Asymptotics and calibration of local volatility models

Abstract

We derive a direct link between local and implied volatilities in the form of a quasilinear degenerate parabolic partial differential equation. Using this equation we establish closed-form asymptotic formulae for the implied volatility near expiry as well as for deep in-and out-of-the-money options. This in turn leads us to propose a new formulation near expiry of the calibration problem for the local volatility model, which we show to be well posed.In the Black-Scholes-Merton model [4,24], it is assumed that the price of a non-dividend paying stock S t follows the lognormal stochastic differential equationwhere t is time, µ and σ are constants and W t is a standard Brownian motion. The parameter σ is called the volatility of the stock S t . It is well-known that the price C(S t , t; K, T ) of a European call option written on S t with strike K and maturity T satisfies the linear parabolic partial differential equationwhere r is the risk-free short-term interest rate. Such options are commonly traded on markets, however σ is not directly observable. Therefore it is common practice to start from the observed prices and invert the closed-form solution to (2) in order to find that constant σ -called implied volatility-for which the solution to (2) agrees with the market price at today's value of the stock. It is widely observed that calls having different strikes and otherwise identical have different implied volatilities. This phenomenon, usually referred to as the smile effect, clearly violates the Black-Scholes-Merton model, since in this framework a constant σ is supposed to determine the dynamics of the underlying stock S t through (1) regardless of options, strikes and maturities.To overcome this difficulty, the model has to be extended. One widely used approach is to consider that the volatility also follows a stochastic diffusion process. Another case is when the volatility σ is not a constant any more but rather a (deterministic) function of the underlying asset and the time. Actually this can be seen as a particular case of the previous approach. These types of models are called local volatility models. For them, the dynamics of the underlying asset is governed by the stochastic differential equationThere are two problems which are relevant in practice. Firstly one needs to compute accurately the implied volatilities of option prices-the pricing problem. If one follows a traditional approach (e.g. solve the PDE corresponding to each model and then invert Black-Scholes formula), this is known to be

Option Pricing in the Moderate Deviations Regime

“…2 Similar results appear in the literature, with different level of mathematical rigor, for other and/or generic diffusion models [2,7,14,35].…”

“…The same is actually true [31] in a generic semimartingale model with diffusive component (with spot volatility σ 0 = √ v 0 > 0): 2) and this translates to the generic ATM implied volatility formula (even in presence of jumps, as long as v 0 > 0) σ 2 imp (0, t) = v 0 + o(1), t ↓ 0. Higher order terms in t will be model dependent.…”

“…The upper summation limit 1/β thus ensures that no irrelevant (i.e., o(1)) terms are contained in the sum. Note that we have 1/β = 2 for β ∈ ( The passage from the (derivatives of the) energy to ATM derivatives of the implied volatility in the short time limit is best conducted via the Berestycki-Busca-Florent (short: BBF) formula that was proved in [2]. (That said, theses relations are also a direct consequence of our expansions, as is pointed out in Section 4.)…”

A General Asymptotic Implied Volatility for Stochastic Volatility Models

“…The smile at zero-order (no dependence on the expiry date) is connected to the geodesic distance on a Riemann surface. This relation between the smile at zero-order and the geodesic distance has already been obtained in [4,5] and used in [2] to compute an asymptotic smile for an equity basket. Starting from this nice geometric result, we show how the first-order correction (linear in the expiry date) depends on an Abelian connection which is a non-trivial function of the drift processes.…”

Coupling index and stocks

“…The number of underlying components of an index is usually large 4 . It is then meaningful to let M tend to infinity.…”