Near-expiry option prices obey stable-law scaling in Lévy markets

When an option is almost expired, even tiny price jumps can decide what it is worth. This paper works out the near-expiry, at-the-money call price and implied volatility for a broad class of exponential Lévy models, where log returns follow a Lévy process. Under mild assumptions that place the process in the small-time domain of attraction of an α-stable law with α between 1 and 2, the authors derive first-order asymptotics as time to maturity goes to zero. They show that the crucial inputs survive a share-measure change, so the needed information can be read directly from how the Lévy measure behaves near the origin. When there is no Brownian component, they obtain new convergence rates of the form t^(1/α)ℓ(t), with ℓ a slowly varying function, and give an example with rate (t / log(1/t))^(1/α). When a Brownian component is present, the jump part becomes lower order and the leading √t behavior is driven entirely by the Gaussian part.

A call option with one day left lives or dies on tiny price jumps. That is the whole game near expiry. This study asks what that price should look like when log returns follow a Lévy process, a jump model with sudden moves. The usual smooth, Gaussian picture misses the wild wiggles traders see. At the money means the strike sits at the current price. The surprise is that stable-looking jumps can give a clean first answer. And when a Brownian part, the smooth random-walk piece, joins in, jumps can fade from view. That split is the most useful twist here.

When jumps set the clock

The analysis gives first-order call-price and implied-volatility asymptotics. That means it finds the leading shape as time to maturity shrinks. The key assumption puts the jump process in the small-time domain of attraction of an α-stable law. Here, α runs between 1 and 2. That law is a simple way to describe wild but scale-friendly jumps. The share measure change is a new probability view. It favors stock-price moves. It keeps the same stable class and the same finite centering constant, the offset that recenters the jump process, finite. So the needed inputs come from the Lévy measure, the jump-size rule, near zero. When there is no Brownian part, the price can converge at t^(1/α) times a slowly varying factor. A slowly varying factor changes very little as time shrinks. One example gives a rate of (t / log(1/t))^(1/α).

How the jump measure gives the answer

The trick is to zoom in on very short times. At that scale, the model asks one question: how many small jumps does the Lévy measure allow near zero? The Lévy measure is the rule that counts jump sizes. Regular variation, a pattern that keeps the same shape when you zoom, tells how that count behaves. The share measure change then checks the same jump law from the stock-holder's view. That move matters because it keeps the stable-domain property and the centering constant, the offset that recenters the jump process, finite. Once those pieces stay in place, the call-price expansion follows from the jump shape alone. No extra distribution guesswork is needed.

Why √t takes over

When a Brownian part is present, the jump story changes fast. The smooth Gaussian part, the ordinary random-walk piece, sets the leading at-the-money call price. The jumps still move the price, but only at lower order. That makes the √t law universal for this class. For traders and model builders, that split is valuable. It says when a fast, smooth noise term dominates, and when the jump law near zero deserves the full spotlight. The same framework also gives implied volatility asymptotics. Implied volatility is the market's back-solved volatility number. So the method does not stop at prices. It also reaches the smile traders quote every day.

What to test next

The non-Brownian example with rate (t / log(1/t))^(1/α) shows that the slow factor ℓ(t) can matter a lot. That makes the near-expiry clock more delicate than a simple power rule. One next test is another exponential Lévy model with the same near-zero jump shape but a different slow factor. The question is whether it keeps the same t^(1/α)ℓ(t) rate. If it does, then the small-time price can be read straight from the jump tail near zero. If it does not, the boundary of the stable-domain picture will be clearer.