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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.,
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 (B) 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 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 articleBennedsen M, Lunde A, Pakkanen MS, 2017,
Hybrid scheme for Brownian semistationary processes
, Finance and Stochastics, Vol: 21, Pages: 931965, ISSN: 14321122We 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 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: 03044149© 2016 Elsevier B.V. We 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 nonLipschitz SDEs
, 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 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 articleDavis M, Lleo S, 2016,
A simple procedure to incorporate predictive models in a continuous time asset allocation
, Quantitative Finance Letters, Vol: 4, Pages: 4046, ISSN: 21649502Stochastic optimisation has found a fertile ground for applications in finance. One of the greatest challenges remains to incorporate a set of scenarios that accurately model the behaviour of financial markets, and in particular their behaviour during crashes and crises, without sacrificing the tractability of the optimal investment policy. This paper shows how to incorporate return predictions and crash predictions as views into continuous time asset allocation models.

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 articleDavis MHA, Lleo S, 2016,
A Simple Procedure for Combining Expert Opinion with Statistical Estimates to Achieve Superior Portfolio Performance
, JOURNAL OF PORTFOLIO MANAGEMENT, Vol: 42, Pages: 4958, ISSN: 00954918 
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 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.

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 articleDavis MHA, Andruszkiewicz G, Lleo S, 2016,
Risksensitive investment in a finitefactor model
, Stochastics, Vol: 89, Pages: 89114, ISSN: 00909491A new jump diffusion regimeswitching model is introduced, which allows for linking jumps in asset prices with regime changes. We prove the existence and uniqueness of the solution to the risksensitive asset management criterion maximization problem in this setting. We provide an ODE for the optimal value function, which may be efficiently solved numerically. Relevant probability measure changes are discussed in the appendix. The recently introduced approach of Klebaner and Liptser (2013) is used to prove the martingale property of the relevant density processes.

Journal articleDavis MHA, 2016,
Verification of internal risk measure estimates
, Statistics and Risk Modeling, Vol: 33, ISSN: 21931402This paper concerns sequential computation of risk measures for financial data and asks how, given a risk measurement procedure, we can tell whether the answers it produces are `correct'. We draw the distinction between `external' and `internal' risk measures and concentrate on the latter, where we observe data in real time, make predictions and observe outcomes. It is argued that evaluation of such procedures is best addressed from the point of view of probability forecasting or Dawid's theory of `prequential statistics' [J. Roy. Statist. Soc. A 147 (1984), 278–292]. We introduce a concept of `calibration' of a risk measure in a dynamic setting, following the precepts of Dawid's weak and strong prequential principles, and examine its application to quantile forecasting (VaR – value at risk) and to mean estimation (applicable to CVaR – expected shortfall). The relationship between these ideas and `elicitability' [J. Amer. Statist. Assoc. 106 (2011), 746–762] is examined. We show in particular that VaR has special properties not shared by any other risk measure. Turning to CVaR we argue that its main deficiency is the unquantifiable tail dependence of estimators. In a final section we show that a simple datadriven feedback algorithm can produce VaR estimates on financial data that easily pass both the consistency test and a further newlyintroduced statistical test for independence of a binary sequence.

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.

Book chapterDavis MHA, 2016,
Modelfree methods in valuation and hedging of derivative securities
, Handbook of PostCrisis Financial Modelling, Editors: Haven, Molyneux, Wilson, Fedotov, Duygun, Publisher: PalgraveMacMillan 
Book chapterDavis MHA, 2016,
A Beaufort scale of predictability
, The Fascination of Probability, Statistics and their Applications, Editors: Podolskij, Stelzer, Thorbjornsen, Veraart, Publisher: Springer, Pages: 419434 
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.
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