Closed-Form Asymptotics for Local Volatility Models

Abstract

We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5,6,8] for short-time asymptotic expansions of heat kernels, and obtain a family of general closed-form approximate solutions for both the pricing kernel and derivative price. A bootstrap scheme allows us to extend our method to large time. We also perform analytic as well as a numerical error analysis, and compare our results to other known methods. 1 arXiv:0910.2309v2 [q-fin.PR] 21 Apr 2010We therefore allow for variable coefficients in both space and time. We assume throughout that a(x) > 0, for x > 0 and that the coefficients a, b, c are smooth functions. The perturbative method introduced in [8] for the study of parabolic equations in arbitrary dimensions was fully justified in the case when a, b, and c and all their derivatives are bounded, and are bounded away from zero: a(x) ≥ γ > 0. In this paper we complete the results of [8] with explicit formulas for the 1D case. Then we numerically test our formulas for the Black-Scholes-Merton and CEV models, obtaining an excellent agreement between our theoretical results and the numerical tests. Both the Black-Scholes-Merton model (1.2) and and the CEV model (1.3) are more general than the models considered in [8] in that their coefficients do not satisfy the assumptions of the paper, yet the numerical tests indicates that the results of that paper are still valid for the more general models considered here. This observation suggests that the theoretical framework of [8] is applicable in greater generality. We plan to study this point in a forthcoming paper.To explain our method, let us recall that, under certain conditions on the operator L and initial value h, described in details in the next section, there exists a smooth function G t (x, y) such that the solution to (1.1) has the representationThe kernel function G t (x, y) in (1.5) is the fundamental solution or the socalled Green function for the problem (1.1).Remark 1.1. Given that G t (x, y) arises in several different contexts, we will call the function G t (x, y) the transition density kernel, pricing kernel, heat kernel, or Green function interchangeably, depending on the context in which the object arises.As mentioned above, except for some very special cases no explicit formulas for G t (x, y) or U (t, x) are available. For the Black-Scholes-Merton model, a change of variables reduces the PDE to a heat equation that can then be solved explicitly. Therefore, exact formulas for the kernel G BSM and the

Approximate solutions to second order parabolic equations. I: Analytic estimates

“…This point will be discussed in detail in [18]. Explicit calculations and concrete, practical applications of our method will be given in [19,20].…”

“…We call the resulting method the Dyson-Taylor commutator method. We think that our method is more accurate and more stable in practical implementations [19,18].…”

Implied Volatility Surface: Construction Methodologies and Characteristics

“…For completion we include a list of relevant papers: [11,14,21,38,122,46,48,49,50,57,63,70,69,66,65,64,67,68,71,73,75,77,78,81,86,87,93,101,103,104,106,112,111,115,118,119,123,125,127,128,133,134,135,137,140,141,16,17,19,22,23,…”

Existence, uniqueness, and global regularity for degenerate elliptic obstacle problems in mathematical finance

“…The accumulation of related research in this field has become vast and we will not attempt to survey it comprehensively here except to note that the hypotheses required by the main theorems in well-known articles such as [56,76] either exclude problems such as (1.4) because their hypotheses are too restrictive or yield conclusions which are not as strong as Theorem 1. 18.…”

“…Recent investigations using probability methods in the mathematical finance literature and focusing on fundamental questions of existence, uniqueness, and global regularity of solutions to the associated parabolic terminal/boundary value problem are due to Bayraktar and Xing [7], Constanzino, Nistor and Mazzucato and their collaborators [18,20], and Ekström, Tysk, and Janson [30,29,52]. 1.2.2.…”