## Publications

54 results found

Raval V, Jacquier A, 2023, The log-moment formula for implied volatility, *Mathematical Finance*, ISSN: 0960-1627

We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. We show that in the absence of arbitrage, if the underlying stock price at time T admits finite log-moments (Formula presented.) for some positive q, the arbitrage-free growth in the left wing of the implied volatility smile for T is less constrained than Lee's bound. The result is rationalized 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 price to admit any negative moment. In this respect, the result can be derived from a model-independent setup. As a byproduct, we relax the moment assumptions on the stock price to provide a new proof of the notorious Gatheral–Fukasawa formula expressing variance swaps in terms of the implied volatility.

Jacquier A, Oumgari M, 2023, Deep Curve-Dependent PDEs for Affine Rough Volatility, *SIAM Journal on Financial Mathematics*, Vol: 14, Pages: 353-382

We introduce a new deep learning-based algorithm to evaluate options in affine rough stochastic volatility models. Viewing the pricing function as the solution to a curve-dependent PDE, depending on forward curves rather than the whole path of the process, we develop a numerical scheme based on deep learning techniques. Numerical simulations suggest that the latter is a promising alternative to classical Monte Carlo simulations.

Assouel A, Jacquier A, Kondratyev A, 2022, A quantum generative adversarial network for distributions, *QUANTUM MACHINE INTELLIGENCE*, Vol: 4, ISSN: 2524-4906

Jacquier A, Pannier A, 2022, Large and moderate deviations for stochastic Volterra systems, *Stochastic Processes and their Applications*, Vol: 149, Pages: 142-187, ISSN: 0304-4149

We provide a unified treatment of pathwise Large and Moderate deviationsprinciples for a general class of multidimensional stochastic Volterraequations with singular kernels, not necessarily of convolution form. Ourmethodology is based on the weak convergence approach by Budhijara, Dupuis and Ellis. We show in particular how this framework encompasses most rough volatility models used in mathematical finance and generalises many recent results in the literature.

Geha M, Jacquier A, Zuric Z, 2021, Large and moderate deviations for importance sampling in the Heston model, *Annals of Operations Research*, ISSN: 0254-5330

We provide a detailed importance sampling analysis for variance reduction instochastic volatility models. The optimal change of measure is obtained using avariety of results from large and moderate deviations: small-time, large-time,small-noise. Specialising the results to the Heston model, we derive manyclosed-form solutions, making the whole approach easy to implement. We supportour theoretical results with a detailed numerical analysis of the variancereduction gains.

Jacquier A, Badikov S, Davis M, 2021, Perturbation analysis of sub/super hedging problems, *Mathematical Finance*, Vol: 31, Pages: 1240-1274, ISSN: 0960-1627

We investigate the links between various no-arbitrage conditions and the existence of pricing functionals in general markets, and prove the Fundamental Theorem of Asset Pricing therein. No-arbitrage conditions, either in this abstract setting or in the case of a market consisting of European Call options, give rise to duality properties of infinite-dimensional sub- and super-hedging problems. With a view towards applications, we show how duality is preserved when reducing these problems over finite-dimensional bases. We also introduce a rigorous perturbation analysis of these linear programing problems, and highlight numerically the influence of smile extrapolation on the bounds of exotic options.

Gerhold S, Jacquier A, Pakkanen M,
et al., 2021, Pathwise large deviations for the rough Bergomi model: Corrigendum, *Journal of Applied Probability*, Vol: 58, Pages: 849-850, ISSN: 0021-9002

This note corrects an error in the definition of the rate function in Jacquier, Pakkanen, and Stone (2018) and slightly simplifies some proofs.

El Amrani M, Jacquier A, Martini C, 2021, Short communication: dynamics of symmetric SSVI smiles and implied volatility bubbles, *SIAM Journal on Financial Mathematics*, Vol: 12, Pages: 1-15, ISSN: 1945-497X

We develop a dynamic version of the SSVI parameterization 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 behavior. 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.

Fontanela F, Jacquier A, Oumgari M, 2021, Short Communication: A Quantum Algorithm for Linear PDEs Arising in Finance, *SIAM JOURNAL ON FINANCIAL MATHEMATICS*, Vol: 12, Pages: SC98-SC114, ISSN: 1945-497X

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Jacquier A, Shi F, 2020, Small-time moderate deviations for the randomised Heston model, *Journal of Applied Probability*, ISSN: 0021-9002

We extend previous large deviations results for the randomised Heston modelto the case of moderate deviations. The proofs involve the G\"artner-Ellistheorem and sharp large deviations tools.

Horvath B, Jacquier A, Tankov P, 2020, Volatility options in rough volatility models, *SIAM Journal on Financial Mathematics*, ISSN: 1945-497X

We discuss the pricing and hedging of volatility options in some roughvolatility models. First, we develop efficient Monte Carlo methods andasymptotic approximations for computing option prices and hedge ratios inmodels where log-volatility follows a Gaussian Volterra process. Whileproviding a good fit for European options, these models are unable to reproducethe VIX option smile observed in the market, and are thus not suitable for VIXproducts. To accommodate these, we introduce the class of modulated Volterraprocesses, and show that they successfully capture the VIX smile.

Jacquier A, Spiliopoulos K, 2020, Pathwise moderate deviations for option pricing, *Mathematical Finance*, Vol: 30, Pages: 426-463, ISSN: 0960-1627

We provide a unifying treatment of pathwise moderate deviations for models commonly used in financial applications, and for related integrated functionals. Suitable scaling allows us to transfer these results into small-time, large-time and tail asymptotics for diffusions, as well as for option prices and realised variances. In passing, we highlight some intuitive relationships between moderate deviations rate functions and their large deviations counterparts; these turn out to be useful for numerical purposes, as large deviations rate functions are often difficult to compute.

Jacquier A, Torricelli L, 2019, Anomalous diffusions in option prices: connecting trade duration and the volatility term structure, Publisher: Elsevier BV

Anomalous diffusions arise as scaling limits of continuous-time random walks (CTRWs) whose innovation times are distributed according to a power law. The impact of a non-exponential waiting time does not vanish with time and leads to different distribution spread rates compared to standard models. In financial modelling this has been used to accommodate for random trade duration in the tick-by-tick price process. We show here that anomalous diffusions are able to reproduce the market behaviour of the implied volatility more consistently than usual Lévy or stochastic volatility models. We focus on two distinct classes of underlying asset models, one with independent price innovations and waiting times, and one allowing dependence between these two components. These two models capture the well-known paradigm according to which shorter trade duration is associated with higher return impact of individual trades. We fully describe these processes in a semimartingale setting leading no-arbitrage pricing formulae, and study their statistical properties. We observe that skewness and kurtosis of the asset returns do not tend to zero as time goes by. We also characterize the large-maturity asymptotics of Call option prices, and find that the convergence rate is slower than in standard Lévy regimes, which in turn yields a declining implied volatility term structure and a slower decay of the skew.

Horvath B, Jacquier A, Lacombe C, 2019, Asymptotic behaviour of randomised fractional volatility models, *Journal of Applied Probability*, ISSN: 0021-9002

We study the asymptotic behaviour of a class of small-noise diffusions drivenby fractional Brownian motion, with random starting points. Different scalingsallow for different asymptotic properties of the process (small-time and tailbehaviours in particular). In order to do so, we extend some results on samplepath large deviations for such diffusions. As an application, we show how theseresults characterise the small-time and tail estimates of the impliedvolatility for rough volatility models, recently proposed in mathematicalfinance.

Jacquier A, Shi F, 2019, The Randomized Heston Model, *SIAM Journal on Financial Mathematics*, Vol: 10, Pages: 89-129, ISSN: 1945-497X

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 terms of the right tail of the initial distribution, and in particular show that an explosion rate of~tγ (γ∈[0,1/2]) for the squared implied volatility-as observed on market data-can be obtained by a suitable choice of randomisation. The proofs are based on large deviations techniques and the theory of regular variations.

Jacquier A, Pakkanen MS, Stone H, 2018, Pathwise large deviations for the rough Bergomi model, *Journal of Applied Probability*, Vol: 55, Pages: 1078-1092, ISSN: 0021-9002

Introduced recently in mathematical finance by Bayer et al. (2016), the rough Bergomi model has proved particularly efficient to calibrate option markets. We investigate some of its probabilistic properties, in particular proving a pathwise large deviations principle for a small-noise version of the model. The exponential function (continuous but superlinear) as well as the drift appearing in the volatility process fall beyond the scope of existing results, and a dedicated analysis is needed.

Gulisashvili A, Horvath B, Jacquier A, 2018, Mass at zero in the uncorrelated SABR model and implied volatility asymptotics, *Quantitative Finance*, Vol: 18, Pages: 1753-1765, ISSN: 1469-7688

We study the mass at the origin in the uncorrelated SABR stochasticvolatility model, and derive several tractable expressions, in particular whentime becomes small or large. As an application--in fact the original motivationfor this paper--we derive small-strike expansions for the implied volatilitywhen the maturity becomes short or large. These formulae, by definitionarbitrage free, allow us to quantify the impact of the mass at zero on existingimplied volatility approximations, and in particular how correct/erroneousthese approximations become.

Alos E, Jacquier A, Leon J, 2018, The implied volatility of Forward-Start options: ATM short-time level, skew and curvature, *Stochastics: An International Journal of Probability and Stochastic Processes*, ISSN: 1744-2508

Using Malliavin Calculus techniques, we derive closed-form expressions forthe at-the-money behaviour of the forward implied volatility, its skew and itscurvature, in general Markovian stochastic volatility models with continuouspaths.

Horvath B, Jacquier AJ, Turfus C, 2018, Analytic Option Prices for the Black-Karasinski Short Rate Model

Jacquier A, Liu H, 2018, Optimal Liquidation in a Level-I Limit Order Book for Large-Tick Stocks, *SIAM JOURNAL ON FINANCIAL MATHEMATICS*, Vol: 9, Pages: 875-906, ISSN: 1945-497X

Guennoun H, Jacquier A, Roome P,
et al., 2018, Asymptotic behaviour of the fractional Heston model, *SIAM Journal on Financial Mathematics*, ISSN: 1945-497X

We consider the fractional Heston model originally proposed by Comte, Coutinand Renault. Inspired by recent ground-breaking work on rough volatility, whichshowed that models with volatility driven by fractional Brownian motion withshort memory allows for better calibration of the volatility surface and morerobust estimation of time series of historical volatility, we provide acharacterisation of the short- and long-maturity asymptotics of the impliedvolatility smile. Our analysis reveals that the short-memory property preciselyprovides a jump-type behaviour of the smile for short maturities, therebyfixing the well-known standard inability of classical stochastic volatilitymodels to fit the short-end of the volatility smile.

Jacquier A, Roome P, 2018, Black-Scholes in a CEV Random Environment, *Mathematics and Financial Economics*

Classical (It\^o diffusions) stochastic volatility models are not able to capture the steepness of small-maturity implied volatility smiles. Jumps, in particular exponential L\'evy 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 volatility for small maturities. The range of rates includes behaviours similar to exponential Levy models and fractional stochastic volatility models.

Jacquier A, Keller-Ressel MKR, 2018, Implied volatility in strict local martingale models, *SIAM Journal on Financial Mathematics*, Vol: 9, Pages: 171-189, ISSN: 1945-497X

We consider implied volatilities in asset pricing models, where the discounted underlying is a strictlocal martingale under the pricing measure. Our main result gives an asymptotic expansion of theright wing of the implied volatility smile and shows that the strict local martingale property can bedetermined from this expansion. This result complements the well-known asymptotic results of Leeand of Benaim and Friz, which apply only to true martingales. This also shows that “price bubbles”in the sense of strict local martingale behavior can in principle be detected by an analysis of impliedvolatility. Finally we relate our results to left-wing expansions of implied volatilities in models withmass at zero by a duality method based on an absolutely continuous measure change.

Horvath B, Jacquier A, Muguruza A, 2017, Functional central limit theorems for rough volatility

We extend Donsker's approximation of Brownian motion to fractional Brownianmotion with Hurst exponent $H \in (0,1)$ and to Volterra-like processes. Someof the most relevant consequences of our `rough Donsker (rDonsker) Theorem' areconvergence results for discrete approximations of a large class of roughmodels. This justifies the validity of simple and easy-to-implement Monte-Carlomethods, for which we provide detailed numerical recipes. We test these againstthe current benchmark Hybrid scheme \cite{BLP15} and find remarkable agreement(for a large range of values of~$H$). This rDonsker Theorem further provides aweak convergence proof for the Hybrid scheme itself, and allows to constructbinomial trees for rough volatility models, the first available scheme (in therough volatility context) for early exercise options such as American orBermudan.

De Marco SDM, Hillairet CH, Jacquier A, 2017, Shapes of implied volatility with positive mass at zero, *SIAM Journal on Financial Mathematics*, Vol: 8, Pages: 709-737, ISSN: 1945-497X

We study the shapes of the implied volatility when the underlying distribution has an atom at zeroand analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of thesmile. We further show that the behaviour at small strikes is uniquely determined by the mass of theatom up to high asymptotic order, under mild assumptions on the remaining distribution on the positivereal line. We investigate the structural di erence with the no-mass-at-zero case, showing how one can{theoretically{distinguish between mass at the origin and a heavy-left-tailed distribution. We numericallytest our model-free results in stochastic models with absorption at the boundary, such as the CEV process,and in jump-to-default models. Note that while Lee's moment formula [25] tells that implied variance is atmost asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as [3,17]do not apply in this setting{essentially due to the breakdown of Put-Call duality.

Jacquier A, Aitor Muguruza AM, Claude Martini CM, 2017, On VIX Futures in the rough Bergomi model, *Quantitative Finance*, Vol: 18, Pages: 45-61, ISSN: 1469-7696

The rough Bergomi model introduced by Bayer et al. [Quant. Finance, 2015, 1–18] has been outperforming conventional Markovian stochastic volatility models by reproducing implied volatility smiles in a very realistic manner, in particular for short maturities. We investigate here the dynamics of the VIX and the forward variance curve generated by this model, and develop efficient pricing algorithms for VIX futures and options. We further analyse the validity of the rough Bergomi model to jointly describe the VIX and the SPX, and present a joint calibration algorithm based on the hybrid scheme by Bennedsen et al. [Finance Stoch., forthcoming].

Badikov SB, Jacquier A, Liu DQ,
et al., 2017, No-arbitrage bounds for the forward smile given marginals, *Quantitative Finance*, Vol: 17, Pages: 1243-1256, ISSN: 1469-7696

We explore the robust replication of forward-start straddles given quoted (Call and Put options) market data. One approach to this problem classically follows semi-infinite linear programming arguments, and we propose a discretisation scheme to reduce its dimensionality and hence its complexity. Alternatively, one can consider the dual problem, consisting in finding optimal martingale measures under which the upper and the lower bounds are attained. Semi-analytical solutions to this dual problem were proposed by Hobson and Klimmek and by Hobson and Neuberger. We recast this dual approach as a finite dimensional linear programme, and reconcile numerically, in the Black-Scholes and in the Heston model, the two approaches.

Chassagneux JFC, Jacquier A, Mihyalov IM, 2016, An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients, *SIAM Journal on Financial Mathematics*, Vol: 7, Pages: 993-1021, ISSN: 1945-497X

We consider the approximation of one-dimensional stochastic differential equations(SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modi-fied explicit Euler-Maruyama discretisation scheme that allows us to prove strongconvergence, with a rate. Under some regularity and integrability conditions, weobtain the optimal strong error rate. We apply this scheme to SDEs widely usedin the mathematical finance literature, including the Cox-Ingersoll-Ross (CIR), the3/2 and the Ait-Sahalia models, as well as a family of mean-reverting processeswith locally smooth coefficients. We numerically illustrate the strong convergenceof the scheme and demonstrate its efficiency in a multilevel Monte Carlo setting.

Gulisashvili AG, Horvath BH, Jacquier A, 2016, On the probability of hitting the boundary for Brownian motions on the SABR plane, *Electronic Communications in Probability*, Vol: 21, Pages: 1-13, ISSN: 1083-589X

Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models–related to the SABR model in mathematical finance–which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods.

Guo GG, Jacquier A, Martini CM,
et al., 2016, Generalized Arbitrage-Free SVI Volatility Surfaces, *SIAM Journal on Financial Mathematics*, Vol: 7, Pages: 619-641, ISSN: 1945-497X

In this paper we propose a generalization of the recent work by Gatheral and Jacquier [J. Gatheral and A. Jacquier, Quant. Finance, 14 (2014), pp. 59--71] on explicit arbitrage-free parameterizations of implied volatility surfaces. We also discuss extensively the notion of arbitrage freeness and Roger Lee's moment formula using the recent analysis by Roper [M. Roper, Arbitrage-Free Implied Volatility Surfaces, preprint, School of Mathematics and Statistics, The University of Sydney, Sydney, New South Wales, Australia, 2010, ŭlhttp://www.maths.usyd.edu.au/u/pubs/publist/preprints/2010/roper-9.pdf]. We further exhibit an arbitrage-free volatility surface different from Gatheral's SVI parameterization.

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