Calibration of SABR Stochastic Volatility Model. Copyright Changwei Xiong November last update: October 17, 2017 TABLE OF CONTENTS

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1 Calibration of SABR Stochastic Volatility Model Copyright Changwei Xiong 2011 November 2011 last update: October 17, 2017 TABLE OF CONTENTS 1. Introduction Asymptotic Solution by Hagan et al Correction to Hagan et al. Solution...4 References...6

2 1. INTRODUCTION SABR model is a CEV model augmented by stochastic volatility that assumes the forward rate evolves under the associated forward (terminal) measure Q T df t,t = α t F β t,t dw t, dα t = να t dz t, dw t dz t = ρdt (1) for time t between initial time s and maturity T. The F t,t is a forward rate process with initial value F s,t = f. The α t is the stochastic volatility with initial value α s = a. The parameter a cannot be observed from the market, however it can be derived analytically from the at-the-money implied volatility as we shall see in due course. The factor ν is known as the volatility of volatility, which adjusts the degree of volatility clustering in time. The parameter β [0,1] controls the relationship between the forward rate and the at-the-money volatility. A β < 1 ( non-lognormal case) leads to skews in the implied volatilities. In the case of β 1, if the market were to move up or down, the level of the at-the-money volatility would not be significantly affected, whereas when β < 1 the volatility increases as forward rate falls (i.e. volatility and forward move in opposite direction). The closer to 0 the more pronounced would be this effect. The correlation parameter ρ plays a similar role as the β does. It defines how the market moves in sync with the volatility dynamics. The model parameters ν, a, β and ρ are all assumed to be deterministic and time invariant. 2. ASYMPTOTIC SOLUTION BY HAGAN ET AL. Using singular perturbation techniques, Hagan et al. [1] provide a closed form asymptotic solution (up to the accuracy of a series expansion) for prices of vanilla instruments. The value of a vanilla option under the SABR model is given by the appropriate Black formula provided that the correct Black implied volatility is used. Given the forward initial value F s,t = f and the expiry time τ = T s, the Black implied volatility can be derived as a function of strike price K from a given set of SABR parameters ν, a, β and ρ by 2

3 (1 β)2a2 σ K = ζ a (1 + ( χ p 2 + ρβνa 4p + (2 3ρ2 )ν 2 p (1 + (1 β)2 q 2 ) ) τ) (2) where p = (fk) 1 β 2, q = ln f νpq, ζ = K a, χ = ln ( 1 2ρζ + ζ2 + ζ ρ ) 1 ρ When K = f, equation (2) reduces to give the at-the-money implied volatility σ f = a f 1 β (1 + ((1 β)2 a 2 p 2 + ρβνa 4p + (2 3ρ2 )ν 2 ) τ) (3) The (3) shows that there exists a relationship ln σ f = ln a (1 β) ln f + (4) It indicates that the value of β can be estimated from a log-log regression of σ f and f with historical data by ignoring terms involving τ. Alternatively, since the parameters β and ρ in SABR model control the distribution function in similar ways (i.e. both control the skewness of the distribution), the redundancy between the two parameters allows one to calibrate the model by fixing β to an assumption (e.g. β = 0.5). The decision is often made on the basis of market experience. The remaining parameters a, ν and ρ have different effects on the volatility curve. The parameter a mainly controls the overall magnitude of the curve, the ν controls how much smile (i.e. convexity) the curve exhibits and the ρ controls the curve s skew. As shown in (3) the parameter a has a functional form with the at-the-money volatility σ f. Inverting the equation gives the value of a as a root of a cubic equation if the ν and ρ are known (in general, the smallest positive root would be taken if there were three real roots) (1 β) 2 τ f 2(1 β) a3 + ρβντ 4f 1 β a2 + (1 + (2 3ρ2 )ν 2 τ ) a σ f f 1 β = 0 (5) This indicates that in SABR model we only need to calibrate ρ and ν to implied volatility surface, providing that the value of β is prescribed and the at-the-money implied volatility σ f is given. The 3

4 calibration is performed at each maturity of the volatility surface by minimizing the objective function defined as a sum of squared residuals (or sum of vega weighted squared residuals) argmin ν,ρ N SABR (σ mkt Ki σ Ki f,ν,a,β,ρ i=1 ) 2 (6) Various nonlinear optimization routines can be used to carry out the calibration, for example, Levenberg-Marquardt method or Nelder-Mead simplex method. 3. CORRECTION TO HAGAN ET AL. SOLUTION The ζ in (2) was defined in equation (A.57c) in [1]. When assuming CEV model for the forward process (i.e. C(F) = F β ), it has the form ζ = ν f a df = ν f C(F) a du K By expanding, we have K u β = ν a f1 β K 1 β 1 β = ν a η, η = f1 β K 1 β 1 β (7) e x = xn n! n=0 e y e y = 2y (1 + y2 3! + y4 5! + ) p (e 1 β 2 q e 1 β 2 q ) = p(1 β)q (1 + (1 β)2 q 2 + ), for y = 1 β q, p = (fk)1 β 2, q = ln f 2 K (8) f α K α = (1 β)pq (1 + (1 β)2 q 2 η = pq (1 + (1 β)2 q 2 Thus ζ can be written as ζ = ν a pq (1 + (1 β)2 q 2 + ) + ) + ) (9) Clearly the expression of ζ used in (2) is just an approximation of (9) truncating all higher order terms of q. This leads to a correction to the original Hagan et al. solution proposed by Obloj [2] in 2008, where 4

5 he uses ζ = ν η in (7) to calculate ζ rather than ζ = ν pq in (2). Plugging the ζ in (9) into the original a a Hagan et al. solution (2), we get the improved implied volatility formula σ K = νq χ (1 + ((1 β)2 a 2 p 2 + ρβνa 4p + (2 3ρ2 )ν 2 ) τ) where p = (fk) 1 β 2, q = ln f K, ζ = ν f 1 β K 1 β a 1 β, χ = ln ( 1 2ρζ + ζ2 + ζ ρ ) 1 ρ Two special cases must be addressed. First when K f, we get the at-the-money volatility lim σ K = a K f f 1 β (1 + ((1 β)2 a 2 f 2(1 β) + ρβνa 4f 1 β + (2 3ρ2 )ν 2 ) τ) (11) knowing that (or simply derived from the series expansion in (8)) (10) f 1 β K 1 β (1 β)k β lim = lim K f (1 β)q K f (1 β) 1 = f 1 β ζ, lim K f χ = 1 K νq lim K f χ = lim a ζ K f χ (1 β)q f 1 β K 1 β = a f 1 β (12) Second when β 1, we have the implied volatility in lognormal SABR lim β 1 σ K = νq χ using the fact that (1 + (ρνa 4 + (2 3ρ2 )ν 2 ) τ) (13) f 1 β K 1 β f 1 β ln f + K 1 β ln K lim = lim = ln f = q lim β 1 1 β β 1 1 K ζ = νq β 1 a (14) 5

6 REFERENCES 1. Hagan, P.; Kumar, D.; Lesniewski, A. and Woodward, D., Managing smile risk, Wilmott Magazine, pages , July Obloj, J., Fine-tune your smile: Correction to Hagan et al., Wilmott Magazine, 35, ,

Risk managing long-dated smile risk with SABR formula

Risk managing long-dated smile risk with SABR formula Claudio Moni QuaRC, RBS November 7, 2011 Abstract In this paper 1, we show that the sensitivities to the SABR parameters can be materially wrong when