The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" display="inline" overflow="scroll"><mml:mi>α</mml:mi></mml:math>-hypergeometric stochastic volatility model

Fonseca, José Da; Martini, Claude

doi:10.1016/j.spa.2015.11.010

Cited by 31 publications

(18 citation statements)

References 26 publications

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“…In this section we will tackle the problem of pricing barrier options under the 2-hypergeometric stochastic volatility -a particular case of the α-hypergeometric stochastic volatility model which was defined by Da Fonseca and Martini [1] as follows:…”

Section: An Asymptotic Expansion Approach To Barrier Option Pricingmentioning

confidence: 99%

“…Like Da Fonseca and Martini [1], we assume that the model is given directly under a risk-neutral measure Q. The deterministic function r(t) represents the (possibly time-dependent) interest rate, while the parameters a and c can be used to set the market price of volatility risk.…”

Section: An Asymptotic Expansion Approach To Barrier Option Pricingmentioning

confidence: 99%

“…It is important to emphasize that the formulation of the α-hypergeometric stochastic volatility model given by Da Fonseca and Martini [1] and by Privault and She [2] does not include the drift term r(t)S t dt. If, as in these two papers, the goal is to price vanilla options, then such a zero interest rate assumption does not entail any loss of generality because the general case of a nonzero interest rate can be reduced to the case r(t) = 0 by rewriting the pricing equation in forward terms (cf.…”

Section: An Asymptotic Expansion Approach To Barrier Option Pricingmentioning

confidence: 99%

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Barrier option pricing under the 2-hypergeometric stochastic volatility model

Sousa

Cruzeiro

Guerra

2018

Journal of Computational and Applied Mathematics

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We investigate the pricing of financial options under the 2-hypergeometric stochastic volatility model. This is an analytically tractable model that reproduces the volatility smile and skew effects observed in empirical market data.Using a regular perturbation method from asymptotic analysis of partial differential equations, we derive an explicit and easily computable approximate formula for the pricing of barrier options under the 2hypergeometric stochastic volatility model. The asymptotic convergence of the method is proved under appropriate regularity conditions, and a multi-stage method for improving the quality of the approximation is discussed. Numerical examples are also provided.A natural way to address this issue is to introduce randomness in the volatility. For this reason, option pricing under stochastic volatility has been the subject of a great deal of research in recent years. Here we focus on the 2-hypergeometric stochastic volatility model, which was introduced by Da Fonseca and Martini [1] as a model which ensures that the volatility is strictly positive -this is an important property which is not present in some other well-established stochastic volatility models. In a very recent paper, Privault and She [2] demonstrated that, under this model, a closed-form asymptotic vanilla option pricing formula can be determined through a regular perturbation method. This is a notable result because their formulas are analytically very simple, which is rarely the case in models with stochastic volatility: as discussed by Zhu [3], the higher complexity of these models usually yields the need for rather sophisticated numerical implementations.The literature on barrier option pricing methods is extensive. Exact closed-form pricing formulas have been derived for only a few models other than that of Black and Scholes, none of which reproduces satisfactorily the market phenomena. (For explicit formulas under one-dimensional models see e.g. Davydov and Linestky [4], Hui and Lo [5]; for an explicit solution under the Heston stochastic volatility model see Lipton [6].) Given the unavailability of explicit formulas, to price barrier options under more complex models one needs to resort to numerical methods. The main approaches are the use of numerical partial differential equation (PDE) techniques and of Monte Carlo methods (we refer the reader to the books of Seydel [7] and of Glasserman [8]), which are often combined with other analytical or numerical techniques (for recent work see for instance Zhang et al. [9], Guardasoni and Sanfelici [10]). Unfortunately, the computational times are nowadays still largely incompatible with the demands of the financial industry.An alternative strategy for pricing under more general models is to derive approximate (or asymptotic) analytical solutions: this has been proposed not only for vanilla options (cf. Privault and She [2]) but also for barrier options, see e.g. Fouque et al. [11], Alos et al. [12]. These asymptotic methods are intrinsically computationally much less expen...

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Section: An Asymptotic Expansion Approach To Barrier Option Pricingmentioning

confidence: 99%

Section: An Asymptotic Expansion Approach To Barrier Option Pricingmentioning

confidence: 99%

Section: An Asymptotic Expansion Approach To Barrier Option Pricingmentioning

confidence: 99%

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Barrier option pricing under the 2-hypergeometric stochastic volatility model

Sousa

Cruzeiro

Guerra

2018

Journal of Computational and Applied Mathematics

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“…In chapter 4, we develop an approximate pricing formula for the European option under the Hypergeometric stochastic volatility model introduced in [DM16] to address the shortcomings associated with pricing option using the Heston stochastic volatility model. We assume that the asset price S t and volatility V t at time t is governed by the following dynamics:…”

Section: Chapter 1 Introduction and Main Resultsmentioning

confidence: 99%

“…It has long been observed that the volatility of a derivatives cannot be a constant but varies with the time to maturity and strike, as such for a more accurate modeling of the asset prices in financial markets, stochastic volatility models are introduced. [DM16], the α-Hypergeometric stochastic volatility model is introduced to ensure that the stochastic volatility is strictly positive. For an asset, the asset price S t and volatility V t at time t is governed by the following dynamics:…”

Section: Introductionmentioning

confidence: 99%

Asymptotic expansions and distribution properties for diffusion processes

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Sequential Itô–Taylor expansions and characteristic functions of stochastic volatility models

Ding,

Cui,

Liu

2023

Journal of Futures Markets

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This study proposes a new approach to derive the characteristic function of a general stochastic volatility model by sequentially utilizing the Itô–Taylor expansions. In particular, our method applies to non‐affine stochastic volatility models with jumps, for which the corresponding characteristic functions do not have closed‐form expressions. Numerically inverting these characteristic functions can yield accurate probability density functions of stochastic volatility models to serve for various pricing and hedging purposes in quantitative finance. The proposed sequential Itô–Taylor expansion allows us to handle derivatives with medium to long maturities. Numerical experiments illustrate the accuracy and effectiveness of our approach.

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The α-hypergeometric stochastic volatility model

Cited by 31 publications

References 26 publications

Barrier option pricing under the 2-hypergeometric stochastic volatility model

Barrier option pricing under the 2-hypergeometric stochastic volatility model

Asymptotic expansions and distribution properties for diffusion processes

Sequential Itô–Taylor expansions and characteristic functions of stochastic volatility models

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