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Effective asymptotic analysis for finance
... Let us first study the action of A n (C) on Schwartz functions, which is induced by the original Heisenberg action. Thus we are in fact confronted with two actions (41) A n (C) × S(R n ) → S(R n ) and H( ) × S(R n ) → S(R n ). ...
... It is often expedient, though, to work with differential subfields of R ∞ , known as Hardy fields [10, §1]: They come with a canonical total order, so are ordered fields; moreover, their germs all have definite limits on the extended real line R∪{±∞}. For example, the Hardy field of logarithmic-exponential functions [10, §1], [41,Ex. 1] is certainly large enough for our modest purposes. ...
The notion of Fourier transformation is described from an algebraic perspective that lends itself to applications in Symbolic Computation. We build the algebraic structures on the basis of a given Heisenberg group (in the general sense of nilquadratic groups enjoying a splitting property); this includes in particular the whole gamut of Pontryagin duality. The free objects in the corresponding categories are determined, and various examples are given. As a first step towards Symbolic Computation, we study two constructive examples in some detail---the Gaussians (with and without polynomial factors) and the hyperbolic secant algebra.
We add some rigour to the work of Henry-Labordère (200916.
Henry-Labordère , P. 2009. Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing, New York, London: Chapman & Hall. View all references; Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing (London and New York: Chapman & Hall)), Lewis (200721.
Lewis, A. (2007) Geometries and Smile Asymptotics for a Class of Stochastic Volatility Models. http://www.optioncity.net (http://www.optioncity.net) (Accessed: 28 May 2011). View all references; Geometries and Smile Asymptotics for a Class of Stochastic Volatility Models. Available at http://www.optioncity.net (accessed 28 May 2011)) and Paulot (200924.
Paulot, L. (2009) Asymptotic implied volatility at the second order with application to the SABR model, Working Paper papers.ssrn.com/sol3/papers.cfm?abstract_id=1413649 (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1413649) (Accessed: 11 June 2011). View all references; Asymptotic implied volatility at the second order with application to the SABR model, Working Paper, Available at papers.ssrn.com/sol3/papers.cfm?abstract_id=1413649 (accessed 11 June 2011)) on the small-time behaviour of a local-stochastic volatility model with zero correlation at leading order. We do this using the Freidlin—Wentzell (FW) theory of large deviations for stochastic differential equations (SDEs), and then converting to a differential 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 diffusion 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 Hölder's inequality, and the implied volatility (using a general result in Roper and Rutkowski (forthcoming27.
Roper , M. and
Rutkowski , M. forthcoming. A note on the behaviour of the Black–Scholes implied volatility close to expiry. International Journal of Theoretical and Applied Finance, View all references, A note on the behavior of the Black–Scholes implied volatility close to expiry, International Journal of Thoretical and Applied Finance). 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.
We derive a direct link between local and implied volatilities in the form of a quasilinear degenerate parabolic partial differential equation. Using this equation we establish closed-form asymptotic formulae for the implied volatility near expiry as well as for deep in- and out-of-the-money options. This in turn leads us to propose a new formulation near expiry of the calibration problem for the local volatility model, which we show to be well posed.
We give an expansion algorithm for germs of exp-log functions at infinity which is correct modulo Schanuels conjecture. We also show how the algorithm can be made generic. More precisely, we reduce the expansion algorithm for exp-log functions depending on parameters to the problem of deciding whether a given system of exp-log equations and inequalities in several variables admits a solution.
Changes in variance, or volatility, over time can be modeled using the approach based on autoregressive conditional heteroscedasticity. Another approach is to model variance as an unobserved stochastic process. Although it is not easy to obtain the exact likelihood function for such stochastic variance models, they tie in closely with developments in finance theory and have certain statistical attractions. This article sets up a multivariate model, discusses its statistical treatment, and shows how it can be modified to capture common movements in volatility in a very natural way. The model is then fitted to daily observations on exchange rates.
We give an algorithm to compute asymptotic expansions of exp-log functions. This algorithm automatically computes the necessary asymptotic scale and does not suffer from problems of indefinite cancellation. In particular, an asymptotic equivalent can always be computed for a given exp-log function. Introduction Exp-log functions are functions obtained from a variable x and the set of rational numbers Q by closure under field operations and the application of exp and log j Delta j. The set of exp-log functions was studied by G. H. Hardy [6], who showed---using different terminology---that their germs at infinity form a totally ordered field. This property makes exp-log functions extremely useful for doing asymptotics. The basic problem of effectively deciding the sign of an exp-log function at infinity (hence of computing limits) remained open for a long time. The advent of computer algebra has revived interest in this question. A first (theoretical) proof that this problem could be...
We present a brief overview of several approaches for calculating the local asymptotic expansion of the heat kernel for Laplace-type operators. The different methods developed in the papers of both authors some time ago are described in more detail.
We investigate PDEs of the form ut=12σ2(t,x)uxx−g(x)u which are associated with the calculation of expectations for a large class of local volatility models. We find nontrivial symmetry groups that can be used to obtain Fourier transforms of fundamental solutions of the PDE. We detail explicit computations in the separable volatility case when σ(t,x)=h(t)(α+βx+γx2), g=0, corresponding to the so called Quadratic Normal Volatility Model. We give financial applications and also show how symmetries can be used to compute first hitting distributions.
We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyze the impact of a mass at zero on at-The-money implied volatility and the overall level of the smile. We further show that the behavior at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difierence 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 numerically test our model-free results in stochastic models with absorption at the boundary, such as the constant elasticity of variance (CEV) process, and in jump-To-default models. Note that while Lee's moment formula [R. W. Lee, Math. Finance, 14 (2004), pp. 469-480] tells us that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as [S. Benaim and P. Friz, Math. Finance, 19 (2009), pp. 1-12] and [A. Gulisashvili,SIAM J. Financial Math., 1 (2010), pp. 609-641] do not apply in this setting|essentially due to the breakdown of put-call duality.
In this note, Black{Scholes implied volatility is expressed in terms of various optimization problems. From these representations, upper and lower bounds are derived which hold uniformly across mon- eyness and call price. Various symmetries of the Black{Scholes formula are exploited to derive new bounds from old. These bounds are used to reprove asymptotic formulas for implied volatility at extreme strikes and/or maturities.
The tail of risk neutral returns can be related explicitly with the wing behaviour of the Black-Scholes implied volatility smile. In situations where precise tail asymptotics are unknown but a moment generating function is available we establish, under easy-to-check Tauberian conditions, tail asymptotics on logarithmic scales. Such asymptotics are enough to make the tail-wing formula (see Benaim and Friz (2008)) work and so we obtain, under generic conditions, a limiting slope when plotting the square of the implied volatility against the log strike, improving a lim sup statement obtained earlier by Lee (2004). We apply these results to time-changed exponential Lévy models and examine several popular models in more detail, both analytically and numerically.
This book presents in depth asymptotic methods for solving parabolic partialdifferential equations at the level suitable for non-mathematicians The focus is on the stochastic description
Although this book is intended for advanced undergraduate or beginning graduate students, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in partial differential equations, mathematical physics, differential geometry, singular perturbations and mathematical finance
This book is about the automatic computation of asymptotic behaviour of functions of a real variable, x. This means that we shall wish to compute limits, and also to give some measure of how rapidly a given function approaches its limit Throughout, we shall assume that our interest is in the behaviour as x → ∞. This is for convenience only, since other cases can be considered by using a transformation of one of the forms x → ±(x – x 0)-1. These give separate treatments of the cases when x tends to x 0 from above and below. We shall want our methods to be at least potentially implementable, and we shall, from time to time, give some consideration to actual implementations. However this book is really about algorithms and the mathematical principles that underlie them, rather than details of implementation.
In a unified model-free framework that includes long-expiry, short-expiry, extreme-strike, and jointly-varying strike-expiry regimes, we find asymptotic implied volatility and implied variance formulas in terms of L, with rigorous error estimates of order 1/L to any given power, where L denotes the absolute log of an option price that approaches zero. Our results therefore sharpen, to arbitrarily-high order of accuracy, the model-free asymptotics of implied volatility in extreme regimes. We then apply these general formulas to particular examples: Levy and Heston.
Using an expansion of the transition density function of a one-dimensional time inhomogeneous diffusion, we obtain the first- and second-order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the first- and second-order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate.
We give an algorithm to compute an asymptotic expansion of multiseries type for the inverse of any given exp-log function. An example of the use of this algorithm to compute asymptotic expansions in combinatorics via the saddle-point method is then treated in detail.
. It would be desirable to have an algorithm to decide equality among the constants which commonly occur in scientific computing.
We do not yet know whether or not this is possible. It is known, however, that if the Schanuel conjecture is true, then equality
is Turing decidable among the closed form numbers, that is, the complex numbers built up from the rationals using field operations,
radicals, exponentials and logarithms. An algorithm based on the Schanuel conjecture is described in this article to decide
equality among these numbers, presented as expressions. The algebraic part of this algorithm is simpler than those which have
been given previously to solve related problems.
Exp-log functions are those obtained from the constant 1 and the variable X by means of arithmetic operations and the function symbols exp() and log∥. This paper gives an explicit algorithm for determining eventual dominance of these functions modulo an oracle for deciding zero equivalence of constant terms, This also provides another proof that the dominance problem for exp-log functions is Turing-reducible to the identity problem for constant terms.
First, we show that implied normal volatility is intimately linked with the
incomplete Gamma function. Then, we deduce an expansion on implied normal
volatility in terms of the time-value of a European call option. Then, we
formulate an equivalence between the implied normal volatility and the
lognormal implied volatility with any strike and any model. This generalizes a
known result for the SABR model. Finally, we adress the issue of the "breakeven
move" of a delta-hedged portfolio.
We examine the asymptotic behaviour of the call price surface and the associated Black-Scholes implied volatility surface in the small time to expiry limit under the condition of no arbitrage. In the final section, we examine a related question of existence of a market model with non-convergent implied volatility. We show that there exist arbitrage free markets in which implied volatility may fail to converge to any value, finite or infinite.
cap. 1. Dérivées -- 2. Primitives et intégrales -- 3. Fonctions élémentaires -- 4. Equations différentielles -- 5. Etude locale des fonctions -- 6. Developpements tayloriens généralises. Formule sommatoire d'Euler-Maclaurin. 7. La fonction gamma
Diss. no. 11432 techn. sc. SFIT Zürich. Literaturverz.
Consider options on a nonnegative underlying random variable with arbitrary distribution. In the absence of arbitrage, we show that at any maturity T, the large-strike tail of the Black-Scholes implied volatility skew is bounded by the square root of 2|x|/T, where x is log-moneyness. The smallest coefficient that can replace the 2 depends only on the number of finite moments in the underlying distribution. We prove the moment formula, which expresses explicitly this model-independent relationship. We prove also the reciprocal moment formula for the small-strike tail, and we exhibit the symmetry between the formulas. The moment formula, which evaluates readily in many cases of practical interest, has applications to skew extrapolation and model calibration.
We study the automatic computation of asymptotic expansions of functional inverses. Based on previous work on asymptotic expansions, we give an algorithm which computes Hardy-field solutions of equations f(y) = x, with f belonging to a large class of functions. D'eveloppements asymptotiques d'inverses fonctionnels R'esum'e Nous 'etudions le calcul automatique de d'eveloppements asymptotiques d'inverses fonctionnels. ` A partir de r'esultats ant'erieurs sur les d'eveloppements asymptotiques, nous donnons un algorithme qui calcule les solutions d"equations f(y) = x, pour des fonctions f appartenant `a une classe 'etendue, et pourvu que ces solutions appartiennent `a un corps de Hardy. To appear in Proceedings ISSAC'92 , P. Wang ed., ACM Press, 1992. Asymptotic Expansions of Functional Inverses Bruno Salvy Algorithms Project, INRIA Rocquencourt, 78153 Le Chesnay Cedex, France John Shackell University of Kent at Canterbury, Canterbury, Kent CT2 7NF, England Abstract We study the autom...
In this paper, we derive a general asymptotic implied volatility at the first-order for any stochastic volatility model using the heat kernel expansion on a Riemann manifold endowed with an Abelian connection. This formula is particularly useful for the calibration procedure. As an application, we obtain an asymptotic smile for a SABR model with a mean-reversion term, called lambda-SABR, corresponding in our geometric framework to the Poincar\'{e} hyperbolic plane. When the lambda-SABR model degenerates into the SABR-model, we show that our asymptotic implied volatility is a better approximation than the classical Hagan-al expression . Furthermore, in order to show the strength of this geometric framework, we give an exact solution of the SABR model with beta=0 or 1. In a next paper, we will show how our method can be applied in other contexts such as the derivation of an asymptotic implied volatility for a Libor market model with a stochastic volatility.
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The asymptotic expansion formula of implied volatility for dynamic sabr model and fx hybrid model
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O. Osajima. The asymptotic expansion formula of implied volatility for dynamic sabr model and fx hybrid
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