Asymptotic Behavior of the Fractional Heston Model

Abstract

We consider the fractional Heston model originally proposed by Comte, Coutin and Renault [12]. Inspired by recent groundbreaking work on rough volatility [2, 6, 24, 26] which showed that models with volatility driven by fractional Brownian motion with short memory allows for better calibration of the volatility surface and more robust estimation of time series of historical volatility, we provide a characterisation of the short-and long-maturity asymptotics of the implied volatility smile. Our analysis reveals that the short-memory property precisely provides a jump-type behaviour of the smile for short maturities, thereby fixing the well-known standard inability of classical stochastic volatility models to fit the short-end of the volatility smile.

Mean–Variance Portfolio Selection Under Volterra Heston Model

“…Classic rough volatility models include fractional Brownian motion (fBm), fractional Ornstein-Uhlenbeck (fOU) process, and rough Bergomi (rBergomi) model. The popularity of the Heston model in the financial market leads to the introduction of the fractional Heston model Guennoun et al (2018) and the rough Heston model El Euch and Rosenbaum (2019). Both are rough versions of the celebrated Heston stochastic volatility model.…”

Portfolio Optimization in Fractional and Rough Heston Models

“…Since Z t ≥ 0 almost surely we obtain that ν t ≥ ν 0 almost surely for all t ≥ 0. It can be shown (see [16]) for t, h ≥ 0 that…”

“…Initiated by the observation that volatility is rough in [14], the current literature takes a new point of view: Rough Heston models, which use a fractional Brownian motion with Hurst index H < 1/2 as driver of the volatility process, incorporating a better fit of implied volatility surfaces as shown in [14], have become very popular; compare e.g. [16,7]. Subsequently many papers concerning option pricing, simulation of paths, asymptotics, and the foundations of fractional and rough environments have emerged; compare [2,8,20,30,11,15] to name a few.…”

“…The outline of our paper is as follows: Section 2 introduces the financial market model and the optimization problem in case of a Hurst parameter H ∈ ( 1 2 , 1). We use here the fractional Riemann-Liouville integral (compare [16]) for the volatility. In Section 3, we provide a finite dimensional approximation of the optimization problem, derive a solution by solving the corresponding Hamilton-Jacobi-Bellman equation, and verify that the obtained solution is indeed optimal.…”

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We consider a fractional version of the Heston volatility model which is inspired by [16]. Within this model we treat portfolio optimization problems for power utility functions.

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