SABR Model Research Articles

Exact Simulation of the SABR Model

The stochastic alpha-beta-rho SABR model becomes popular in the financial industry because it is capable of providing good fits to various types of implied volatility curves observed in the marketp...

Implementation of the Free Boundary SABR Model

This paper implements the free boundary SABR model to interpolate and extrapolate swaptions prices in order to derive the probability density function for the Swap Rate.

Perturbative Solution of the SABR Eikonal Equation

Abstract In this paper we give a new derivation of the leading order term of the implied volatility in the SABR model. As is widely know, Hagan et al. obtained an analytic formula for the implied volatility via asymptotic expansion. It has been argued that the leading order term of this expansion contains an error and some authors have recalculated it using different methods. Here we give another derivation using standard perturbation theory of the underlying eikonal equation. Our result agrees with that found by Berestycki et al.

Asset pricing under general collateralization

We consider the valuation of collateralized derivative contracts such as bond option or Caplet contracts. We allow for posting different collaterals such as securities or cash for the derivatives and its hedges. The pricing is based on modeling the joint evolution of collateral rate and the spread between collaterals. The Hull–White models are applied to collateral rate and spread to generate the closed pricing formula for zero coupon bond option. We also derive the pricing formula for Caplet under the Libor Market model and SABR model framework.

Fast Quantization of Stochastic Volatility Models

Recursive Marginal Quantization (RMQ) allows fast approximation of solutions to stochastic differential equations in one-dimension. When applied to two factor models, RMQ is inefficient due to the fact that the optimization problem is usually performed using stochastic methods, e.g., Lloyd's algorithm or Competitive Learning Vector Quantization. In this paper, a new algorithm is proposed that allows RMQ to be applied to two-factor stochastic volatility models, which retains the efficiency of gradient-descent techniques. By margining over potential realizations of the volatility process, a significant decrease in computational effort is achieved when compared to current quantization methods. Additionally, techniques for modelling the correct zero-boundary behaviour are used to allow the new algorithm to be applied to cases where the previous methods would fail. The proposed technique is illustrated for European options on the Heston and Stein-Stein models, while a more thorough application is considered in the case of the popular SABR model, where various exotic options are also priced.

Bartlett's Delta in the SABR Model

We refine the analysis of hedging strategies for options under the SABR model. In particular, we provide a theoretical justification of the empirical observation that the modified delta (“Bartlett’s delta”) introduced there provides a more accurate and robust hedging strategy than the conventional SABR delta hedge.

Asset Pricing Under General Collateralization

We consider the valuation of collateralized derivative contracts such as bond option or Caplet contracts. We allow for posting different collaterals such as securities or cash for the derivatives and its hedges. The pricing is based on modeling the joint evolution of collateral rate and the spread between collaterals. The Hull-White models are applied to collateral rate and spread to generate the closed pricing formula for zero coupon bond option. We also derive the pricing formula for Caplet under the Libor Market model and SABR model framework.

Asymptotics for the Euler-Discretized Hull-White Stochastic Volatility Model

We consider the stochastic volatility model dSt = σtStdWt,dσt = ωσtdZt, with (Wt,Zt) uncorrelated standard Brownian motions. This is a special case of the Hull-White and the β=1 (log-normal) SABR model, which are widely used in financial practice. We study the properties of this model, discretized in time under several applications of the Euler-Maruyama scheme, and point out that the resulting model has certain properties which are different from those of the continuous time model. We study the asymptotics of the time-discretized model in the n→∞ limit of a very large number of time steps of size τ, at fixed \(\beta =\frac 12\omega ^{2}\tau n^{2}\) and \(\rho ={\sigma _{0}^{2}}\tau \), and derive three results: i) almost sure limits, ii) fluctuation results, and iii) explicit expressions for growth rates (Lyapunov exponents) of the positive integer moments of St. Under the Euler-Maruyama discretization for (St,logσt), the Lyapunov exponents have a phase transition, which appears in numerical simulations of the model as a numerical explosion of the asset price moments. We derive criteria for the appearance of these explosions.

A weak approximation with Malliavin weights for local stochastic volatility model

This paper introduces a new efficient and practical weak approximation for option price under local stochastic volatility model as marginal expectation of stochastic differential equation, using iterative asymptotic expansion with Malliavin weights. The explicit Malliavin weights for SABR model are shown. Numerical experiments confirm the validity of our discretization with a few time steps.

Functional Analytic (Ir-)Regularity Properties of SABR-type Processes

The SABR model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in market practice, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR these are rarely satisfied. We examine here regularity properties of the model from this perspective with view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller-properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups, and identify suitable Banach - and Hilbert spaces for these. Finally, we comment on the short time and large time asymptotic behaviour of SABR-type processes beyond the heat-kernel framework.

Small-Time Asymptotics under Local-Stochastic Volatility with a Jump-to-Default: Curvature and the Heat Kernel Expansion

We compute a sharp small-time estimate for implied volatility under a general uncorrelated local-stochastic volatility model. For this we use the Bellaiche \cite{Bel81} heat kernel expansion combined with Laplace's method to integrate over the volatility variable on a compact set, and (after a gauge transformation) we use the Davies \cite{Dav88} upper bound for the heat kernel on a manifold with bounded Ricci curvature to deal with the tail integrals. If the correlation $\rho < 0$, our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for $\beta=1, \rho \le 0$. \bl{For uncorrelated stochastic volatility models, our results also include a SABR-type model with $\beta=1$ and an affine mean-reverting drift, and the exponential Ornstein-Uhlenbeck model.} We later augment the model with a single jump-to-default with intensity $\lm$, which produces qualitatively different behaviour for the short-maturity smile; in particular, for $\rho=0$, log-moneyness $x > 0$, the implied volatility increases by $\lm f(x) t +o(t) $ for some function $f(x)$ which blows up as $x \searrow 0$. Finally, we compare our result with the general asymptotic expansion in Lorig, Pagliarani \& Pascucci \cite{LPP15}, and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov \& Spector \cite{AS12} for the case $\rho=0$.

An Efficient Second Order Discretization Using Smart Malliavin Weight and Quasi Monte Carlo Method for Option Pricing

This paper shows an efficient second order discretization scheme of expectations of stochastic differential equations. We introduce smart Malliavin weight which is given by a simple polynomials of Brownian motions as an improvement of the scheme of Yamada (2017). A new quasi Monte Carlo simulation is proposed to attain an efficient option pricing scheme. Numerical examples for the SABR model are shown to illustrate the validity of the scheme.

Pricing via recursive quantization in stochastic volatility models

We provide the first recursive quantization-based approach for pricing options in the presence of stochastic volatility. This method can be applied to any model for which an Euler scheme is available for the underlying price process and it allows one to price vanillas, as well as exotics, thanks to the knowledge of the transition probabilities for the discretized stock process. We apply the methodology to some celebrated stochastic volatility models, including the Stein and Stein [Rev. Financ. Stud. 1991, (4), 727–752] model and the SABR model introduced in Hagan et al. [Wilmott Mag., 2002, 84–108]. A numerical exercise shows that the pricing of vanillas turns out to be accurate; in addition, when applied to some exotics like equity-volatility options, the quantization-based method overperforms by far the Monte Carlo simulation.

A Weak Approximation with Malliavin Weights for Local Stochastic Volatility Model

This paper introduces a new efficient and practical weak approximation for option price under local stochastic volatility model as marginal expectation of stochastic differential equation, using iterative asymptotic expansion with Malliavin weights. The explicit Malliavin weights for SABR model are shown. Numerical experiments confirm the validity of our discretization with a few time steps.

Volatility Smile and Risk Neutral Density for FX Options: An Example for the USDMXN

The objective of this paper is to provide the main guidelines to correctly handle the FX market conventions in order to build a consistent Volatility Smile. The models detailed for the Volatility Smile are: Vanna Volga, SABR and a Quadratic Polynomial in Delta. Hopefully, further research can incorporate these models instead of the one introduced by Malz. Also, it is shown how to estimate the Risk Neutral Density for the Vanna Volga and SABR models, which can be used for risk assessment or even to analyze monetary policy or interventions in the FX market.

On a one time-step Monte Carlo simulation approach of the SABR model: Application to European options

In this work, we propose a one time-step Monte Carlo method for the SABR model. We base our approach on an accurate approximation of the cumulative distribution function of the time-integrated variance (conditional on the SABR volatility), using Fourier techniques and a copula. Resulting is a fast simulation algorithm which can be employed to price European options under the SABR dynamics. Our approach can thus be seen as an alternative to Hagan’s analytic formula for short maturities that may be employed for model calibration purposes.

Gorilla In Our Midst

Patrick S. Hagan: SABR model, the LGM model, auto-calibration, internal and external adjustors, and MC spline-based stripping methods. Brute intellect in the Western lowlands of quant finance.

LEVERAGED ETF IMPLIED VOLATILITIES FROM ETF DYNAMICS

AbstractThe growth of the exchange‐traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). We study the relationship between the ETF and LETF implied volatility surfaces when the underlying ETF is modeled by a general class of local‐stochastic volatility models. A closed‐form approximation for prices is derived for European‐style options whose payoffs depend on the terminal value of the ETF and/or LETF. Rigorous error bounds for this pricing approximation are established. A closed‐form approximation for implied volatilities is also derived. We also discuss a scaling procedure for comparing implied volatilities across leverage ratios. The implied volatility expansions and scalings are tested in three settings: Heston, limited constant elasticity of variance (CEV), and limited SABR; the last two are regularized versions of the well‐known CEV and SABR models.

On a One Time-Step SABR Simulation Approach: Application to European Options

In this work, we propose a one time-step Monte Carlo method for the SABR model. We base our approach on an accurate approximation of the cumulative distribution function of the integrated variance (conditional on the SABR volatility process), using Fourier techniques and a copula. Resulting is a fast simulation algorithm which can be employed to price European options under the SABR dynamics. Our approach can thus be seen as an alternative to Hagan's analytic formula for short maturities that may be employed for model calibration purposes.

Complete Analytical Solution of the SABR Model for Fixed-Income Option Pricing And Value-at-Risk Problems - A Probability Density Function Approach

The first ever explicit formulation of the concept of an option’s probability density function has been introduced in our publications “Breakthrough in Understanding Derivatives and Option Based Hedging - Marginal and Joint Probability Density Functions of Vanilla Options - True Value-at-Risk and Option Based Hedging Strategies” and “Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems. A Probability Density Function Approach” (see links: http://ssrn.com/abstract=2489601 and http://ssrn.com/abstract=2546430). In this paper we report similar unique results for pricing options in the presence of stochastic volatility (SABR model), enabling complete analytical resolution of all problems associated with options considered within the SABR Model. Analogous results for the Heston Model for stochastic volatility have been reported earlier (see links: http://ssrn.com/abstract=2549033; http://ssrn.com/abstract=2609143 and http://ssrn.com/abstract=2605948).Our discovery of the probability density function for options with stochastic volatility within the SABR model enables exact analytical closed-form representations of their expected values (prices) for the first time without depending on approximate numerical methods. We demonstrate by means of these reference prices that approximate numerical methods introduce substantial errors, even as high as 200-300% in some ranges of parameters. Expected value is the first moment. All higher moments are as easily represented in analytical closed-form based on our probability density function, but are not calculable by extensions of any numerical methods now used to represent the first moment. Our formulation of the density function for options with stochastic volatility of the underlying securities within the SABR model is expressive enough to enable derivation for the first time ever of corollary analytical closed-form results Value-At-Risk (“VAR”) characteristics. These VAR characteristics are probabilities that options or portfolios of options together with the underlying securities will be below or above any set of thresholds at termination or at any time prior to termination. Such assessments are absolutely out of reach of current published methods for treating options within the SABR model.All numerical evaluations based on our analytical closed-form results are practically instantaneous and absolutely accurate.