Asymptotics of Forward Implied Volatility

Jacquier, Antoine; Roome, Patrick

doi:10.2139/ssrn.2185075

Cited by 16 publications

(43 citation statements)

References 64 publications

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“…Under some conditions on the parameters, it was shown in [47] that the smooth behaviour of the pointwise limit lim τ ↑∞ τ −1 log E(e uX (t) τ ) yielded an asymptotic behaviour for the forward smile as σ In particular for t = 0 (spot smiles), they recovered the result in [27] (also under some restrictions on the parameters). Interestingly, the limiting large-maturity forward smile v ∞ 0 does not depend on the forward-start date t. A number of practitioners (see eg.…”

Section: Xt T≥0mentioning

confidence: 67%

Large-Maturity Regimes of the Heston Forward Smile

Jacquier

Roome

2014

SSRN Journal

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Abstract. We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and different powers of the maturity come into play. As a by-product of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVI-type formulae. The proofs are based on extensions and refinements of sharp large deviations theory, in particular in cases where standard convexity arguments fail.

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Section: Xt T≥0mentioning

confidence: 67%

Large-Maturity Regimes of the Heston Forward Smile

Jacquier

Roome

2014

SSRN Journal

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“…In the literature on implied volatility asymptotics, the moment generating function of the stock price has proved to be an extremely useful tool to obtain sharp estimates. This is obviously the case for the wings of the smile (small and large strikes) via Roger Lee's formula, mentioned in Section 2.1.1, but also to describe short-and large-maturity asymptotics, as developed for instance in [37] or [39], via the use of (a refined version of) the Gärtner-Ellis theorem. As shown in Section 2.1.1, the moment generating function of a stock price satisfying (2.1) is fully determined by that of the random variable V.…”

Section: 3mentioning

confidence: 99%

“…Our intention (as mentioned in Section 1) is to use it as a building block for more advanced models (such as stochastic volatility models where the initial variance is sampled from a continuous distribution) so that we are able to better match steep small-maturity observed smiles. In these types of more sophisticated models, the large-time behaviour is governed more from the chosen stochastic volatility model rather than the choice of distribution for the initial variance (see [39,40] for examples), especially if the variance process possesses some ergodic properties. This also suggests to use this class of models to introduce two different time scales: one to match the small-time smile (the distribution for the initial variance) and one to match the medium-to large-time smile (the chosen stochastic volatility model).…”

Section: Proof One Could Prove the Statement Directly By Computing Tmentioning

confidence: 99%

“…For a given martingale process e X , a forward-start option with reset date t, maturity t + τ and strike e k is worth, at inception, E(exp(X t+τ − X t ) − e k ) + . In the Black-Scholes model, the stationarity property of the increments imply that this option is simply equal to a standard Call option on e X (started at X 0 = 0) with strike e k and maturity τ ; therefore, one can define the forward implied volatility σ t,τ (k), similarly to the standard implied volatility (see [39] for more details). Suppose now that the log stock price process X satisfies the following SDE:…”

Section: Proof One Could Prove the Statement Directly By Computing Tmentioning

confidence: 99%

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Black-Scholes in a CEV Random Environment: A New Approach to Smile Modelling

Jacquier

Roome

2015

SSRN Journal

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Abstract. Classical (Itô diffusions) stochastic volatility models are not able to capture the steepness of smallmaturity implied volatility smiles. Jumps, in particular exponential Lévy and affine models, which exhibit small-maturity exploding smiles, have historically been proposed to remedy this (see [60] for an overview). A recent breakthrough was made by Gatheral, Jaisson and Rosenbaum [30], who proposed to replace the Brownian driver of the instantaneous volatility by a short-memory fractional Brownian motion, which is able to capture the short-maturity steepness while preserving path continuity. We suggest here a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Lévy models and fractional stochastic volatility models.

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“…An alternative modeling is the use of Levy processes proposed for instance in [4]. Recently, Jacquier and Roome [12] provided an expansion formula of the forward implied volatility using calculations based on the forward characteristic function and large deviations techniques. Such an enthusiasm for the stochastic volatility models or more generally for two or more factors models in the literature can be explained by the potential availability of closed formulas using the (semi) explicit computation of the forward characteristic function owing to the tower property for conditional expectations.…”

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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Asymptotics of Forward Implied Volatility

Cited by 16 publications

References 64 publications

Large-Maturity Regimes of the Heston Forward Smile

Large-Maturity Regimes of the Heston Forward Smile

Black-Scholes in a CEV Random Environment: A New Approach to Smile Modelling

Forward implied volatility expansion in time-dependent local volatility models

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