Functional Analytic (Ir-)Regularity Properties of Sabr-Type Processes

Abstract

Abstract. 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 Banachand 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.

On the probability of hitting the boundary for Brownian motions on the SABR plane

“…Reversing the arguments presented in [10,18], the probability P(τ X 0 < τ Y 0 ) coincides with the probability that the process X hits zero over the time horizon [0, ∞). Indeed, through (3.3), the time change (3.2) converts the Brownian motion Y into a geometric Brownian motion Y started at y 0 > 0, so that the (a.s. finite) point τ Y 0 is mapped to τ Y 0 = ∞.…”

“…Interestingly however, the large-time behaviour remains invariant under some transformations affecting β, while local properties (such as the density) can be translated from one case to another, reflecting the 'phase transition' occurring in the above three cases. As observed in [10], the constraints ρ = 0 or β = 0 are the only parameter configurations where certain advantageous regularity properties of (1.2) are valid. In fact these are the only cases for which (1.2) can be written as a Brownian motion on some weighted 3 manifold.…”

“…The joint distribution of hitting times of zero of two (correlated) Brownian motions without drift was first established by Iyengar [20], and refined by Metzler [24] (see also [4] for further results on hitting times of correlated Brownian motions). We also borrow some ideas from [10], where Hobson's construction for the normal SABR model [18, Example 5.2] is extended to (1.1) for general β ∈ [0, 1]. This indeed follows from the observation that stochastic time change methods, going back to Volkonskii [32], can still be applied to the Brownian motion on the SABR plane.…”

Dirichlet Forms and Finite Element Methods for the SABR Model

“…We then propose a Gelfand triplet of spaces for our finite element discretization, consisting of a space V of admissible functions (the domain of the SABR-Dirichlet form), its dual space V * and a pivotal Hilbert space H, containing V. In Section 2.2 we briefly recall some relevant existing results to prove wellposedness of the SABR-pricing problem on the triplet V ⊂ H ⊂ V * , and conclude the existence of a unique weak solution to the variational formulation of the Kolmogorov partial differential equations on these spaces. We furthermore derive in this section a non-symmetric Dirichlet form for the SABR model, thereby extending the results of [22] on Dirichlet forms on SABR-type models to the non-symmetric case. In Section 3 we present the finite element discretization of the weak solution of the equation examined in the previous sections.…”

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

“…Since much of the popularity of the SABR model is due to the tractability of its asymptotic formula, one should aim at preserving it while taking into account the mass at zero. The parameter sets ρ = 0 or β = 0 are the most tractable, and in fact (as observed in [13]) the only ones where certain advantageous regularity properties of the SABR process can be expected. We therefore concentrate here on the singular part of the distribution for these regimes, that is, we study the probability P(X T = 0) and provide tractable formulae and asymptotic approximations.…”