Short-Expiry Vol Asymptotics — The Medvedev-Scaillet Expansion

At very short maturities, the implied vol surface is dominated by the shape of the return distribution near the current spot. A striking empirical fact: the ATM skew of SPX options grows approximately as

1/T

1/\sqrt{T}

T \to 0

, while diffusion-only models (local vol, Heston, SABR) predict a skew that grows at most as

1/\sqrt{T}

. This discrepancy is the fingerprint of jumps — and the Medvedev-Scaillet expansion makes it precise.

The expansion gives the leading-order implied vol for any stochastic vol model in the short- $T$ limit, expressed in terms of the model's local vol function. It shows why the diffusion skew is bounded by $1/\sqrt{T}$ , and why the SVJ (stochastic vol + jumps) model can exceed this bound, explaining the empirical short-dated skew.

Learning objectives. By the end of this note you should be able to:

state the Medvedev-Scaillet leading-order implied vol and skew for a general stochastic vol model;
explain why the diffusion ATM skew is bounded as $1/(2\sqrt{T})$ at short maturities;
describe the jump correction to the short-maturity skew and how it enables matching the empirical $1/T$ skew.

In this lesson

2.1 The Medvedev-Scaillet expansion
2.2 The ATM skew at short maturities
2.3 The SVJ correction
2.4 What the expansion implies for model choice

Before you start

The SABR Model — Stochastic Alpha, Beta, Rho, and the Hagan Approximation

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