Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

Abstract

We study the mass at the origin in the uncorrelated SABR stochastic volatility model, and derive several tractable expressions, in particular when time becomes small or large. As an application-in fact the original motivation for this paper-we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage free, allow us to quantify the impact of the mass at zero on existing implied volatility approximations, and in particular how correct/erroneous these approximations become.

On the Euler–Maruyama Scheme for Degenerate Stochastic Differential Equations with Non-sticky Condition

“…are uniformly integrable, thus we conclude (13), (14) and (15). Recall that σ 1 (y) 2 := lim inf x→y σ(x) 2 for y ∈ R. Using Fatou's lemma and (13), we have for any 0 ≤ r < t ≤ T ,…”

“…Note that one may use Itô's formula for Y t = |X t | 2(1−α) /(1 − α) 2 for "some" t ≥ 0, in order to prove Y satisfies the equation dY t = 2 √ Y t dW t + 1−2α 1−α dt, (see, e.g. [17] and [14]). The above computation shows that this is true for t < inf{s > 0; X s = 0}.…”

Approximate Solutions to Second-Order Parabolic Equations: evolution systems and discretization

“…Here the list of papers that one could quote is truly Gargantuan, but let nevertheless mention the papers [6,7,13,14,17,22,25,26] among earlier papers most closely related to our work [12,10,9,11] (in chronological order), in which we have developed the Dyson-Taylor commutator method used in this paper. Let us mention also the more recent papers [16,18,20,21,23,24,33,45], where the reader will be able to find further references. Some general related monographs include [15,27,30].…”

Dirichlet Forms and Finite Element Methods for the SABR Model

“…However, it seems that these refinements have not fully resolved the arbitrage issue near the origin. Recent results [29,38,39] focus on the singular part of the distribution and suggest an explanation for the irregularities appearing at interest rates near zero; and [38,39] provides a means to regularize Hagan's asymptotic formula at low strikes for specific parameter configurations, based on tail asymptotics derived in [26,37].…”

“…Although the exact distribution of the CEV process is available [55], simulation of the full SABR model based on it can in many cases become involved and expensive. In fact, exact formulas decomposing the SABR-distribution into a CEV part and a volatility part are only available in restricted parameter regimes, see [6,32,48] for the absolutely continuous part and [38,39] for the singular part of the distribution. A simple space transformation (see (1.5) below) makes some numerical approximation results for the CIR model (the perhaps most well-understood degenerate diffusion) applicable to certain parameter regimes of the SABR process.…”