Large-Maturity Regimes of the Heston Forward Smile

Jacquier, Antoine; Roome, Patrick

doi:10.2139/ssrn.2515501

Cited by 7 publications

(7 citation statements)

References 73 publications

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“…Indeed, the “volatility of volatility” parameter ν enables us to control the smile, the correlation parameter ρ to deal with the skew, and the initial volatility V 0 to fix the at‐the‐money volatility level; see Forde, Jacquier, and Lee (), Gatheral (), Janek, Kluge, Weron, and Wystup (), and Poon (). Furthermore, as observed in markets and in contrast to local volatility models, in the Heston model the volatility smile moves in the same direction as the underlying and the forward smile does not flatten with time; see Gatheral (), Jacquier and Roome (, ), and Pascucci and Mazzon (). There is an explicit formula for the characteristic function of the asset log‐price; see Heston (). From this formula, efficient numerical methods have been developed, allowing for instantaneous model calibration and pricing of derivatives; see Albrecher, Mayer, Schoutens, and Tistaert (), Carr and Madan (), Kahl and Jäckel (), and Lewis ().…”

Section: Introductionmentioning

confidence: 86%

The characteristic function of rough Heston models

Euch

Rosenbaum

2018

Mathematical Finance

264

270

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It has been recently shown that rough volatility models, where the volatility is driven by a fractional Brownian motion with small Hurst parameter, provide very relevant dynamics in order to reproduce the behavior of both historical and implied volatilities. However, due to the non-Markovian nature of the fractional Brownian motion, they raise new issues when it comes to derivatives pricing. Using an original link between nearly unstable Hawkes processes and fractional volatility models, we compute the characteristic function of the log-price in rough Heston models. In the classical Heston model, the characteristic function is expressed in terms of the solution of a Riccati equation.Here, we show that rough Heston models exhibit quite a similar structure, the Riccati equation being replaced by a fractional Riccati equation. K E Y W O R D Sfractional Brownian motion, fractional Riccati equation, Hawkes processes, limit theorems, rough Heston models, rough volatility models Here the parameters , , 0 , and are positive, and and are two Brownian motions with correlation coefficient , that is, ⟨dW , dB ⟩ = dt.The popularity of this model is probably due to three main reasons:Mathematical Finance. 2019;29:3-38. wileyonlinelibrary.com/journal/mafi

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Section: Introductionmentioning

confidence: 86%

The characteristic function of rough Heston models

Euch

Rosenbaum

2018

Mathematical Finance

264

270

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“…Indeed, the "volatility of volatility" parameter ν enables us to control the smile, the correlation parameter ρ to deal with the skew, and the initial volatility V 0 to fix the at-the-money volatility level, see [15,17,30,38]. Furthermore, as observed in markets and in contrast to local volatility models, in Heston model, the volatility smile moves in the same direction as the underlying and the forward smile does not flatten with time, see [17,26,27,37].…”

Section: Introductionmentioning

confidence: 99%

The characteristic function of rough Heston models

Euch¹,

Rosenbaum²

2016

Preprint

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“…Our intention (as mentioned in Section 1) is to use it as a building block for more advanced models (such as stochastic volatility models where the initial variance is sampled from a continuous distribution) so that we are able to better match steep small-maturity observed smiles. In these types of more sophisticated models, the large-time behaviour is governed more from the chosen stochastic volatility model rather than the choice of distribution for the initial variance (see [48,49] for examples), especially if the variance process possesses some ergodic properties. This also suggests to use this class of models to introduce two different time scales: one to match the small-time smile (the distribution for the initial variance) and one to match the medium-to large-time smile (the chosen stochastic volatility model).…”

Section: 41mentioning

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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Large-Maturity Regimes of the Heston Forward Smile

Cited by 7 publications

References 73 publications

The characteristic function of rough Heston models

The characteristic function of rough Heston models

The characteristic function of rough Heston models

Black–Scholes in a CEV random environment

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