The Randomised Heston Model

Jacquier, Antoine; Shi, Fangwei

doi:10.2139/ssrn.2829920

Cited by 5 publications

(10 citation statements)

References 57 publications

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“…The case α = 0 is comparable to classical stochastic volatility models in the absence of time-dependent or randomised parameters, or jump processes. Some admirable recent efforts in the randomised case are Mechkov (2016) and Jacquier and Shi (2017), and for jumps Mechkov (2015). On the contrary, when α = −0.43, the explosions of skew and smile as t → 0 are precisely as observed in practice.…”

Section: Implied Volatility Estimatorsmentioning

confidence: 90%

Turbocharging Monte Carlo pricing for the rough Bergomi model

McCrickerd

Pakkanen

2018

Quantitative Finance

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The rough Bergomi model, introduced by Bayer, Friz and Gatheral (2016), is one of the recent rough volatility models that are consistent with the stylised fact of implied volatility surfaces being essentially time-invariant, and are able to capture the term structure of skew observed in equity markets. In the absence of analytical European option pricing methods for the model, we focus on reducing the runtime-adjusted variance of Monte Carlo implied volatilities, thereby contributing to the model's calibration by simulation. We employ a novel composition of variance reduction methods, immediately applicable to any conditionally log-normal stochastic volatility model. Assuming one targets implied volatility estimates with a given degree of confidence, thus calibration RMSE, the results we demonstrate equate to significant runtime reductions-roughly 20 times on average, across different correlation regimes.

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Section: Implied Volatility Estimatorsmentioning

confidence: 90%

Turbocharging Monte Carlo pricing for the rough Bergomi model

McCrickerd

Pakkanen

2018

Quantitative Finance

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“…This is obviously the case for the wings of the smile (small and large strikes) via Roger Lee's formula, mentioned in Section 2.1.1, but also to describe short-and large-maturity asymptotics, as developed for instance in [44] or [48], via the use of (a refined version of) the Gärtner-Ellis theorem. In [51], the authors used this property to study a generalised version of the Heston model, where the starting value of the instantaneous volatility is randomised according to some distribution. It it closed to the present model, yet does not supersede it, and makes full use of the knowledge of the moment generating function of the Heston model.…”

Section: Model and Main Resultsmentioning

confidence: 99%

Black–Scholes in a CEV random environment

Jacquier

Roome

2018

Math Finan Econ

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Classical (Itô diffusions) stochastic volatility models are not able to capture the steepness of smallmaturity implied volatility smiles. Jumps, in particular exponential Lévy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see [65] for an overview), and more recently rough volatility models [2,33]. We suggest here a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Lévy models and fractional stochastic volatility models. IntroductionWe propose a simple model with continuous paths for stock prices that allows for small-maturity explosion of the implied volatility smile. It is indeed a well-documented fact on Equity markets (see for instance [34,Chapter 5]) that standard (Itô) stochastic models with continuous paths are not able to capture the observed steepness of the left wing of the smile when the maturity becomes small. To remedy this, several authors have suggested the addition of jumps, either in the form of an independent Lévy process or within the more general framework of affine diffusions. Jumps (in the stock price dynamics) imply an explosive behaviour for the smallmaturity smile and are better able to capture the observed steepness of the small-maturity implied volatility smile. In particular, Tankov [65] showed that, for exponential Lévy models with Lévy measure supported on the whole real line, the squared implied volatility smile explodes as σ 2 τ (k) ∼ −k 2 /(2τ log τ ), as the maturity τ tends to zero, where k represents the log-moneyness. Such a small-maturity behaviour of the smile is not only captured by jump-based models, but rough volatility (non-Markovian) models, where the stochastic volatility component is driven by a fractional Brownian motion, are in fact also able to reflect this property of the data.In a series of papers several authors [2,7,31,33,37,39,47] have indeed proved that, when the Hurst index of the fractional Brownian motion lies within (0, 1/2), then the implied volatility explodes at a rate of τ H−1/2 as the maturity τ tends to zero.In this paper we propose an alternative framework: we suppose that the stock price follows a standard Black-Scholes model; however the instantaneous variance, instead of being constant, is sampled from a continuous distribution. We first derive some general properties, interesting from a financial modelling point of view, and devote a particular attention to a particular case of it, where the variance is generated from independent CEV dynamics. Assume that interest rates and dividends are null, and let S denote the stock price process starting at S 0 = 1, the solution to the stochastic differential equation dS τ = S τ √ VdW τ , for τ ≥ 0, where W is a standard Brownian motion. Here, V is a random variable, which we assume to be distributed as V ∼ Y t , for s...

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“…In [34], the authors chose a non-central chi-squared initial distribution and showed that, under some appropriate rescaling, the implied volatility blows up at speed t −1/2 . Jacquier and Shi [35], also inspired by [42], pushed this analysis further by studying the impact of the random initial data on the short-time explosion of the smile.…”

Section: Large Deviations and Implied Volatility Asymptoticsmentioning

confidence: 99%

Asymptotic Behavior of the Fractional Heston Model

Guennoun¹,

Jacquier²,

Roome³

et al. 2018

SIAM J. Finan. Math.

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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.

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The Randomised Heston Model

Cited by 5 publications

References 57 publications

Turbocharging Monte Carlo pricing for the rough Bergomi model

Turbocharging Monte Carlo pricing for the rough Bergomi model

Black–Scholes in a CEV random environment

Asymptotic Behavior of the Fractional Heston Model

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