The characteristic function of rough Heston models

By Omar El Euch, Mathieu Rosenbaum. 2016

341 citations recorded by the source corpus.

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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/39f2e43bb2a9839031b0835bce69afd959aed7ff

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Citations in the corpus

No‐arbitrage implies power‐law market impact and rough volatility (2018, methodology): “This is done using the characteristic function of Hawkes processes in the spirit of [11].”
Markovian structure of the Volterra Heston model (2018, methodology): “This model nests as special cases the Heston model for K ≡ 1, and the rough Heston model of [8], obtained by setting K(t) = t α−1 Γ(α) for α ∈ (12 , 1) and g0(t) = V0 + ∫ t”
Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case (2018, background): “ R E gl(x)e−tfl(x)λl 0(dx). • Heston-like or Barndorﬀ-Nielsen-Shephard-like structures can be analogously constructed again by choosing appropriate Banach spaces Y. For instance the rough Heston case [15] can be treated with Y = Cb(R + ∪ {∞};R2) and the equations dλ1 t(dx) = −xλ1 t(dx)dt+ν(dx) q h1,”
Volatility is rough (2014, neutral): “3 of Fukasawa (2011), as an example of the application of his martingale expansion, Fukasawa shows that a stochastic volatility model where the volatility is driven by fractional Brownian motion with Hurst exponent H generating an ATM volatility skew of the form ψ(τ) ∼ τ H−1/2, at least for small τ ”
Rough volatility: Evidence from option prices (2017, background): “Other developments about rough volatility models can be found in [2, 3, 8, 9, 10, 12, 14, 18, 21].”
Perfect hedging in rough Heston models (2017, methodology): “Furthermore, efficient numerical pricing procedures for vanilla options can be easily designed from it, see [10].”
A weak solution theory for stochastic Volterra equations of convolution type (2019, background): “This type of equation appears in multiple applications, for example turbulence (Barndorff-Nielsen and Schmiegel, 2008), energy markets (BarndorffNielsen et al., 2013), and rough volatility modeling in finance (El Euch and Rosenbaum, 2019; Gatheral, Jaisson and Rosenbaum, 2018).”
Short-time near-the-money skew in rough fractional volatility models (2017, methodology): “Previous work has thus focused on finding efficient Monte Carlo simulation schemes [BFG16, BLP17, BFG17] or – in the special case of the Rough Heston model – on an explicit formula for the characteristic function of the logprice (see [ER16]), thus in this particular model making pricing amenable to ”