Asymptotic and non asymptotic approximations for option valuation

Bompis, Romain; Gobet, Emmanuel

doi:10.1142/9789814436434_0004

Cited by 20 publications

(30 citation statements)

References 37 publications

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“…The slope of ATM implied volatility depends on both the correlation and the slope of local volatility, which are therefore interpreted as skew parameters. There might be a competition between

σ_{0}^{false(1 false)}

and ρ in the calibration procedures. (ii)For pure local volatility models (i.e.,

ξ_{trueprefixsup} = 0

), we retrieve the results of theorem 22 in Bompis and Gobet (). (iii)For pure Heston models (i.e.,

{scriptM}_{1} false(σ false) = 0

), we recover the expansion given in theorem 2.5 of Forde and Jacquier (). In the case of zero correlation, the approximation formula becomes, for short maturity

\begin{matrix} true \\ right σ_{I} (x_{0}, T, k) & left \approx \overset{σ}{true} - \frac{C_{3, T}^{s}}{16 \overset{σ}{true} T} - \frac{C_{3, T}^{s}}{4 {\bar{σ}}^{3} T^{2}} + \frac{C_{3, T}^{s}}{4 {\bar{σ}}^{5} T^{3}} m^{2} \approx \overset{σ}{true} - \frac{ξ_{0}^{2} σ_{0} T}{24 \sqrt{v_{0}}} ([] \frac{σ_{0}^{2} v_{0} T}{4} + 1) + \frac{ξ_{0}^{2}}{24 σ_{0} v_{0}^{\frac{3}{2}}} m^{2} . \end{matrix}

We have obtained that an uncorrelated Heston model induces symmetric smile with respecct to the moneyness.…”

Section: Expansion Formulas For the Implied Volatilitymentioning

confidence: 83%

Analytical approximations of local‐Heston volatility model and error analysis

Bompis

Gobet

2017

Mathematical Finance

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This paper studies the expansion of an option price (with bounded Lipschitz payoff) in a stochastic volatility model including a local volatility component. The stochastic volatility is a square root process, which is widely used for modeling the behavior of the variance process (Heston model). The local volatility part is of general form, requiring only appropriate growth and boundedness assumptions. We rigorously establish tight error estimates of our expansions, using Malliavin calculus. The error analysis, which requires a careful treatment because of the lack of weak differentiability of the model, is interesting on its own. Moreover, in the particular case of call–put options, we also provide expansions of the Black–Scholes implied volatility that allow to obtain very simple formulas that are fast to compute compared to the Monte Carlo approach and maintain a very competitive accuracy.

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“…The slope of ATM implied volatility depends on both the correlation and the slope of local volatility, which are therefore interpreted as skew parameters. There might be a competition between

σ_{0}^{false(1 false)}

and ρ in the calibration procedures. (ii)For pure local volatility models (i.e.,

ξ_{trueprefixsup} = 0

), we retrieve the results of theorem 22 in Bompis and Gobet (). (iii)For pure Heston models (i.e.,

{scriptM}_{1} false(σ false) = 0

), we recover the expansion given in theorem 2.5 of Forde and Jacquier (). In the case of zero correlation, the approximation formula becomes, for short maturity

\begin{matrix} true \\ right σ_{I} (x_{0}, T, k) & left \approx \overset{σ}{true} - \frac{C_{3, T}^{s}}{16 \overset{σ}{true} T} - \frac{C_{3, T}^{s}}{4 {\bar{σ}}^{3} T^{2}} + \frac{C_{3, T}^{s}}{4 {\bar{σ}}^{5} T^{3}} m^{2} \approx \overset{σ}{true} - \frac{ξ_{0}^{2} σ_{0} T}{24 \sqrt{v_{0}}} ([] \frac{σ_{0}^{2} v_{0} T}{4} + 1) + \frac{ξ_{0}^{2}}{24 σ_{0} v_{0}^{\frac{3}{2}}} m^{2} . \end{matrix}

We have obtained that an uncorrelated Heston model induces symmetric smile with respecct to the moneyness.…”

Section: Expansion Formulas For the Implied Volatilitymentioning

confidence: 83%

Analytical approximations of local‐Heston volatility model and error analysis

Bompis

Gobet

2017

Mathematical Finance

Self Cite

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“…If one additionally chooses

false(\overset{x}{true}, \overset{y}{true} false) = false(x, y false)

, then the implied volatility approximation given in Lorig et al. () is equivalent to the implied volatility expansion given in Bompis and Gobet (). However, the expansion presented here and in Lorig et al.…”

Section: Implied Volatilitymentioning

confidence: 99%

Leveraged Etf Implied Volatilities From Etf Dynamics

Leung

Lorig

Pascucci

2016

Mathematical Finance

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The growth of the exchange-traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). We study the relationship between the ETF and LETF implied volatility surfaces when the underlying ETF is modeled by a general class of local-stochastic volatility models. A closed-form approximation for prices is derived for European-style options whose payoff depends on the terminal value of the ETF and/or LETF. Rigorous error bounds for this pricing approximation are established. A closed-form approximation for implied volatilities is also derived. We also discuss a scaling procedure for comparing implied volatilities across leverage ratios.The implied volatility expansions and scalings are tested in three well-known settings: CEV, Heston and SABR.

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“…which is the approximation of the vanilla third order implied volatility expansion given in [5,Theorem 22]. The additional term π k F (a; x ′ avg ) T +ti ti due to the forward start is therefore interpreted as a forward bias.…”

Section: Third Order Forward Implied Volatility Expansionmentioning

confidence: 99%

“…To achieve this, we start from the vanilla implied volatility approximation provided in [5,Theorem 22] and then we use a conditioning expectation argument to express the price (1) of the forward start option as an expectation of the Black-Scholes price function with a stochastic volatility argument involving the local volatility function frozen at X ti , plus an error. Then we perform a volatility expansion to consider the local volatility function frozen at the log-spot x 0 .…”

Section: Introductionmentioning

confidence: 99%

Forward implied volatility expansion in time-dependent local volatility models

Bompis

Hok

2014

ESAIM: ProcS

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Abstract. We introduce an analytical approximation to efficiently price forward start options on equity in time-dependent local volatility models as the forward start date, the maturity or the volatility coefficient are small. We use a conditional expectation argument to represent the price as an expectation of a Black-Scholes formula computed with a stochastic implied volatility depending on the value of the equity at the forward date. Then we perform a volatility expansion to derive an analytical approximation of the forward implied volatility with a precise error estimate. We also illustrate the accuracy of the formula with some numerical experiments. Some results and tools of this work were presented at the conference SMAI 2013 in the mini-symposium "Méthodes asymptotiques en finance".Résumé. Nous introduisons une approximation analytique afin d'évaluer efficacement les optionsà départ différé dites forward start dans les modèlesà volatilité locale qui dépend du temps quand la date forward, la maturité ou le coefficient de volatilité sont petits. Nous utilisons un argument d'espérance conditionnelle pour représenter le prix comme l'espérance d'une formule de Black-Scholes calculée avec une volatilité implicite stochastique qui dépend de la valeur de l'actionà la date forward. Ensuite, nous effectuons un développement en volatilité pour obtenir une approximation analytique de la volatilité implicite forward avec une estimation précise de l'erreur. Nous illustronségalement la précision de notre formule avec quelques expériences numériques. Certains résultats et outils de ce travail ontété présentés au congrès SMAI 2013 dans le mini-symposium "Méthodes asymptotiques en finance".

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Asymptotic and non asymptotic approximations for option valuation

Cited by 20 publications

References 37 publications

Analytical approximations of local‐Heston volatility model and error analysis

Analytical approximations of local‐Heston volatility model and error analysis

Leveraged Etf Implied Volatilities From Etf Dynamics

Forward implied volatility expansion in time-dependent local volatility models

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