The 4/2 Stochastic Volatility Model: A Unified Approach for the Heston and the 3/2 Model

Grasselli, Martino

doi:10.1111/mafi.12124

Cited by 132 publications

(113 citation statements)

References 42 publications

(98 reference statements)

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“…(2.1)], we get the dynamics of the price process S = e X in the 3/2-model under the local martingale measure, and hence of X = log(S). We stress that, denoting by κ G , θ G , σ G the parameters in [15,Eq. (2.2)], the relation between λ , κ and σ in (5.17) and κ G , θ G , σ G is…”

Section: The 3/2-modelmentioning

confidence: 99%

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Semi-static variance-optimal hedging in stochastic volatility models with Fourier representation

2019

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In a financial market model, we consider the variance-optimal semi-static hedging of a given contingent claim, a generalization of the classic variance-optimal hedging. To obtain a tractable formula for the expected squared hedging error and the optimal hedging strategy, we use a Fourier approach in a general multidimensional semimartingale factor model. As a special case, we recover existing results for variance-optimal hedging in affine stochastic volatility models. We apply the theory to set up a variance-optimal semi-static hedging strategy for a variance swap in both the Heston and the 3/2-model, the latter of which is a non-affine stochastic volatility model. * Corresponding author. Mail: Paolo.Di_Tella@tu-dresden.de. † MKR thanks Johannes Muhle-Karbe for early discussions on the idea of "Variance-Optimal Semi-Static Hedging". We acknowledge funding from the German Research Foundation (DFG) under grant ZUK 64 (all authors) and KE 1736/1-1 (MKR, MH) using the orthogonal component L in the GKW decomposition (2.3).Moreover, notice that, if L ∈ H 2 0 is orthogonal to L 2 (S) in the Hilbert space sense, then L is also orthogonal to S, i. e. LS is a martingale starting at zero and, in particular, L, S = 0. Therefore, from (2.3) we can compute the optimal strategy ϑ * by S, H = ϑ * · S, S = · 0

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Section: The 3/2-modelmentioning

confidence: 99%

“…To compute the conditional expectation E[e zX t |V t ] we apply the results of [15]. If (R t ) t∈[0,T ] denotes the volatility process of [15] (cf. [15, Eq.…”

Section: The 3/2-modelmentioning

confidence: 99%

Semi-static variance-optimal hedging in stochastic volatility models with Fourier representation

2019

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“…In other words, the process X is a function of a CIR process Y , where this function is the sum of functions of ( A 1) and ( A 2) types. This can be seen as the generalization of the model introduced by Grasselli (), where the stochastic volatility is

a / \sqrt{Y} + b \sqrt{Y}

and follows a (2, 0) mixture model in our terminology. The process Y represents the underlying factor for the optimal stopping problem.…”

Section: Pricing the American Vix Call Under The Generalized Mixture mentioning

confidence: 99%

“…The

(3 / 2, 1 / 2)

mixture model is obtained when

ν = μ = 1

. The (2, 0) mixture model examined by Grasselli () is obtained when

ν = μ = 1 / 2

.…”

Section: Pricing the American Vix Call Under The Generalized Mixture mentioning

confidence: 99%

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On American VIX options under the generalized 3/2 and 1/2 models

Detemple

Kitapbayev

2017

Mathematical Finance

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In this paper, we extend the 3/2-model for VIX studied by Goard and Mazur (2013) and introduce the generalized 3/2 and 1/2 classes of volatility processes. Under these models, we study the pricing of European and American VIX options and, for the latter, we obtain an early exercise premium representation using a free-boundary approach and local time-space calculus. The optimal exercise boundary for the volatility is obtained as the unique solution to an integral equation of Volterra type. We also consider a model mixing these two classes and formulate the corresponding optimal stopping problem in terms of the observed factor process. The price of an American VIX call is then represented by an early exercise premium formula. We show the existence of a pair of optimal exercise boundaries for the factor process and characterize them as the unique solution to a system of integral equations

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The mean‐reverting 4/2 stochastic volatility model: Properties and financial applications

Escobar‐Anel

Gong

2020

Appl Stoch Models Bus & Ind

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This article defines and studies a stochastic process that combines two important stylized facts of financial data: reversion to the mean, and a flexible generalized stochastic volatility process: the 4/2 process. This work is motivated by the modeling of at least two financial asset classes: commodities and volatility indexes. We provide analytical expressions for the conditional characteristic functions and closed-form approximations to relevant cases, in particular a mean-reverting Heston stochastic volatility model. Our results describe feasible changes of measure with the final aim of pricing financial derivatives. The empirical analysis and the estimation methodology confirm the need of such a model in several examples from the targeted asset classes. Applications to option pricing corroborate the substantial impact on the implied volatility surfaces of the new parameters.

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The 4/2 Stochastic Volatility Model: A Unified Approach for the Heston and the 3/2 Model

Cited by 132 publications

References 42 publications

Semi-static variance-optimal hedging in stochastic volatility models with Fourier representation

Semi-static variance-optimal hedging in stochastic volatility models with Fourier representation

On American VIX options under the generalized 3/2 and 1/2 models

The mean‐reverting 4/2 stochastic volatility model: Properties and financial applications

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