Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top

2015 | Book

Probability Distribution in the SABR Model of Stochastic Volatility Read chapter Read first chapter

Large Deviations and Asymptotic Methods in Finance

Editors: Peter K. Friz, Jim Gatheral, Archil Gulisashvili, Antoine Jacquier, Josef Teichmann

Publisher: Springer International Publishing

Book Series : Springer Proceedings in Mathematics & Statistics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Login to get access
insite
SEARCH

About this book

Topics covered in this volume (large deviations, differential geometry, asymptotic expansions, central limit theorems) give a full picture of the current advances in the application of asymptotic methods in mathematical finance, and thereby provide rigorous solutions to important mathematical and financial issues, such as implied volatility asymptotics, local volatility extrapolation, systemic risk and volatility estimation. This volume gathers together ground-breaking results in this field by some of its leading experts.

Over the past decade, asymptotic methods have played an increasingly important role in the study of the behaviour of (financial) models. These methods provide a useful alternative to numerical methods in settings where the latter may lose accuracy (in extremes such as small and large strikes, and small maturities), and lead to a clearer understanding of the behaviour of models, and of the influence of parameters on this behaviour.

Graduate students, researchers and practitioners will find this book very useful, and the diversity of topics will appeal to people from mathematical finance, probability theory and differential geometry.

Table of Contents

Frontmatter
Probability Distribution in the SABR Model of Stochastic Volatility
Abstract
We study the SABR model of stochastic volatility (Wilmott Mag, 2003 [10]). This model is essentially an extension of the local volatility model (Risk 7(1):18–20 [4], Risk 7(2):32–39, 1994 [6]), in which a suitable volatility parameter is assumed to be stochastic. The SABR model admits a large variety of shapes of volatility smiles, and it performs remarkably well in the swaptions and caps/floors markets. We refine the results of (Wilmott Mag, 2003 [10]) by constructing an accurate and efficient asymptotic form of the probability distribution of forwards. Furthermore, we discuss the impact of boundary conditions at zero forward on the volatility smile. Our analysis is based on a WKB type expansion for the heat kernel of a perturbed Laplace-Beltrami operator on a suitable hyperbolic Riemannian manifold.
Patrick Hagan, Andrew Lesniewski, Diana Woodward
Asymptotic Implied Volatility at the Second Order with Application to the SABR Model
Abstract
We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kernel expansion. Furthermore, the first correction in the heat kernel expansion gives the second order correction for implied volatility, which we also give exactly at all strikes. As an application, we compute this asymptotic expansion at order 2 for the SABR model and compare it to the original formula.
Louis Paulot
Unifying the BGM and SABR Models: A Short Ride in Hyperbolic Geometry
Abstract
In this paper, using a geometric method introduced in (Henry-Labordère Large Deviations and Asymptotic Methods in Finance (2015) [12]) and initiated by (Avellaneda et al. Risk Mag. (2002) [4]), we derive an asymptotic swaption implied volatility at the first-order for a general stochastic volatility Libor Market Model. This formula is useful to quickly calibrate a model to a full swaption matrix. We apply this formula to a specific model where the forward rates are assumed to follow a multi-dimensional CEV process correlated to a SABR process. For a caplet, this model degenerates to the classical SABR model and our asymptotic swaption implied volatility reduces naturally to the Hagan-al formula (Hagan et al. Willmott Mag. 88–108 (2002) [11]). The geometry underlying this model is the hyperbolic manifold \({\mathbb H}^{n+1}\) with n the number of Libor forward rates.
Pierre Henry-Labordère
Second Order Expansion for Implied Volatility in Two Factor Local Stochastic Volatility Models and Applications to the Dynamic $$\lambda $$ -Sabr Model
Abstract
Using an expansion of the transition density function of a two dimensional time inhomogeneous diffusion, we obtain the first and second order terms in the short time asymptotics of the local volatility function in a family of time inhomogeneous local-stochastic volatility models. With the local volatility function at our disposal, we show how recent results (Gatheral et al., Math. Financ. 22:591–620, 2012, [28]) for one dimensional diffusions can be applied to also determine expansions for call prices as well as for the implied volatility. The results are worked out in detail in the case of the dynamic Sabr model, thus generalizing earlier work by Hagan et al. (Wilmott Mag. 84–108, 2003, [31]), Hagan and Lesniewski (Springer Proceedings in Mathematics and Statistics, vol. 110, 2015, [32]) and by Henry-Labordère (Springer Proceedings in Mathematics and Statistics, vol. 110, 2015, Geometry, and Modeling in Finance. Chapman & Hall/CRC Financial Mathematics Series, 2008, [39, 40]).
Gérard Ben Arous, Peter Laurence
General Asymptotics of Wiener Functionals and Application to Implied Volatilities
Abstract
In the present paper, we give an asymptotic expansion of probability density for a component of general diffusion models. Our approach is based on infinite dimensional analysis on the Malliavin calculus and Kusuoka-Stroock’s asymptotic expansion theory for general Wiener functionals (Kusuoka and Stroock, J. Funct. Anal. 99:1–74, 1991 [12]). The initial term of the expansion is given by the geodesic distance and we calculate it by solving Hamilton’s equation. We apply our approach to obtain asymptotic expansion formulae for implied volatilities in general diffusion models, e.g. CEV and SABR model.
Yasufumi Osajima
Implied Volatility of Basket Options at Extreme Strikes
Abstract
In the paper, we characterize the asymptotic behavior of the implied volatility of a basket call option at large and small strikes in a variety of settings with increasing generality. First, we obtain an asymptotic formula with an error bound for the left wing of the implied volatility, under the assumption that the dynamics of asset prices are described by the multidimensional Black-Scholes model. Next, we find the leading term of asymptotics of the implied volatility in the case where the asset prices follow the multidimensional Black-Scholes model with time change by an independent increasing stochastic process. Finally, we deal with a general situation in which the dependence between the assets is described by a given copula function. In this setting, we obtain a model-free tail-wing formula that links the implied volatility to a special characteristic of the copula called the weak lower tail dependence function.
Archil Gulisashvili, Peter Tankov
Small-Time Asymptotics for the At-the-Money Implied Volatility in a Multi-dimensional Local Volatility Model
Abstract
We consider a basket or spread option based on a multi-dimensional local volatility model. Bayer and Laurence (Commun. Pure. Appl. Math., 67(10), 2014, [5]) derived highly accurate analytic formulas for prices and implied volatilities of such options when the options are not at the money. We now extend these results to the ATM case. Moreover, we also derive similar formulas for the local volatility of the basket.
Christian Bayer, Peter Laurence
A Remark on Gatheral’s ‘Most-Likely Path Approximation’ of Implied Volatility
Abstract
We give a new proof of the representation of implied volatility as a time-average of weighted expectations of local or stochastic volatility. With this proof we clarify the question of existence of ‘forward implied variance’ in the original derivation of Gatheral, who introduced this representation in his book ‘The Volatility Surface’.
Martin Keller-Ressel, Josef Teichmann
Implied Volatility from Local Volatility: A Path Integral Approach
Abstract
Assuming local volatility, we derive an exact Brownian bridge representation for the transition density; an exact expression for the transition density in terms of a path integral then follows. By Taylor-expanding around a certain path, we obtain a generalization of the heat kernel expansion of the density which coincides with the classical one in the time-homogeneous case, but is more accurate and natural in the time inhomogeneous case. As a further application of our path integral representation, we obtain an improved most-likely-path approximation for implied volatility in terms of local volatility.
Tai-Ho Wang, Jim Gatheral
Extrapolation Analytics for Dupire’s Local Volatility
Abstract
We consider wing asymptotics of local volatility surfaces. While our recent paper in the journal Risk (De Marco et al. Risk 2:82–87, 2013, [3]) discusses our approximation formula from a practical and numerical perspective, the present paper focuses on rigorous proofs of the approximations. We apply the saddle point method (Heston model) and Hankel contour integration (variance gamma model).
Peter Friz, Stefan Gerhold
The Gärtner-Ellis Theorem, Homogenization, and Affine Processes
Abstract
We obtain a first order extension of the large deviation estimates in the Gärtner-Ellis theorem. In addition, for a given family of measures, we find a special family of functions having a similar Laplace principle expansion up to order one to that of the original family of measures. The construction of the special family of functions mentioned above is based on heat kernel expansions. Some of the ideas employed in the paper come from the theory of affine stochastic processes. For instance, we provide an explicit expansion with respect to the homogenization parameter of the rescaled cumulant generating function in the case of a generic continuous affine process. We also compute the coefficients in the homogenization expansion for the Heston model that is one of the most popular stock price models with stochastic volatility.
Archil Gulisashvili, Josef Teichmann
Asymptotics for $$d$$ -Dimensional Lévy-Type Processes
Abstract
We consider a general \(d\)-dimensional Lévy-type process with killing. Combining the classical Dyson series approach with a novel polynomial expansion of the generator \(\mathcal {A}(t)\) of the Lévy-type process, we derive a family of asymptotic approximations for transition densities and European-style options prices. Examples of stochastic volatility models with jumps are provided in order to illustrate the numerical accuracy of our approach. The methods described in this paper extend the results from Corielli et al. (SIAM J Financ Math 1:833–867, 2010, [4]), Pagliarani and Pascucci (Int. J. Theor. Appl. Financ. 16(8):1–35, 2013, [20]) to Lorig et al. (Analytical expansions for parabolic equations, 2013, [13]) for Markov diffusions to Markov processes with jumps.
Matthew Lorig, Stefano Pagliarani, Andrea Pascucci
Asymptotic Expansion Approach in Finance
Abstract
This paper provides a survey on an asymptotic expansion approach to valuation and hedging problems in finance. The asymptotic expansion is a widely applicable methodology for analytical approximations of expectations of certain Wiener functionals. Hence not only academic researchers but also practitioners have been applying the scheme to a variety of problems in finance such as pricing and hedging derivatives under high-dimensional stochastic environments. The present note gives an overview of the approach.
Akihiko Takahashi
On Small Time Asymptotics for Rough Differential Equations Driven by Fractional Brownian Motions
Abstract
We survey existing results concerning the study in small times of the density of the solution of a rough differential equation driven by fractional Brownian motions. We also slightly improve existing results and discuss some possible applications to mathematical finance.
Fabrice Baudoin, Cheng Ouyang
On Singularities in the Heston Model
Abstract
In this note we provide characterization of the singularities of the Heston characteristic function. In particular, we show that all the singularities are pure imaginary.
Vladimir Lucic
On the Probability Density Function of Baskets
Abstract
The state price density of a basket, even under uncorrelated Black–Scholes dynamics, does not allow for a closed form density. (This may be rephrased as statement on the sum of lognormals and is especially annoying for such are used most frequently in Financial and Actuarial Mathematics.) In this note we discuss short time and small volatility expansions, respectively. The method works for general multi-factor models with correlations and leads to the analysis of a system of ordinary (Hamiltonian) differential equations. Surprisingly perhaps, even in two asset Black–Scholes situation (with its flat geometry), the expansion can degenerate at a critical (basket) strike level; a phenomena which seems to have gone unnoticed in the literature to date. Explicit computations relate this to a phase transition from a unique to more than one “most-likely” paths (along which the diffusion, if suitably conditioned, concentrates in the afore-mentioned regimes). This also provides a (quantifiable) understanding of how precisely a presently out-of-money basket option may still end up in-the-money.
Christian Bayer, Peter K. Friz, Peter Laurence
On Small-Noise Equations with Degenerate Limiting System Arising from Volatility Models
Abstract
The one-dimensional SDE with non Lipschitz diffusion coefficient
$$\begin{aligned} \textit{dX}_{t} = b(X_{t})\textit{dt} + \sigma X_{t}^{\gamma } \textit{dB}_{t}, \quad X_{0}=x, \quad \gamma <1 \end{aligned}$$
(1)
  is widely studied in mathematical finance. Several works have proposed asymptotic analysis of densities and implied volatilities in models involving instances of (1), based on a careful implementation of saddle-point methods and (essentially) the explicit knowledge of Fourier transforms. Recent research on tail asymptotics for heat kernels (Deuschel et al. Comm. in Pure and Applied Math., 67(1):40–82, 2014, [11]) suggests to work with the rescaled variable \( X^{\varepsilon }:=\varepsilon ^{1/(1-\gamma )} X\): while allowing to turn a space asymptotic problem into a small-\(\varepsilon \) problem, the process \(X^{\varepsilon }\) satisfies a SDE in Wentzell–Freidlin form (i.e. with driving noise \(\varepsilon \textit{dB}\)). We prove a pathwise large deviation principle for the process \(X^{\varepsilon }\) as \(\varepsilon \rightarrow 0\). As it will be seen, the limiting ODE governing the large deviations admits infinitely many solutions, a non-standard situation in the Wentzell–Freidlin theory. As for applications, the \(\varepsilon \)-scaling allows to derive leading order asymptotics for path functionals: while on the one hand the resulting formulae are confirmed by the CIR-CEV benchmarks, on the other hand the large deviation approach (i) applies to equations with a more general drift term and (ii) potentially opens the way to heat kernel analysis for higher-dimensional diffusions involving (1) as a component.
Giovanni Conforti, Stefano De Marco, Jean-Dominique Deuschel
Long Time Asymptotics for Optimal Investment
Abstract
This survey reviews portfolio selection problem for long-term horizon. We consider two objectives: (i) maximize the probability for outperforming a target growth rate of wealth process (ii) minimize the probability of falling below a target growth rate. We study the asymptotic behavior of these criteria formulated as large deviations control problems, that we solve by duality method leading to ergodic risk-sensitive portfolio optimization problems. Special emphasis is placed on linear factor models where explicit solutions are obtained.
Huyên Pham
Systemic Risk and Default Clustering for Large Financial Systems
Abstract
As it is known in the finance risk and macroeconomics literature, risk-sharing in large portfolios may increase the probability of creation of default clusters and of systemic risk. We review recent developments on mathematical and computational tools for the quantification of such phenomena. Limiting analysis such as law of large numbers and central limit theorems allow to approximate the distribution in large systems and study quantities such as the loss distribution in large portfolios. Large deviations analysis allow us to study the tail of the loss distribution and to identify pathways to default clustering. Sensitivity analysis allows to understand the most likely ways in which different effects, such as contagion and systematic risks, combine to lead to large default rates. Such results could give useful insights into how to optimally safeguard against such events.
Konstantinos Spiliopoulos
Estimation of Volatility Functionals: The Case of a $$\sqrt{n}$$ Window
Abstract
We consider a multidimensional Itô semimartingale regularly sampled on [0, t] at high frequency \(1/\Delta _n\), with \(\Delta _n\) going to zero. The goal of this paper is to provide an estimator for the integral over [0, t] of a given function of the volatility matrix, with the optimal rate \(1/\sqrt{\Delta _n}\) and minimal asymptotic variance. To achieve this, we use spot volatility estimators based on observations within time intervals of length \(k_n\Delta _n\). In [5], this was done with \(k_n\rightarrow \infty \) and \(k_n \sqrt{\Delta _n}\rightarrow 0\), and a central limit theorem was given after suitable de-biasing. Here we do the same with the choice \(k_n\asymp 1/\sqrt{\Delta _n}\). This results in a smaller bias, although more difficult to eliminate.
Jean Jacod, Mathieu Rosenbaum
Metadata
Title
Large Deviations and Asymptotic Methods in Finance
Editors
Peter K. Friz
Jim Gatheral
Archil Gulisashvili
Antoine Jacquier
Josef Teichmann
Copyright Year
2015
Electronic ISBN
978-3-319-11605-1
Print ISBN
978-3-319-11604-4
DOI
https://doi.org/10.1007/978-3-319-11605-1