We investigate the asymptotic behaviour of the Bachelier implied volatility tails, extending the large-strike results established for the Black-Scholes implied volatility. Exploiting the theory of regular variation, we derive explicit expressions for the Bachelier implied volatility in the wings of the smile, directly linking them to the tail decay of the underlying returns' distribution. Furthermore, we establish a rigorous connection between the analyticity strip of the characteristic function and the asymptotic slope of the volatility smile. Moreover, we show that if the implied variance grows linearly for large absolute moneyness degree, the underlying returns' distribution must exhibit exponential tail decay, and the corresponding characteristic function is analytic in a horizontal strip of the complex plane. These findings characterise valid models based solely on the observable asymptotic behaviour of the smile.
What kind of implied volatility extrapolation is appropriate? Roger Lee proved that the Black-Scholes implied variance can not grow faster than linearly in log-moneyness. This paper investigates what happens in the Bachelier (or Normal) implied volatility world, making sure to cover the various aspects of vanilla option arbitrages.
We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential Lévy models. This expansion applies to both small and large maturities and is based solely on the properties of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques, and allows us to recover (in particular) many results for the spot implied volatility smile. In passing we (i) show that the forward-start date ha…
We correct a mistake in the published version of our paper. Our new conclusion is that the "implied leverage effect" for single stocks is underestimated by option markets for short maturities and overestimated for long maturities, while it is always overestimated for OEX options, except for the shortest maturities where the revised theory and data match perfectly.
We develop a dynamic version of the SSVI parameterisation for the total implied variance, ensuring that European vanilla option prices are martingales, hence preventing the occurrence of arbitrage, both static and dynamic. Insisting on the constraint that the total implied variance needs to be null at the maturity of the option, we show that no model--in our setting--allows for such behaviour. This naturally gives rise to the concept of implied volatility bubbles, whereby trading in an arbitrage-free way is only possible during part of the life of the contract, but not all the way until expiry…
The paper studies estimation of parameters of diffusion market models from historical data. The standard definition of implied volatility for these models presents its value as an implicit function of several parameters, including the risk-free interest rate. In reality, the risk free interest rate is unknown and need to be forecasted, because the option price depends on its future curve. Therefore, the standard implied volatility is {\it conditional}: it depends on the future values of the risk free rate. We study two implied parameters: the implied volatility and the implied average cumulati…
We study a Markov-Functional (MF) interest-rate model with Uncertain Volatility Displaced Diffusion (UVDD) digital mapping, which is consistent with the volatility-smile phenomenon observed in the option market. We first check the impact of pricing Bermudan swaptions by the model. Next, we also investigate the future smiles implied by the MF models and the smile dynamics implied by the UVDD model. Finally, we conduct hedging simulations against Bermudan swaptions to test extensively the hedge performance of this smile-consistenet MF model.
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 par…
We study in details the skew of stock option smiles, which is induced by the so-called leverage effect on the underlying -- i.e. the correlation between past returns and future square returns. This naturally explains the anomalous dependence of the skew as a function of maturity of the option. The market cap dependence of the leverage effect is analyzed using a one-factor model. We show how this leverage correlation gives rise to a non-trivial smile dynamics, which turns out to be intermediate between the "sticky strike" and the "sticky delta" rules. Finally, we compare our result with stock o…
We introduce a new class of local volatility models. Within this framework, we obtain expressions for both (i) the price of any European option and (ii) the induced implied volatility smile. As an illustration of our framework, we perform specific pricing and implied volatility computations for a CEV-like example. Numerical examples are provided.
We review and illustrate how the volatility smile translates into a probability distribution, the market-implied probability distribution representing believes priced in. The effects of changes in the smile are examined. Special attention is given to the effects of slope, which might appear at first counter-intuitive. We then show how Bayesian methods can be used to deal with sparse real market data. With each skew in a parametric model we associate a probability. This is illustrated with an example, for which multivariate parameter distributions are derived. We introduce the fuzzy smile (or f…
We construct a data-driven statistical indicator for quantifying the tail risk perceived by the EURGBP option market surrounding Brexit-related events. We show that under lognormal SABR dynamics this tail risk is closely related to the so-called martingale defect and provide a closed-form expression for this defect which can be computed by solving an inverse calibration problem. In order to cope with the the uncertainty which is inherent to this inverse problem, we adopt a Bayesian statistical parameter estimation perspective. We probe the resulting posterior densities with a combination of op…
We provide explicit conditions on the distribution of risk-neutral log-returns which yield sharp asymptotic estimates on the implied volatility smile. We allow for a variety of asymptotic regimes, including both small maturity (with arbitrary strike) and extreme strike (with arbitrary bounded maturity), extending previous work of Benaim and Friz [Math. Finance 19 (2009), 1-12]. We present applications to popular models, including Carr-Wu finite moment logstable model, Merton's jump diffusion model and Heston's model.
We study here the large-time behaviour of all continuous affine stochastic volatility models (in the sense of Keller-Ressel) and deduce a closed-form formula for the large-maturity implied volatility smile. Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals pathological behaviours of the asymptotic smile. In particular, we show that the condition assumed in Gatheral and Jacquier under which the Heston implied volatility converges to the SVI parameterisation is necessary and sufficient.
In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-of-the-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter configurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent an…
We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential Lévy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent Lévy measure. Using techniques from regular perturbation theory and Fourier analysis, we derive a series expansion for the price of a European-style option. We also provide precise conditions under which this series expansion converges to the exact price. Additionally, for a certain subclass of assets in our modeling framework, we…
Classical (Itô diffusions) stochastic volatility models are not able to capture the steepness of small-maturity 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 \cite{Tank} for an overview), and more recently rough volatility models \cite{AlosLeon, Fukasawa}. 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 volatil…
In the Black-Scholes context we consider the probability distribution function (PDF) of financial returns implied by volatility smile and we study the relation between the decay of its tails and the fitting parameters of the smile. We show that, considering a scaling law derived from data, it is possible to get a new fitting procedure of the volatility smile that considers also the exponential decay of the real PDF of returns observed in the financial markets. Our study finds application in the Risk Management activities where the tails characterization of financial returns PDF has a central r…
We present an explicit hedging strategy, which enables to prove arbitrageness of market incorporating at least two assets depending on the same random factor. The implied Black-Scholes volatility, computed taking into account the form of the graph of the option price, related to our strategy, demonstrates the "skewness" inherent to the observational data.
We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a non-constant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal-Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short end of the implied volatility surface and sheds …
Exponential Lévy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfaces have been analyzed in some detail. In the non-asymptotic regimes, option prices are described by the Lewis-Lipton formula which allows one to represent them as Fourier integrals; the prices can be trivially expressed in terms of their implied volatility. Recently, attempts at calculating the asymptotic limi…
We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that when the underlying stock price martingale admits finite log-moments E[|log(S)|^q] for some positive q, the arbitrage-free growth in the left wing of the implied volatility smile is less constrained than Lee's bound. The result is rationalised by a market trading discretely monitored variance swaps wherein the payoff is a function of squared log-returns, and requires no assumption for the underlying martingale to admit any negative moment. In this respect, the result can derived from a model-independ…
We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can--theoretically--distinguish between mass at the origin and a heavy-left-tailed distributi…
It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment $s_+$ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: $σ_{BS}( k,T)^{2}T\sim Ψ(s_+-1) \times k$ (Roger Lee's moment formula). Motivated by recent "tail-wing" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Dragulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined expan…
Implied volatility is at the very core of modern finance, notwithstanding standard option pricing models continue to derive option prices starting from the joint dynamics of the underlying asset price and the spot volatility. These models often cause difficulties: no closed formulas for prices, demanding calibration techniques, unclear maps between spot and implied volatility. Inspired by the practice of using implied volatility as quoting system for option prices, models for the joint dynamics of the underlying asset price and the implied volatility have been proposed to replace standard opti…
We consider the at-the-money strike derivative of implied volatility as the maturity tends to zero. Our main results quantify the behavior of the slope for infinite activity exponential Lévy models including a Brownian component. As auxiliary results, we obtain asymptotic expansions of short maturity at-the-money digital call options, using Mellin transform asymptotics. Finally, we discuss when the at-the-money slope is consistent with the steepness of the smile wings, as given by Lee's moment formula.
In this work, we aim to gain a better understanding of the volatility smile observed in options markets through microsimulation (MS). We adopt two types of active traders in our MS model: speculators and arbitrageurs, and call and put options on one underlying asset. Speculators make decisions based on their expectations of the asset price at the option expiration time. Arbitrageurs trade at different arbitrage opportunities such as violation of put-call parity. Difference in liquidity among options is also included. Notwithstanding its simplicity, our model can generate implied volatility (IV…
This paper gives an arbitrage-free prediction for future prices of an arbitrary co-terminal set of options with a given maturity, based on the observed time series of these option prices. The statistical analysis of such a multi-dimensional time series of option prices corresponding to $n$ strikes (with $n$ large, e.g. $n\geq 40$) and the same maturity, is a difficult task due to the fact that option prices at any moment in time satisfy non-linear and non-explicit no-arbitrage restrictions. Hence any $n$-dimensional time series model also has to satisfy these implicit restrictions at each time…
For any strictly positive martingale $S = \exp(X)$ for which $X$ has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential Lévy model (Merton), one infinite activity exponential Lévy model (Variance Gamma), and one stochastic volatility model (Heston). Finally, we illustrate how our expansio…
Gulisashvili et al. [Quant. Finance, 2018, 18(10), 1753-1765] provide a small-time asymptotics for the mass at zero under the uncorrelated stochastic-alpha-beta-rho (SABR) model by approximating the integrated variance with a moment-matched lognormal distribution. We improve the accuracy of the numerical integration by using the Gauss--Hermite quadrature. We further obtain the option price by integrating the constant elasticity of variance (CEV) option prices in the same manner without resorting to the small-strike volatility smile asymptotics of De Marco et al. [SIAM J. Financ. Math., 2017, 8…
The implied volatility skew has received relatively little attention in the literature on short-term asymptotics for financial models with jumps, despite its importance in model selection and calibration. We rectify this by providing high-order asymptotic expansions for the at-the-money implied volatility skew, under a rich class of stochastic volatility models with independent stable-like jumps of infinite variation. The case of a pure-jump stable-like Lévy model is also considered under the minimal possible conditions for the resulting expansion to be well defined. Unlike recent results for …
Let $σ_t(x)$ denote the implied volatility at maturity $t$ for a strike $K=S_0 e^{xt}$, where $x\in\bbR$ and $S_0$ is the current value of the underlying. We show that $σ_t(x)$ has a uniform (in $x$) limit as maturity $t$ tends to infinity, given by the formula $σ_\infty(x)=\sqrt{2}(h^*(x)^{1/2}+(h^*(x)-x)^{1/2})$, for $x$ in some compact neighbourhood of zero in the class of affine stochastic volatility models. The function $h^*$ is the convex dual of the limiting cumulant generating function $h$ of the scaled log-spot process. We express $h$ in terms of the functional characteristics of the …
We propose a randomised version of the Heston model-a widely used stochastic volatility model in mathematical finance-assuming that the starting point of the variance process is a random variable. In such a system, we study the small-and large-time behaviours of the implied volatility, and show that the proposed randomisation generates a short-maturity smile much steeper (`with explosion') than in the standard Heston model, thereby palliating the deficiency of classical stochastic volatility models in short time. We precisely quantify the speed of explosion of the smile for short maturities in…
Pricing options on illiquid assets using liquid market benchmarks: an application to energy markets
Enel Global Energy and Commodity Management（Enel全球能源与商品管理部）；Dipartimento di Matematica, Università di Roma Tor Vergata（罗马托尔维加塔大学数学系）；INdAM-GNAMPA, Research Unit at Tor Vergata（Tor Vergata INdAM-GNAMPA研究组）；Dipartimento di Economia e Finanza, Università di Roma Tor Vergata（罗马托尔维加塔大学经济与金融系）
It is well know that, in the short maturity limit, the implied volatility approaches the integral harmonic mean of the local volatility with respect to log-strike, see [Berestycki et al., Asymptotics and calibration of local volatility models, Quantitative Finance, 2, 2002]. This paper is dedicated to a complementary model-free result: an arbitrage-free implied volatility in fact is the harmonic mean of a positive function for any fixed maturity. We investigate the latter function, which is tightly linked to Fukasawa's invertible map $f_{1/2}$ [Fukasawa, The normalizing transformation of the i…
We consider the asymptotic behavior of the implied volatility in stochastic asset price models with atoms. In such models, the asset price distribution has a singular component at zero. Examples of models with atoms include the constant elasticity of variance model, jump-to-default models, and stochastic models described by processes stopped at the first hitting time of zero. For models with atoms, the behavior of the implied volatility at large strikes is similar to that in models without atoms. On the other hand, the behavior of the implied volatility at small strikes is influenced significa…
We characterize the behaviour of the Rough Heston model introduced by Jaisson\&Rosenbaum \cite{JR16} in the small-time, large-time and $α\to 1/2$ (i.e. $H\to 0$) limits. We show that the short-maturity smile scales in qualitatively the same way as a general rough stochastic volatility model (cf.\ \cite{FZ17}, \cite{FGP18a} et al.), and the rate function is equal to the Fenchel-Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in \cite{ER19}, but with the drift and mean reversion terms removed. The solution to this VIE satisf…
We revisit the problem of pricing options with historical volatility estimators. We do this in the context of a generalized GARCH model with multiple time scales and asymmetry. It is argued that the reason for the observed volatility risk premium is tail risk aversion. We parametrize such risk aversion in terms of three coefficients: convexity, skew and kurtosis risk premium. We propose that option prices under the real-world measure are not martingales, but that their drift is governed by such tail risk premia. We then derive a fair-pricing equation for options and show that the solutions can…
Using the large deviation principle (LDP) for a re-scaled fractional Brownian motion $B^H_t$ where the rate function is defined via the reproducing kernel Hilbert space, we compute small-time asymptotics for a correlated fractional stochastic volatility model of the form $dS_t=S_tσ(Y_t) (\barρ dW_t +ρdB_t), \,dY_t=dB^H_t$ where $σ$ is $α$-Hölder continuous for some $α\in(0,1]$; in particular, we show that $t^{H-\frac{1}{2}} \log S_t $ satisfies the LDP as $t\to0$ and the model has a well-defined implied volatility smile as $t \to 0$, when the log-moneyness $k(t)=x t^{\frac{1}{2}-H}$. Thus the …
We compute a sharp small-time estimate for implied volatility under a general uncorrelated local-stochastic volatility model. For this we use the Bellaiche \cite{Bel81} heat kernel expansion combined with Laplace's method to integrate over the volatility variable on a compact set, and (after a gauge transformation) we use the Davies \cite{Dav88} upper bound for the heat kernel on a manifold with bounded Ricci curvature to deal with the tail integrals. If the correlation $ρ< 0$, our approach still works if the drift of the volatility takes a specific functional form and there is no local volati…
