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  • Book chapter
    Cass T, Litterer C, Lyons T, 2012,

    Rough Paths on Manifolds

    , New trends in stochastic analysis and related topics, 33–88,, Editors: Zhao, Truman, Publisher: World Scientific Publishing Company, Pages: 33-88, ISBN: 9789814360913

    The volume is dedicated to Professor David Elworthy to celebrate his fundamental contribution and exceptional influence on stochastic analysis and related fields.

  • Journal article
    Forde M, Lee R, 2012,

    The small-time smile and term structure of implied volatility under the Heston model

    , SIAM Journal of Financial Mathematics

    We characterise the asymptotic smile and term structure of implied volatility in the Heston modelat small maturities. Using saddlepoint methods we derive a small-maturity expansion formula for call optionprices, which we then transform into a closed-form expansion (including the leading-order and correction terms)for implied volatility. This refined expansion reveals the relationship between the small-expiry smile and allHeston parameters (including the pair in the volatility drift coefficient), sharpening the leading-order resultof [Forde, Jacquier, `Small-time asymptotics for implied volatility under the Heston model', IJTAF, 12(6): 861-876, 2009] which found the relationship between the zero-expiry smile and the diffusion coefficients.

  • Journal article
    Jacquier A, Keller-Ressel M, Mijatovic A, 2012,

    Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models

    , Stochastics-An International Journal of Probability and Stochastic Processes

    Let $\sigma_t(x)$ denote the implied volatility at maturity $t$ for a strike$K=S_0 e^{xt}$, where $x\in\bbR$ and $S_0$ is the current value of theunderlying. We show that $\sigma_t(x)$ has a uniform (in $x$) limit as maturity$t$ tends to infinity, given by the formula$\sigma_\infty(x)=\sqrt{2}(h^*(x)^{1/2}+(h^*(x)-x)^{1/2})$, for $x$ in somecompact neighbourhood of zero in the class of affine stochastic volatilitymodels. The function $h^*$ is the convex dual of the limiting cumulantgenerating function $h$ of the scaled log-spot process. We express $h$ in termsof the functional characteristics of the underlying model. The proof of thelimiting formula rests on the large deviation behaviour of the scaled log-spotprocess as time tends to infinity. We apply our results to obtain the limitingsmile for several classes of stochastic volatility models with jumps used inapplications (e.g. Heston with state-independent jumps, Bates withstate-dependent jumps and Barndorff-Nielsen-Shephard model).

  • Journal article
    Forde M, Jacquier A, Mijatovic A, 2011,

    A note on essential smoothness in the Heston model

    , FINANCE AND STOCHASTICS, Vol: 15, Pages: 781-784, ISSN: 0949-2984
  • Journal article
    Cass T, Litterer C, 2011,

    On the error estimate for cubature on Wiener space

  • Journal article
    Cont R, Larrard AD, 2011,

    Price dynamics in a Markovian limit order market

    , SIAM Journal on Financial Mathematics, Vol: 4, Pages: 1-25, ISSN: 1945-497X

    We propose and study a simple stochastic model for the dynamics of a limitorder book, in which arrivals of market order, limit orders and ordercancellations are described in terms of a Markovian queueing system. Throughits analytical tractability, the model allows to obtain analytical expressionsfor various quantities of interest such as the distribution of the durationbetween price changes, the distribution and autocorrelation of price changes,and the probability of an upward move in the price, {\it conditional} on thestate of the order book. We study the diffusion limit of the price process andexpress the volatility of price changes in terms of parameters describing thearrival rates of buy and sell orders and cancelations. These analytical resultsprovide some insight into the relation between order flow and price dynamics inorder-driven markets.

  • Journal article
    Bingham NH, Ostaszewski AJ, 2011,

    Homotopy and the Kestelman-Borwein-Ditor Theorem

    , CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, Vol: 54, Pages: 12-20, ISSN: 0008-4395
  • Journal article
    Czichowsky C, Westray N, Zheng H, 2011,

    Convergence in the semimartingale topology and constrained portfolios

    , SEMINAIRE DE PROBABILITES XLIII, Pages: 395-412, ISSN: 0075-8434
  • Journal article
    Bingham NH, Ostaszewski AJ, 2011,

    Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski

    , MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, Vol: 150, Pages: 1-22, ISSN: 0305-0041
  • Journal article
    Lin J, Liang G, Wu S, Zheng Het al., 2011,

    The valuation of the basket CDSs in the primary-subsidiary model

    , ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH, Vol: 28, Pages: 213-238, ISSN: 0217-5959
  • Journal article
    Cass T, Friz P, 2011,

    Malliavin calculus and rough paths

    , Bulletin des Sciences Mathematiques, Vol: 6-7, Pages: 542-556
  • Journal article
    Jacquier A, Forde M, 2011,

    Small-time asymptotics for an uncorrelated local-stochastic volatility model

    , Applied Mathematical Finance

    We refine the work of Henry-Labordere, Lewis and Paulot on the small-time behaviour of a local-stochastic volatility model with zero correlation at leading order. We do this using the Freidlin-Wentzell theory of large deviations for SDEs, and then converting to a di®erential geometry problem of computing the shortest geodesic from a point to a vertical line on a Riemmanian manifold, whose metric is induced by the inverse of the diffsion coefficient. The solution to this variable endpoint problem is obtained using a transversality condition, where the geodesic is perpendicular to the vertical line under the aforementioned metric. We then establish the corresponding small-time asymptotic behaviour for call options using Holder's inequality, and the implied volatility. We also derive a series expansion for the implied volatility in the small-maturity limit, in powers of the log-moneyness, and we show how to calibrate such a model to the observed implied volatility smile in the small-maturity limit.

  • Journal article
    Forde M, Jacquier A, Mijatovic A, 2010,

    Asymptotic formulae for implied volatility in the Heston model

    , PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 466, Pages: 3593-3620, ISSN: 1364-5021
  • Journal article
    Xu G, Zheng H, 2010,

    Basket options valuation for a local volatility jump-diffusion model with the asymptotic expansion method

    , INSURANCE MATHEMATICS & ECONOMICS, Vol: 47, Pages: 415-422, ISSN: 0167-6687
  • Journal article
    Bingham NH, Ostaszewski AJ, 2010,

    Regular variation without limits

    , JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, Vol: 370, Pages: 322-338, ISSN: 0022-247X
  • Journal article
    Bingham NH, Ostaszewski AJ, 2010,

    Topological regular variation: III. Regular variation

    , TOPOLOGY AND ITS APPLICATIONS, Vol: 157, Pages: 2024-2037, ISSN: 0166-8641
  • Journal article
    Bingham NH, Ostaszewski AJ, 2010,

    Topological regular variation: II. The fundamental theorems

    , TOPOLOGY AND ITS APPLICATIONS, Vol: 157, Pages: 2014-2023, ISSN: 0166-8641
  • Journal article
    Bingham NH, Ostaszewski AJ, 2010,

    Topological regular variation: I. Slow variation

    , TOPOLOGY AND ITS APPLICATIONS, Vol: 157, Pages: 1999-2013, ISSN: 0166-8641
  • Journal article
    Bingham NH, Fry JM, Kiesel R, 2010,

    Multivariate elliptic processes

    , STATISTICA NEERLANDICA, Vol: 64, Pages: 352-366, ISSN: 0039-0402
  • Book chapter
    Bingham N, 2010,

    Kingman, category and combinatorics

    , Probability and Mathematical Genetics, Editors: Bingham, Goldie, Publisher: Cambridge University Press, Pages: 135-168, ISBN: 9780521145770

    Focussing on the work of Sir John Kingman, one of the world's leading researchers in probability and mathematical genetics, this book touches on the important areas of these subjects in the last 50 years. Leading authorities give a unique insight into a wide range of currently topical problems. Papers in probability concentrate on combinatorial and structural aspects, in particular exchangeability and regeneration. The Kingman coalescent links probability with mathematical genetics and is fundamental to the study of the latter. This has implications across the whole of genomic modelling including the Human Genome Project. Other papers in mathematical population genetics range from statistical aspects including heterogeneous clustering, to the assessment of molecular variability in cancer genomes. Further papers in statistics are concerned with empirical deconvolution, perfect simulation, and wavelets. This book will be warmly received by established experts as well as their students and others interested in the content.

  • Journal article
    Cass T, Friz P, 2010,

    Densities for rough differential equations under Hörmander’s condition

    , Annals of Mathematics, Vol: 171, Pages: 2115-2141, ISSN: 0003-486X
  • Journal article
    Bingham NH, Ostaszewski AJ, 2010,

    AUTOMATIC CONTINUITY VIA ANALYTIC THINNING

    , PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 138, Pages: 907-919, ISSN: 0002-9939
  • Journal article
    Gatheral J, Jacquier A, 2010,

    Convergence of Heston to SVI

    , Quantitative Finance, Vol: 8, Pages: 1129-1132

    In this short note, we prove by an appropriate change of variables that theSVI implied volatility parameterization presented in Gatheral's book and thelarge-time asymptotic of the Heston implied volatility agree algebraically,thus confirming a conjecture from Gatheral as well as providing a simplerexpression for the asymptotic implied volatility in the Heston model. We showhow this result can help in interpreting SVI parameters.

  • Journal article
    Bingham NH, Ostaszewski AJ, 2010,

    BEYOND LEBESGUE AND BAIRE II: BITOPOLOGY AND MEASURE-CATEGORY DUALITY

    , COLLOQUIUM MATHEMATICUM, Vol: 121, Pages: 225-238, ISSN: 0010-1354
  • Journal article
    Bingham NH, Ostaszewski AJ, 2010,

    Normed versus topological groups: Dichotomy and duality

    , DISSERTATIONES MATHEMATICAE, Pages: 5-+, ISSN: 0012-3862
  • Journal article
    Jacquier A, Forde M, 2010,

    Robust approximations for pricing Asian options and volatility swaps under stochastic volatility

    , Applied Mathematical Finance, Vol: 17, Pages: 241-259

    We show that if the discounted Stock price process is a continuous martingale, then there is a simple relationship linking the variance of the terminal Stock price and the variance of its arithmetic average. We use this to establish a model-independent upper bound for the price of a continuously sampled fixed-strike arithmetic Asian call option, in the presence of non-zero time-dependent interest rates (Theorem 1.2). We also propose a model-independent lognormal moment-matching procedure for approximating the price of an Asian call, and we show how to apply these approximations under the Black–Scholes and Heston models (subsection 1.3). We then apply a similar analysis to a time-dependent Heston stochastic volatility model, and we show how to construct a time-dependent mean reversion and volatility-of-variance function, so as to be consistent with the observed variance swap curve and a pre-specified term structure for the variance of the integrated variance (Theorem 2.1). We characterize the small-time asymptotics of the first and second moments of the integrated variance (Proposition 2.2) and derive an approximation for the price of a volatility swap under the time-dependent Heston model (Equation (52)), using the Brockhaus–Long approximation (Brockhaus, and Long, 2000). We also outline a bootstrapping procedure for calibrating a piecewise-linear mean reversion level and volatility-of-volatility function (Subsection 2.3.2).

  • Book chapter
    Jacquier A, Martini C, 2010,

    Uncertain Volatility Model

    , Encyclopedia of Quantitative Finance, Editors: Cont, ISBN: 9780470057568

    Mille bravos!" —Dr Bruno Dupire (Bloomberg L.P.) The Encyclopedia of Quantitative Finance is a major reference work designed to provide a comprehensive coverage of essential topics related to the quantitative modelling of financial ...

  • Book chapter
    Cass T, Qian Z, Tudor J, 2010,

    Non-Linear Evolution Equations Driven by Rough Paths

  • Journal article
    Xu G, Zheng H, 2009,

    Approximate basket options valuation for a jump-diffusion model

    , INSURANCE MATHEMATICS & ECONOMICS, Vol: 45, Pages: 188-194, ISSN: 0167-6687
  • Journal article
    Zheng H, Jiang L, 2009,

    Basket CDS pricing with interacting intensities

    , FINANCE AND STOCHASTICS, Vol: 13, Pages: 445-469, ISSN: 0949-2984

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