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Journal articleCass T, Lim N, 2019,
A StratonovichSkorohod integral formula for Gaussian rough paths
, Annals of Probability, Vol: 47, Pages: 160, ISSN: 00911798Given a Gaussian process X, its canonical geometric rough path lift X, and a solution Y to the rough differential equation (RDE) dYt=V(Yt)∘dXt, we present a closedform correction formula for ∫Y∘dX−∫YdX, that is, the difference between the rough and Skorohod integrals of Y with respect to X. When X is standard Brownian motion, we recover the classical StratonovichtoItô conversion formula, which we generalize to Gaussian rough paths with finite pvariation, p<3, and satisfying an additional natural condition. This encompasses many familiar examples, including fractional Brownian motion with H>13. To prove the formula, we first show that the Riemannsum approximants of the Skorohod integral converge in L2(Ω) by using a novel characterization of the Cameron–Martin norm in terms of higherdimensional Young–Stieltjes integrals. Next, we append the approximants of the Skorohod integral with a suitable compensation term without altering the limit, and the formula is finally obtained after a rebalancing of terms.

Journal articleGulisashvili A, Horvath B, Jacquier A, 2018,
Mass at zero in the uncorrelated SABR model and implied volatility asymptotics
, Quantitative Finance, Vol: 18, Pages: 17531765, ISSN: 14697688We 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 applicationin fact the original motivationfor this paperwe derive smallstrike 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.

Journal articleGuennoun H, Jacquier A, Roome P, et al., 2014,
Asymptotic behaviour of the fractional Heston model
, SIAM Journal on Financial Mathematics, ISSN: 1945497XWe consider the fractional Heston model originally proposed by Comte, Coutinand Renault. Inspired by recent groundbreaking 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 longmaturity asymptotics of the impliedvolatility smile. Our analysis reveals that the shortmemory property preciselyprovides a jumptype behaviour of the smile for short maturities, therebyfixing the wellknown standard inability of classical stochastic volatilitymodels to fit the shortend of the volatility smile.

Journal articleDavis M, Obłój J, Siorpaes P, 2018,
Pathwise stochastic calculus with local times
, Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, Vol: 54, Pages: 121, ISSN: 02460203We study a notion of local time for a continuous path, defined as a limit ofsuitable discrete quantities along a general sequence of partitions of the timeinterval. Our approach subsumes other existing definitions and agrees with theusual (stochastic) local times a.s. for paths of a continuous semimartingale.We establish pathwise version of the It\^oTanaka, change of variables andchange of time formulae. We provide equivalent conditions for existence ofpathwise local time. Finally, we study in detail how the limiting objects, thequadratic variation and the local time, depend on the choice of partitions. Inparticular, we show that an arbitrary given nondecreasing process can beachieved a.s. by the pathwise quadratic variation of a standard Brownian motionfor a suitable sequence of (random) partitions; however, such degeneratebehavior is excluded when the partitions are constructed from stopping times.

Journal articleBennedsen M, Lunde A, Pakkanen MS, 2017,
Hybrid scheme for Brownian semistationary processes
, Finance and Stochastics, Vol: 21, Pages: 931965, ISSN: 09492984We introduce a simulation scheme for Brownian semistationary processes, whichis based on discretizing the stochastic integral representation of the processin the time domain. We assume that the kernel function of the process isregularly varying at zero. The novel feature of the scheme is to approximatethe kernel function by a power function near zero and by a step functionelsewhere. The resulting approximation of the process is a combination ofWiener integrals of the power function and a Riemann sum, which is why we callthis method a hybrid scheme. Our main theoretical result describes theasymptotics of the mean square error of the hybrid scheme and we observe thatthe scheme leads to a substantial improvement of accuracy compared to theordinary forward Riemannsum scheme, while having the same computationalcomplexity. We exemplify the use of the hybrid scheme by two numericalexperiments, where we examine the finitesample properties of an estimator ofthe roughness parameter of a Brownian semistationary process and study MonteCarlo option pricing in the rough Bergomi model of Bayer et al. (2015),respectively.

Journal articleDe Marco SDM, Hillairet CH, Jacquier A, 2017,
Shapes of implied volatility with positive mass at zero
, SIAM Journal on Financial Mathematics, Vol: 8, Pages: 709737, ISSN: 1945497XWe 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 atthemoney 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 nomassatzero case, showing how one can{theoretically{distinguish between mass at the origin and a heavylefttailed distribution. We numericallytest our modelfree results in stochastic models with absorption at the boundary, such as the CEV process,and in jumptodefault models. Note that while Lee's moment formula [25] tells that implied variance is atmost asymptotically linear in logstrike, other celebrated results for exact smile asymptotics such as [3,17]do not apply in this setting{essentially due to the breakdown of PutCall duality.

Journal articleCass T, Ogrodnik M, 2017,
Tail estimates for Markovian rough paths
, Annals of Probability, Vol: 45, Pages: 24772504, ISSN: 00911798The accumulated local pvariation functional [Ann. Probab. 41 (213) 3026–3050] arises naturally in the theory of rough paths in estimates both for solutions to rough differential equations (RDEs), and for the higherorder terms of the signature (or Lyons lift). In stochastic examples, it has been observed that the tails of the accumulated local pvariation functional typically decay much faster than the tails of classical pvariation. This observation has been decisive, for example, for problems involving Malliavin calculus for Gaussian rough paths [Ann. Probab. 43 (2015) 188–239].All of the examples treated so far have been in this Gaussian setting that contains a great deal of additional structure. In this paper, we work in the context of Markov processes on a locally compact Polish space E, which are associated to a class of Dirichlet forms. In this general framework, we first prove a betterthanexponential tail estimate for the accumulated local pvariation functional derived from the intrinsic metric of this Dirichlet form. By then specialising to a class of Dirichlet forms on the step ⌊p⌋ free nilpotent group, which are subelliptic in the sense of Fefferman–Phong, we derive a better than exponential tail estimate for a class of Markovian rough paths. This class includes the examples studied in [Probab. Theory Related Fields 142 (2008) 475–523]. We comment on the significance of these estimates to recent papers, including the results of Ni Hao [Personal communication (2014)] and Chevyrev and Lyons [Ann. Probab. To appear].

Journal articlePakkanen MS, Sottinen T, Yazigi A, 2017,
On the conditional small ball property of multivariate Lévydriven moving average processes
, Stochastic Processes and their Applications, Vol: 127, Pages: 749782, ISSN: 03044149We study whether a multivariate Lévydriven moving average process can shadow arbitrarily closely any continuous path, starting from the present value of the process, with positive conditional probability, which we call the conditional small ball property. Our main results establish the conditional small ball property for Lévydriven moving average processes under natural nondegeneracy conditions on the kernel function of the process and on the driving Lévy process. We discuss in depth how to verify these conditions in practice. As concrete examples, to which our results apply, we consider fractional Lévy processes and multivariate Lévydriven Ornstein–Uhlenbeck processes.

Journal articleChassagneux JFC, Jacquier A, Mihyalov IM, 2016,
An explicit Euler scheme with strong rate of convergence for financial SDEs with nonLipschitz coefficients
, SIAM Journal on Financial Mathematics, Vol: 7, Pages: 9931021, ISSN: 1945497XWe consider the approximation of onedimensional stochastic differential equations(SDEs) with nonLipschitz drift or diffusion coefficients. We present a modified explicit EulerMaruyama 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 CoxIngersollRoss (CIR), the3/2 and the AitSahalia models, as well as a family of meanreverting processeswith locally smooth coefficients. We numerically illustrate the strong convergenceof the scheme and demonstrate its efficiency in a multilevel Monte Carlo setting.

Journal articleBingham NH, Gashi B, 2016,
Voronoi means, moving averages, and power series
, Journal of Mathematical Analysis and Applications, Vol: 449, Pages: 682696, ISSN: 10960813We introduce a nonregular generalisation of the Nörlund mean, and show its equivalence with a certain moving average. The Abelian and Tauberian theorems establish relations with convergent sequences and certain power series. A strong law of large numbers is also proved.

Journal articleCont R, Ananova A, 2016,
Pathwise integration with respect to paths of finite quadratic variation
, Journal de Mathematiques Pures et Appliquees, Vol: 107, Pages: 737757, ISSN: 00217824We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of nonanticipative Riemann sums for gradienttype integrands.We show that the integral satisfies a pathwise isometry property, analogous to the wellknown Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition for regular functionals of an irregular path with nonvanishing quadratic variation, as a unique sum of a pathwise integral and a component with zero quadratic variation.

Journal articleGulisashvili 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: 113, ISSN: 1083589XStarting from the hyperbolic Brownian motion as a timechanged Brownian motion, we explore a set of probabilistic models–related to the SABR model in mathematical finance–which can be obtained by geometrypreserving 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 timechange methods.

Journal articleCass T, Driver BK, Lim N, et al., 2016,
On the integration of weakly geometric rough paths
, Journal of the Mathematical Society of Japan, Vol: 68, Pages: 15051524, ISSN: 00255645We close a gap in the theory of integration for weakly geometric rough paths in the infinitedimensional setting. We show that theintegral of a weakly geometric rough path against a sufficiently regular one form is, once again, a weakly geometric rough path.

Journal articleLukkarinen J, Pakkanen MS, 2016,
Arbitrage without borrowing or short selling?
, Mathematics and Financial Economics, Vol: 11, Pages: 263274, ISSN: 18629679We show that a trader, who starts with no initial wealth and is not allowedto borrow money or short sell assets, is theoretically able to attain positivewealth by continuous trading, provided that she has perfect foresight of future asset prices, given by a continuous semimartingale. Such an arbitrage strategy can be constructed as a process of finite variation that satisfies a seemingly innocuous selffinancing condition, formulated using a pathwiseRiemannStieltjes integral. Our result exemplifies the potential intricacies offormulating economically meaningful selffinancing conditions in continuoustime, when one leaves the conventional arbitragefree framework.

Journal articleGuo GG, Jacquier A, Martini CM, et al., 2016,
Generalized ArbitrageFree SVI Volatility Surfaces
, SIAM Journal on Financial Mathematics, Vol: 7, Pages: 619641, ISSN: 1945497XIn this paper we propose a generalization of the recent work by Gatheral and Jacquier [J. Gatheral and A. Jacquier, Quant. Finance, 14 (2014), pp. 5971] on explicit arbitragefree 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, ArbitrageFree 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/roper9.pdf]. We further exhibit an arbitragefree volatility surface different from Gatheral's SVI parameterization.

Journal articlePakkanen MS, Réveillac A, 2016,
Functional limit theorems for generalized variations of the fractional Brownian sheet
, Bernoulli, Vol: 22, Pages: 16711708, ISSN: 13507265We prove functional central and noncentral limit theorems for generalizedvariations of the anisotropic dparameter fractional Brownian sheet (fBs) forany natural number d. Whether the central or the noncentral limit theoremapplies depends on the Hermite rank of the variation functional and on thesmallest component of the Hurst parameter vector of the fBs. The limitingprocess in the former result is another fBs, independent of the original fBs,whereas the limit given by the latter result is an Hermite sheet, which isdriven by the same white noise as the original fBs. As an application, wederive functional limit theorems for power variations of the fBs and discusswhat is a proper way to interpolate them to ensure functional convergence.

Journal articleCont R, Kukanov A, 2016,
Optimal order placement in limit order markets
, Quantitative Finance, Vol: 17, Pages: 2139, ISSN: 14697696To execute a trade, participants in electronic equity markets may choose to submit limit orders or market orders across various exchanges where a stock is traded. This decision is influenced by the characteristics of the order flow and queue sizes in each limit order book, as well as the structure oftransaction fees and rebates across exchanges. We propose a quantitativeframework for studying this order placement problem by formulating it as a convex optimization problem. This formulation allows to study how the interplay between the state of order books, the fee structure, order flow properties and preferences of a trader determine the optimal placement decision. In the case of a single exchange, we derive an explicit solution for the optimal split between limit and market orders. For the general problem of order placement across multiple exchanges, we propose a stochastic algorithm for computing the optimal policy and study the sensitivity of the solution to various parameters using a numerical implementation of the algorithm.

Journal articleCont R, Wagalath L, 2016,
Risk management for whales
, Risk London Risk Magazine Limited, ISSN: 09528776We propose framework for modeling portfolio risk which integrates market risk with liquidation costs which may arise in stress scenarios. Our model provides a systematic method for computing liquidationadjusted risk measures for a portfolio. Calculation of Liquidationadjusted VaR (LVaR) for sample portfolios reveals a substantial impact of liquidation costs on portfolio risk for portfolios with large concentrated positions.

Journal articleDe Marco S, Jacquier A, Roome P, 2016,
Two examples of non strictly convex large deviations
, Electronic Communications in Probability, Vol: 21, Pages: 112, ISSN: 1083589XWe present two examples of a large deviations principle where the rate function is not strictly convex. This is motivated by a model used in mathematical finance (the Heston model), and adds a new item to the zoology of non strictly convex large deviations.

Journal articleAmini H, Cont R, Minca A, 2016,
Resilience to Contagion in Financial Networks
, Mathematical Finance, Vol: 26, Pages: 329365, ISSN: 09601627Propagation of balancesheet or cashflow insolvency across financialinstitutions may be modeled as a cascade process on a network representingtheir mutual exposures. We derive rigorous asymptotic results for the magnitudeof contagion in a large financial network and give an analytical expression forthe asymptotic fraction of defaults, in terms of network characteristics. Ourresults extend previous studies on contagion in random graphs to inhomogeneousdirected graphs with a given degree sequence and arbitrary distribution ofweights. We introduce a criterion for the resilience of a large financialnetwork to the insolvency of a small group of financial institutions andquantify how contagion amplifies small shocks to the network. Our resultsemphasize the role played by "contagious links" and show that institutionswhich contribute most to network instability in case of default have both largeconnectivity and a large fraction of contagious links. The asymptotic resultsshow good agreement with simulations for networks with realistic sizes.

Journal articleJacquier A, roome PR, 2016,
Largematurity regimes of the Heston forward smile
, Stochastic Processes and Their Applications, Vol: 126, Pages: 10871123, ISSN: 03044149We provide a full characterisation of the largematurity 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 higherorder terms highly depends on the parameters, and diff erent powers of the maturity come into play. As a byproduct of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVItype formulae. The proofs are based on extensions and re finements of sharp large deviations theory, in particular in cases where standard convexity arguments fail.

Journal articleCont R, Wagalath L, 2016,
INSTITUTIONAL INVESTORS AND THE DEPENDENCESTRUCTURE OF ASSET RETURNS
, International Journal of Theoretical & Applied Finance, Vol: 19, ISSN: 17936322We propose a model of a financial market with multiple assets that takes into accountthe impact of a large institutional investor rebalancing its positions so as to maintaina fixed allocation in each asset. We show that feedback effects can lead to significantexcess realized correlation between asset returns and modify the principal componentstructure of the (realized) correlation matrix of returns. Our study naturally links, ina quantitative manner, the properties of the realized correlation matrix — correlationbetween assets, eigenvectors and eigenvalues — to the sizes and trading volumes oflarge institutional investors. In particular, we show that even starting with uncorrelated“fundamentals”, fund rebalancing endogenously generates a correlation matrix of returnswith a first eigenvector with positive components, which can be associated to the market,as observed empirically. Finally, we show that feedback effects flatten the differencesbetween the expected returns of assets and tend to align them with the returns of theinstitutional investor’s portfolio, making this benchmark fund more difficult to beat, notbecause of its strategy but precisely because of its size and market impact.

BookCont R, Bally V, Caramellino L, 2016,
Stochastic Integration by Parts and Functional Itô Calculus
, Publisher: Birkhäuser, ISBN: 9783319271286This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012).The notes of the course by Vlad Bally, coauthored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavin's work on abstract Wiener spaces. The results are applied to prove absolute continuity and regularity results of the density for a broad class of random processes.Rama Cont's notes provide an introduction to the Functional Itô Calculus, a nonanticipative functional calculus that extends the classical Itô calculus to pathdependent functionals of stochastic processes. This calculus leads to a new class of pathdependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forwardbackward stochastic differential equations.This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to practitioners in mathematical finance.

Journal articleBingham NH, Ostaszewski AJ, 2015,
Beurling moving averages and approximate homomorphisms
, Indagationes Mathematicae, Vol: 27, Pages: 601633, ISSN: 00193577 
Journal articleCass T, Driver BK, Litterer C, 2015,
Constrained Rough Paths
, Proceedings of the London Mathematical Society, Vol: 111, Pages: 14711518, ISSN: 1460244XWe introduce a notion of rough paths on embedded submanifolds and demonstratethat this class of rough paths is natural. On the way we develop a notion ofrough integration and an efficient and intrinsic theory of rough differentialequations (RDEs) on manifolds. The theory of RDEs is then used to constructparallel translation along manifold valued rough paths. Finally, this frameworkis used to show there is a one to one correspondence between rough paths on addimensional manifold and rough paths on ddimensional Euclidean space. Thislast result is a rough path analogue of Cartan's development map and itsstochastic version which was developed by Eeels and Elworthy and Malliavin.

Journal articleJacquier A, Haba FH, 2015,
Asymptotic arbitrage in the Heston model
, International Journal of Theoretical and Applied Finance, Vol: 18, ISSN: 02190249In this paper, we introduce a new form of asymptotic arbitrage, which we call a partial asymptotic arbitrage, halfway between those of Follmer & Schachermayer (2007) and Kabanov & Kramkov (1998). In the context of the Heston model, we establish a precise link between the set of equivalent martingale measures, the ergodicity of the underlying variance process and this partial asymptotic arbitrage. In contrast to Follmer & Schachermayer (2007), our result does not assume a suitable condition on the stock price process to allow for (partial) asymptotic arbitrage.

Journal articleCont R, LU Y, 2015,
Weak approximation of martingale representations
, Stochastic Processes and Their Applications, Vol: 126, Pages: 857882, ISSN: 03044149We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are obtained by computing a directional derivative of the weak Euler scheme and yield a consistent estimator for the integrand in the martingale representation formula for any squareintegrable functional of the solution of an SDE with pathdependent coefficients. Explicit convergence rates are derived for functionals which are Lipschitzcontinuous in the supremum norm. Our results require neither the Markov property, nor any differentiability conditions on the functional or the coefficients of the stochastic differential equations involved.

Journal articleCont R, 2015,
The end of the waterfall: Default resources of central counterparties
, Journal of Risk Management in Financial Institutions, Vol: 8, Pages: 365389, ISSN: 17528887Central counterparties (CCPs) have become pillars of the new global financial architecture following the financial crisis of 2008. The key role of CCPs in mitigating counterparty risk and contagion has in turn cast them as systemically important financial institutions whoseeventual failure may lead to potentially serious consequences for financial stability, andprompted discussions on CCP risk management standards and safeguards for recovery andresolutions of CCPs in case of failure. We contribute to the debate on CCP default resourcesby focusing on the incentives generated by the CCP loss allocation rules for the CCP and itsmembers and discussing how the design of loss allocation rules may be used to align theseincentives in favor of outcomes which benefit financial stability. After reviewing theingredients of the CCP loss waterfall and various proposals for loss recovery provisions forCCPs, we examine the risk management incentives created by different ingredients in theloss waterfall and discuss possible approaches for validating the design of the waterfall.We emphasize the importance of CCP stress tests and argue that such stress tests need toaccount for the interconnectedness of CCPs through common members and crossmarginagreements. A key proposal is that capital charges on assets held against CCP Default Fundsshould depend on the quality of the risk management of the CCP, as assessed throughindependent stress tests.

Journal articleJacquier A, Lorig M, 2015,
From characteristic functions to implied volatility expansions
, Advances in Applied Probability, Vol: 47, Pages: 837857, ISSN: 14756064For any strictly positive martingale S with an analytically tractable 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 log(K/S0). We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three wellknown martingale models: one finite activity exponential Levy model (Merton), one infinite activity exponential Levy model (Variance Gamma), and one stochastic volatility model (Heston). We show how this technique can be extended to compute approximate forward implied volatilities and we implement this extension in the Heston setting. Finally, we illustrate how our expansion can be used to perform a modelfree calibration of the empirically observed implied volatility surface.

Journal articleCont R, Bentata A, 2015,
Forward equations for option prices in semimartingale models
, Finance and Stochastics, Pages: 617651, ISSN: 14321122We derive a forward partial integrodifferential equation for prices of calloptions in a model where the dynamics of the underlying asset under the pricing measure is described by a possibly discontinuous semimartingale. A uniquenesstheorem is given for the solutions of this equation. This result generalizesDupire's forward equation to a large class of nonMarkovian models with jumps.
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