reveals that H ∈ (0, 1/2) (in fact H ≈ 0.1 in their calibration results), indicating short memory of the volatility, thereby contradicting decades of time series analyses. By considering a specific fractional uncorrelated volatility model, directly inspired by the fractional version of the Heston model [15,34], Guennoun, Jacquier and Roome [32] provide a theoretical justification of this result. They show in particular that, when H ∈ (0, 1/2), the implied volatility explodes as σ 2 τ (k) ∼ y 0 τ H−1/2 /Γ(H + 3/2) as τ tends to zero (where y 0 is the initial instantaneous variance).…”

“…In this framework, sub-and super-hedging strategies (corresponding to best and worst case scenarios) are usually derived via the Black-Scholes-Barenblatt equation, and Fouque and Ren [27] recently provided approximation results when the two bounds become close to each other. One can also, at least formally, look at (2.1) from the perspective of fractional stochastic volatility models, first proposed by Comte et al in [14], and later developed and revived in [15,28,30,32]. In these models, standard stochastic volatility models are generalised by replacing the Brownian motion driving the instantaneous volatility by a fractional Brownian motion.…”

Maximum Quasi-likelihood Estimation in Fractional Levy Stochastic Volatility Model

“…Thus the model here captures the two stylized facts of long memory and jumps. Continuous time long memory process in volatility received some attention, see Comte and Renault [10] where they used fractional Ornstein-Uhlenbeck process as the stochastic volatility process and Comte, Coutin and Renault [11] where they used fractional CIR square root process as the stochastic volatility process. Marquardt [12] studied fractional Levy processes and applied it to long memory moving average processes.…”