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We introduce a new class of continuous-time models of the stochastic volatility of asset prices. The models can simultaneously incorporate roughness and slowly decaying autocorrelations, including proper long memory, which are two stylized…

Statistical Finance · Quantitative Finance 2021-01-06 Mikkel Bennedsen , Asger Lunde , Mikko S. Pakkanen

A motivating question in this paper is whether a sensible investment strategy may systematically contain long positions in out-of-the-money European calls with short expiry. Here we consider a very simple trading strategy for calls. The…

Mathematical Finance · Quantitative Finance 2014-10-07 Jarno Talponen

We derive a new, exact and transparent expansion for option smiles, which lends itself both to analytical approximation and, perhaps more importantly, to congenial numerical treatments. We show that the skew and the curvature of the smile…

Pricing of Securities · Quantitative Finance 2012-04-25 L. De Leo , V. Vargas , S. Ciliberti , J. -P. Bouchaud

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. Using a suitable representation of the fractional part,…

Portfolio Management · Quantitative Finance 2019-05-17 Nicole Bäuerle , Sascha Desmettre

This paper investigates asymptotically optimal importance sampling (IS) schemes for pricing European call options under the Heston stochastic volatility model. We focus on two distinct rare-event regimes where standard Monte Carlo methods…

Mathematical Finance · Quantitative Finance 2025-11-26 Yun-Feng Tu , Chuan-Hsiang Han

We present small-time implied volatility asymptotics for Realised Variance (RV) and VIX options for a number of (rough) stochastic volatility models via large deviations principle. We provide numerical results along with efficient and…

Mathematical Finance · Quantitative Finance 2020-11-03 Chloe Lacombe , Aitor Muguruza , Henry Stone

We study the mean escape time in a market model with stochastic volatility. The process followed by the volatility is the Cox Ingersoll and Ross process which is widely used to model stock price fluctuations. The market model can be…

Statistical Mechanics · Physics 2009-11-11 Giovanni Bonanno , Davide Valenti , Bernardo Spagnolo

In a recent article the authors obtained a formula which relates explicitly the tail of risk neutral returns with the wing behavior of the Black Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but…

Probability · Mathematics 2007-05-23 Shalom Benaim , Peter Friz

We model the dynamics of asset prices and associated derivatives by consideration of the dynamics of the conditional probability density process for the value of an asset at some specified time in the future. In the case where the price…

Pricing of Securities · Quantitative Finance 2011-11-14 Damir Filipović , Lane P. Hughston , Andrea Macrina

We discuss the pricing and hedging of volatility options in some rough volatility models. First, we develop efficient Monte Carlo methods and asymptotic approximations for computing option prices and hedge ratios in models where…

Pricing of Securities · Quantitative Finance 2019-01-31 Blanka Horvath , Antoine Jacquier , Peter Tankov

In this article we look at stochastic processes with uncertain parameters, and consider different ways in which information is obtained when carrying out observations. For example we focus on the case of a the random evolution of a traded…

Mathematical Finance · Quantitative Finance 2024-07-08 Will Hicks

We evaluate a co-evolutionary calibration framework for the Heston model in which a genetic algorithm (GA) over parameters is coupled to an evolving neural inverse map from option surfaces to parameters. While GA-history sampling can reduce…

Pricing of Securities · Quantitative Finance 2025-12-04 Julian Gutierrez

We perform a classification of the Lie point symmetries for the Black--Scholes--Merton Model for European options with stochastic volatility, $\sigma$, in which the last is defined by a stochastic differential equation with an…

Analysis of PDEs · Mathematics 2016-05-04 A. Paliathanasis , K. Krishnakumar , K. M. Tamizhmani , P. G. L. Leach

We analyze the VIX futures market with a focus on the exchange-traded notes written on such contracts, in particular we investigate the VXX notes tracking the short-end part of the futures term structure. Inspired by recent developments in…

Mathematical Finance · Quantitative Finance 2021-06-15 Martino Grasselli , Andrea Mazzoran , Andrea Pallavicini

Stochastic volatility models based on Gaussian processes, like fractional Brownian motion, are able to reproduce important stylized facts of financial markets such as rich autocorrelation structures, persistence and roughness of sample…

Probability · Mathematics 2022-05-10 Eduardo Abi Jaber

A general method to construct recombinant tree approximations for stochastic volatility models is developed and applied to the Heston model for stock price dynamics. In this application, the resulting approximation is a four tuple Markov…

Computational Finance · Quantitative Finance 2016-08-14 Erdinç Akyıldırım , Yan Dolinsky , H. Mete Soner

In equity and foreign exchange markets the risk-neutral dynamics of the underlying asset are commonly represented by stochastic volatility models with jumps. In this paper we consider a dense subclass of such models and develop analytically…

Pricing of Securities · Quantitative Finance 2010-10-11 Aleksandar Mijatović , Martijn Pistorius

In this note we consider splitting methods based on linear multistep methods and stabilizing corrections. To enhance the stability of the methods, we employ an idea of Bruno & Cubillos (2016) who combine a high-order extrapolation formula…

Numerical Analysis · Mathematics 2017-09-05 Willem Hundsdorfer , Karel in 't Hout

Fractional stochastic volatility models have been widely used to capture the non-Markovian structure revealed from financial time series of realized volatility. On the other hand, empirical studies have identified scales in stock price…

Mathematical Finance · Quantitative Finance 2019-01-25 Jean-Pierre Fouque , Ruimeng Hu

We propose a convolution-FFT method for pricing European options under the Heston model that leverages a continuously differentiable representation of the joint characteristic function. Unlike existing Fourier-based methods that rely on…

Computational Finance · Quantitative Finance 2025-12-08 Xiang Gao , Cody Hyndman