Arbitrage-Free SABR

Abstract

Smile risk is often managed using the explicit implied volatility formulas developed for the SABR model [1]. These asymptotic formulas are not exact, and this can lead to arbitrage for low strike options. Here we provide an alternate method for pricing options under the SABR model: We use asymptotic techniques to reduce the SABR model from two dimensions to one dimension. This leads to an effective one-dimensional forward equation for the probability density which has the same asymptotic order of accuracy as the explicit implied volatility formulas. We obtain arbitrage-free option prices by numerically solving this PDE. The implied volatilities obtained from the numerical solutions closely match the explicit implied volatility curves, apart from a boundary layer at very low rates. For very low-rate environments, or for very low strikes, the implied absolute (normal) volatility dips downward, closely matching market observations. We also show how negative rates can be accommodated by replacing the F factor with (F + a) .

Bartlett's Delta in the SABR Model

“…This claim was supported there by empirical and numerical arguments, see also [1], [5], and [6]. The purpose of this note is to provide a theoretical justification of this claim.…”

“…More evidence is described in [2], [1], [5] (for interest rate options), and in [6] (for equity options). Figure 1 shows the classic SABR delta corresponding to three different calibrations of the same smile curve: β = 0 (black line), β = 0.5 (red line), and β = 1 (green line).…”

The COS method for option valuation under the SABR dynamics

“…This formula, also known as the Hagan formula, may however sometimes lead to arbitrage possibilities for low strikes, as one can observe by examining the corresponding PDF [36]. The Hagan formula is not accurate for pricing options with long maturities [1,18,36]. As no analytic expression for the bivariate ChF of the SABR model is available, it may be difficult to calculate accurate reference values for our purposes.…”

“…In the last few decades, many pricing methods have been introduced, for example, convenient formulas [3,17,19], Monte Carlo methods [5,15,20], finite difference methods [10,18,22], quadrature methods [7] and also Fourier methods [6,12,25,26,28,33]. A variety of option pricing methods is compared in [37], including several Fourier methods.…”

“…Due to the closed-form expression of the implied volatility, the Hagan formula [17], the SABR model is a widely used stochastic local volatility model in the financial world. Many pricing methods for SABR-like models have been presented in the literature, like [2,9,16,18,27], however, many of those methods are mainly accurate for valuing European options with short maturities under the SABR model. In [1] a pricing method is given, which is accurate for options with long maturities, but this method is not applicable for all parameter sets [1].…”