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SABR Model
  • Advanced Analytics for the SABR Model In this paper, we present advanced analytical formulas for SABR model option pricing. The first technical result consists of a new exact formula for the zero correlation case. This closed form is a simple 2D integration of elementary functions, particularly attractive for numerical implementation. The second result is an effective approximation of the general correlation case. We use a map to the zero correlation case having a nice behavior on strike edges. The map formulas are easily implemented and do not contain any numerical integration. These formulas are important in volatility surface construction and CMS products replication because they provide correct behavior for far strikes and reduced approximation error. The latter is also helpful for dynamic SABR models. , A.Antonov, M.Spector (2012)
  • Series Expansion of the SABR Joint Density Under the SABR stochastic volatility model, pricing and hedging contracts that are sensitive to forward smile risk (e.g., forward starting options, barrier options) require the joint transition density. In this paper, we address this problem by providing closed-form representations, asymptotically, of the joint transition density. Specifically, we construct an expansion of the joint density through a hierarchy of parabolic equations after applying total volatility-of-volatility scaling and a near-Gaussian coordinate transformation. We then established an existence result to characterize the truncation error and provide explicit joint density formulas for the first three orders. Our approach inherits the same spirit of a small total volatility-of-volatility assumption as in in the original SABR analysis. Our results for the joint transition density serve as a basis for managing forward smile risk. Through numerical experiments, we illustrate the accuracy of our expansion in terms of joint density, marginal density, probability mass and implied volatilities for European call options , Q.Wu (2010)
  • Fitting the Smile, Smart Parameters for SABR and Heston In this paper we revisit the problem of calibrating stochastic volatility models. By finding smart initial parameters, we improve robustness of Levenberg-Marquardt. Applying this technique to the SABR and Heston models reduces calibration time by more than 90% compared to global optimization techniques such as Simplex or Differential Evolution. , P.Gauthier, P.H.Y, Rivaille (2009)
  • Asymptotic Implied Volatility at the Second Order with Application to the SABR Model 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. , L.Paulot (2009)
  • Joining the SABR and Libor models together Fabio Mercurio and Massimo Morini propose a Libor market model consistent with SABR dynamics and develop approximations that allow for the use of the SABR formula with modified inputs. They verify that the approximations are acceptably precise, imply good fitting of market data and produce regular Libor rate parameters. They finally show that the correct assessment of the no-arbitrage volatility drift leads to a more sensible pricing of derivatives not included in the calibration set. , Mercurio, Morini (2009)
  • Sensible Sensitivities for the SABR Model In this note we describe how to obtain reasonable Vegas for European securities priced off a single maturity SABR model calibrated to market data. We first introduce our notations and state what our hedging problem is. We then start by recalling the standard Jacobian methodology when the number of SABR parameters and volatility calibration data match. We then relax this assumption and show how to produce hedging ratios which bear financial intuition when the SABR model becomes underparameterized. The methodology can be easily generalized to all single maturity underparameterized models. , Messaoud Chibane, Hong Miao, Chenghai Xu (2009)
  • LIBOR market model with SABR style stochastic volatility , P.Hagan, A.Lesniewski (2008)
  • Local Time for the SABR Model: Connection with the Complex Black Scholes and Application to CMS and Spread Options , E.Benhamou, O.Croissant (2008)
  • A Stochastic Volatility Alternative to SABR , L.C.G.Rogers, L.A.M.Veraart (2008)
  • Effective Parameters for Stochastic Volatility Models , Z.Wang (2007)
  • Unifying the Bgm and Sabr Models: a Short Ride in Hyperbolic Geometry , P.Henry-Labordere (2007)
  • A Time-Homogeous, SABR-Consistent Extension of the LMM: Calibration And Numerical Results , R.Rebonato (2007)
  • Fine-tune your smile: Correction to Hagan et al , J.Obloj (2007)
  • No-Arbitrage Dynamics for a Tractable SABR Term Structure Libor Model , M.Morini, F.Mercurio (2007)
  • The Asymptotic Expansion Formula of Implied Volatility for Dynamic SABR Model and FX Hybrid Model , Y.Osajima (2007)
  • A Note on the SABR Model The SABR model (Hagan et al. 2002) has emerged in the last years as a reference stochastic volatility framework for modelling the swaption implied volatility at different strikes. However, many implications of the model behaviour in a market context have so far been overlooked. , M.Morini, F.Mercurio (2006)
  • Managing Smile Risk Market smiles and skews are usually managed by using local volatility models a la Dupire. We discover that the dynamics of the market smile predicted by local vol models is opposite of observed market behavior: when the price of the underlying decreases, local vol models predict that the smile shifts to higher prices; when the price increases, these models predict that the smile shifts to lower prices. Due to this contradiction between model and market, delta and vega hedges derived from the model can be unstable and may perform worse than naive Black-Scholes’ hedges. To eliminate this problem, we derive the SABR model, a stochastic volatility model in which the forward value satisfies dF= a Fˆβ dW1, da = ν a dW2 and the forward F and volatility a are correlated: dW1dW2 = ρdt. We use singular perturbation techniques to obtain the prices of European options under the SABR model, and from these prices we obtain explicit, closed-form algebraic formulas for the implied volatility as functions of today’s forward price f = F(0) and the strike K. These formulas immediately yield the market price, the market risks, including vanna and volga risks, and show that the SABR model captures the correct dynamics of the smile. We apply the SABR model to USD interest rate options, and find good agreement between the theoretical and observed smiles. , P.S.Hagan, D.Kumar, A.Lesniewski, D.E.Woodwar (2004)





















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