Shapes of Implied Volatility with Positive Mass at Zero

Abstract

We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one cantheoretically-distinguish between mass at the origin and a heavy-left-tailed distribution. We numerically test our model-free results in stochastic models with absorption at the boundary, such as the CEV process, and in jump-to-default models. Note that while Lee's moment formula [25] tells that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as [3,17] do not apply in this setting-essentially due to the breakdown of Put-Call duality.

The Log Moment formula for implied volatility

“…Lee also proved a symmetric right-wing formula, but we omit its presentation as we shall not require it here. This left-wing behaviour of the smile left two unresolved issues however: if S T has a strictly positive mass at the origin, then Lee's expression is not able to distinguish it from a mass-less distribution with fat tails; this was tackled in [16]. The second issue is that in fact no information about the moments of S T is really available in the market, and the so-called Power options [7] are rarely traded.…”

“…Lee's formulation however does not provide further details. In the case of a strictly positive mass at the origin, De Marco, Hillairet and Jacquier [16,Theorem 3.6] proved that…”

“…It serves not only to infer directly observed information about the implied volatility smile into constraints on model parameters but also to provide arbitrage-free solutions to the extrapolation problem (how to evaluate options for strikes outside the observed range). Recent refinements have led to a deeper understanding of the information contained in the implied volatility smile, determining whether the probability of default of the underlying could be inferred [16] or the potential lack of martingality of the latter [23]. These have complemented the otherwise exhaustive literature on the asymptotic behaviour of stochastic models in Finance, a thorough review of which can be consulted in [19].…”

Equity Option Implied Probability of Default and Equity Recovery Rate

“…Consider a stock that has a current price of with a positive risk‐neutral default probability of PD prior to some time T . Then, as the stock is worthless in the case of default, Moreover, as De Marco, Hillairet, and Jacquier () show can be calculated from call options using the identity of Breeden and Litzenberger () and is given by Then, a digital contract that pays a unit currency at time T if default happens prior to time T and pays zero otherwise is given by and can be replicated in terms of call options and cash, as follows: …”

Effective Asymptotics Analysis for Finance

“…Subsequently, the volatility smile has been studied in a fairly general way, with a minimum of hypotheses on the probabilistic distribution of the assets [3,4,8,16,25]. This has made it possible to isolate intrinsic behaviours that are shared by a large number of models in the study of volatility smiles.…”