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  • Journal article
    Cass T, Lim N, 2019,

    A Stratonovich-Skorohod integral formula for Gaussian rough paths

    , Annals of Probability, Vol: 47, Pages: 1-60, ISSN: 0091-1798

    Given 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 closed-form 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 Stratonovich-to-Itô conversion formula, which we generalize to Gaussian rough paths with finite p-variation, 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 Riemann-sum approximants of the Skorohod integral converge in L2(Ω) by using a novel characterization of the Cameron–Martin norm in terms of higher-dimensional 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 article
    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.

  • Journal article
    Guennoun H, Jacquier A, Roome P, Shi Fet al.,

    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.

  • Journal article
    Davis 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: 1-21, ISSN: 0246-0203

    We 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\^o-Tanaka, 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 non-decreasing 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 article
    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.

  • Journal article
    Cass T, Ogrodnik M, 2017,

    Tail estimates for Markovian rough paths

    , Annals of Probability, Vol: 45, Pages: 2477-2504, ISSN: 0091-1798

    The accumulated local p-variation 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 higher-order terms of the signature (or Lyons lift). In stochastic examples, it has been observed that the tails of the accumulated local p-variation functional typically decay much faster than the tails of classical p-variation. 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 better-than-exponential tail estimate for the accumulated local p-variation 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 sub-elliptic 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 article
    Bennedsen M, Lunde A, Pakkanen MS, 2017,

    Hybrid scheme for Brownian semistationary processes

    , Finance and Stochastics, Vol: 21, Pages: 931-965, ISSN: 1432-1122

    We 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 Riemann-sum scheme, while having the same computationalcomplexity. We exemplify the use of the hybrid scheme by two numericalexperiments, where we examine the finite-sample 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 article
    Pakkanen MS, Sottinen T, Yazigi A, 2017,

    On the conditional small ball property of multivariate Lévy-driven moving average processes

    , Stochastic Processes and their Applications, Vol: 127, Pages: 749-782, ISSN: 0304-4149

    © 2016 Elsevier B.V. We study whether a multivariate Lévy-driven 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 Lévy-driven moving average processes under natural non-degeneracy 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 Lévy-driven Ornstein–Uhlenbeck processes.

  • Journal article
    Chassagneux JFC, Jacquier A, Mihyalov IM, 2016,

    An explicit Euler scheme with strong rate of convergence for non-Lipschitz SDEs

    , 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.

  • Journal article
    Bingham NH, Gashi B, 2016,

    Voronoi means, moving averages, and power series

    , Journal of Mathematical Analysis and Applications, Vol: 449, Pages: 682-696, ISSN: 1096-0813

    We introduce a non-regular 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 article
    Cont R, Ananova A, 2016,

    Pathwise integration with respect to paths of finite quadratic variation

    , Journal de Mathematiques Pures et Appliquees, Vol: 107, Pages: 737-757, ISSN: 0021-7824

    We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands.We show that the integral satisfies a pathwise isometry property, analogous to the well-known 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 non-vanishing quadratic variation, as a unique sum of a pathwise integral and a component with zero quadratic variation.

  • Journal article
    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.

  • Journal article
    Cass T, Driver BK, Lim N, Litterer Cet al., 2016,

    On the integration of weakly geometric rough paths

    , Journal of the Mathematical Society of Japan, Vol: 68, Pages: 1505-1524, ISSN: 0025-5645

    We close a gap in the theory of integration for weakly ge-ometric rough paths in the infinite-dimensional 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 article
    Lukkarinen J, Pakkanen MS, 2016,

    Arbitrage without borrowing or short selling?

    , Mathematics and Financial Economics, Vol: 11, Pages: 263-274, ISSN: 1862-9679

    We 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 self-financing condition, formulated using a pathwiseRiemann-Stieltjes integral. Our result exemplifies the potential intricacies offormulating economically meaningful self-financing conditions in continuoustime, when one leaves the conventional arbitrage-free framework.

  • Journal article
    Guo GG, Jacquier A, Martini CM, Neufcourt LNet 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.

  • Journal article
    Cont R, Kukanov A, 2016,

    Optimal order placement in limit order markets

    , Quantitative Finance, Vol: 17, Pages: 21-39, ISSN: 1469-7696

    To 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 article
    Davis M, Lleo S, 2016,

    A simple procedure to incorporate predictive models in a continuous time asset allocation

    , Quantitative Finance Letters, Vol: 4, Pages: 40-46, ISSN: 2164-9502

    Stochastic 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 article
    Cont R, Wagalath L, 2016,

    Risk management for whales

    , Risk -London- Risk Magazine Limited-, ISSN: 0952-8776

    We 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 liquidation-adjusted risk measures for a portfolio. Calculation of Liquidation-adjusted VaR (LVaR) for sample portfolios reveals a substantial impact of liquidation costs on portfolio risk for portfolios with large concentrated positions.

  • Journal article
    Davis 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: 49-58, ISSN: 0095-4918
  • Journal article
    De Marco S, Jacquier A, Roome P, 2016,

    Two examples of non strictly convex large deviations

    , Electronic Communications in Probability, Vol: 21, Pages: 1-12, ISSN: 1083-589X

    We 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 article
    Amini H, Cont R, Minca A, 2016,

    Resilience to Contagion in Financial Networks

    , Mathematical Finance, Vol: 26, Pages: 329-365, ISSN: 0960-1627

    Propagation of balance-sheet or cash-flow 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 article
    Cont R, Wagalath L, 2016,

    INSTITUTIONAL INVESTORS AND THE DEPENDENCESTRUCTURE OF ASSET RETURNS

    , International Journal of Theoretical & Applied Finance, Vol: 19, ISSN: 1793-6322

    We 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 article
    Pakkanen MS, Réveillac A, 2016,

    Functional limit theorems for generalized variations of the fractional Brownian sheet

    , Bernoulli, Vol: 22, Pages: 1671-1708, ISSN: 1350-7265

    We prove functional central and non-central limit theorems for generalizedvariations of the anisotropic d-parameter fractional Brownian sheet (fBs) forany natural number d. Whether the central or the non-central 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 article
    Davis MHA, Andruszkiewicz G, Lleo S, 2016,

    Risk-sensitive investment in a finite-factor model

    , Stochastics, Vol: 89, Pages: 89-114, ISSN: 0090-9491

    A new jump diffusion regime-switching 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 risk-sensitive 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 article
    Davis MHA, 2016,

    Verification of internal risk measure estimates

    , Statistics and Risk Modeling, Vol: 33, ISSN: 2193-1402

    This 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 data-driven feedback algorithm can produce VaR estimates on financial data that easily pass both the consistency test and a further newly-introduced statistical test for independence of a binary sequence.

  • Book
    Cont R, Bally V, Caramellino L, 2016,

    Stochastic Integration by Parts and Functional Itô Calculus

    , Publisher: Birkhäuser, ISBN: 978-3-319-27128-6

    This 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, co-authored 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 non-anticipative functional calculus that extends the classical Itô calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward 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 chapter
    Davis MHA, 2016,

    Model-free methods in valuation and hedging of derivative securities

    , Handbook of Post-Crisis Financial Modelling, Editors: Haven, Molyneux, Wilson, Fedotov, Duygun, Publisher: Palgrave-MacMillan
  • Book chapter
    Davis MHA, 2016,

    A Beaufort scale of predictability

    , The Fascination of Probability, Statistics and their Applications, Editors: Podolskij, Stelzer, Thorbjornsen, Veraart, Publisher: Springer, Pages: 419-434
  • Journal article
    Bingham NH, Ostaszewski AJ, 2015,

    Beurling moving averages and approximate homomorphisms

    , Indagationes Mathematicae, Vol: 27, Pages: 601-633, ISSN: 0019-3577
  • Journal article
    Cass T, Driver BK, Litterer C, 2015,

    Constrained Rough Paths

    , Proceedings of the London Mathematical Society, Vol: 111, Pages: 1471-1518, ISSN: 1460-244X

    We 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 ad-dimensional manifold and rough paths on d-dimensional 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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