Does the Hurst index matter for option prices under fractional volatility?

Funahashi, Hideharu; Kijima, Masaaki

doi:10.1007/s10436-016-0289-1

Cited by 17 publications

(22 citation statements)

References 22 publications

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“…Recent empirical evidence in particular shows that stochastic volatility is often rough with rapidly decaying correlations at the origin (see Gatheral et al (2016)). In Funahashi and Kijima (2017) it was shown numerically that the implied volatility correction for a rough fractional stochastic volatility model tends to the correction associated with the Markov case in the regime of long time to maturity. This is consistent with the analytic result derived in this paper where we consider a fast mean reverting rough volatility.…”

Section: Introductionmentioning

confidence: 99%

Option pricing under fast-varying and rough stochastic volatility

Garnier

Sølna

2018

Ann Finance

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Recent empirical studies suggest that the volatilities associated with financial time series exhibit short-range correlations. This entails that the volatility process is very rough and its autocorrelation exhibits sharp decay at the origin. Another classic stylistic feature often assumed for the volatility is that it is mean reverting. In this paper it is shown that the price impact of a rapidly mean reverting rough volatility model coincides with that associated with fast mean reverting Markov stochastic volatility models. This reconciles the empirical observation of rough volatility paths with the good fit of the implied volatility surface to models of fast mean reverting Markov volatilities. Moreover, the result conforms with recent numerical results regarding rough stochastic volatility models. It extends the scope of models for which the asymptotic results of fast mean reverting Markov volatilities are valid. The paper concludes with a general discussion of fractional volatility asymptotics and their interrelation. The regimes discussed there include fast and slow volatility factors with strong or small volatility fluctuations and with the limits not commuting in general. The notion of a characteristic term structure exponent is introduced, this exponent governs the implied volatility term structure in the various asymptotic regimes.

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Section: Introductionmentioning

confidence: 99%

Option pricing under fast-varying and rough stochastic volatility

Garnier

Sølna

2018

Ann Finance

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“…In this study, we extend the fractional volatility model proposed in [4] by introducing another factor of rough volatility. Through numerical experiments, we demonstrate that, when one of the Hurst indexes in fractional volatility is larger than 1/2 (volatility persistence) and the other is smaller than 1/2 (rough volatility), our model can explain both the slower decay of the smile amplitude decline and the term structure of the at-the-money volatility skew observed in the options market, simultaneously.…”

Section: Discussionmentioning

confidence: 99%

“…See [4] for the detailed derivation. Now, we introduce the market prices of risks η t andη i t to define the standard BMs W t and W i t under a martingale measure Q. Namely, define:…”

Section: The Setupmentioning

confidence: 99%

“…In this section, we derive an approximation formula for option prices by applying the technique used in [4] for the fractional volatility model (7). To this end, we expand the underlying asset into a sum of iterative stochastic integrals with deterministic integrands.…”

Section: Approximation Formulamentioning

confidence: 99%

“…Inspired by the model proposed in [4], we consider the following two-factor fractional volatility model. Namely, the underlying asset price S t and its volatility σ t = σ(X 1 t , X 2 t ) are modeled by the stochastic differential equations (SDEs): By applying the integration-by-parts formula and changing the order of integration, the volatility can be rewritten as:…”

Section: The Setupmentioning

confidence: 99%

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A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

Funahashi

Kijima

2017

Fractal Fract

Self Cite

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Abstract:In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index H > 1/2, (ii) is proven to be satisfied by a rough volatility model with H < 1/2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist.

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

Bezborodov

Persio

Mishura

2018

Methodol Comput Appl Probab

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We consider the pricing problem related to payoffs that can have discontinuities of polynomial growth. The asset price dynamic is modeled within the Black and 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, the latter one being impossible to be evaluated analytically. The second approach consists in considering the conditional expectation with respect to the entire trajectory of the fractional Brownian motion (fBm). Then we derive a closed formula which involves only integral functional depending on the fBm trajectory, to evaluate the price; finally we discretize the fBm and estimate the rate of convergence of the associated numerical scheme to the option price. In both cases the rate of convergence is the same and equals n −rH , where n is a number of the points of discretization, H is the Hurst index of fBm, and r is the Hölder exponent of volatility function. The third method consists in calculating the density of the integral functional depending on the trajectory of the fBm via Malliavin calculus also providing the option price in terms of the associated probability density.

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Does the Hurst index matter for option prices under fractional volatility?

Cited by 17 publications

References 22 publications

Option pricing under fast-varying and rough stochastic volatility

Option pricing under fast-varying and rough stochastic volatility

A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

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

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