CGMY Call Prices Split into a t^{1/Y} Term and a Linear Term

When an option is about to expire, its price can behave in two distinct waves instead of one simple line. This paper studies that short-time behavior for at-the-money call options in the CGMY model, a pure-jump model with activity parameter Y between 1 and 2. The key result is a two-term expansion: the normalized ATM call price satisfies c(t,0) = d1 t^{1/Y} + d2 t + o(t) as t goes to 0. The first coefficient, d1, is the familiar stable-limit term from the domain of attraction of a symmetric Y-stable law. The second, d2, is given by an explicit integral involving the characteristic exponent and the limiting stable exponent. The authors also extract higher-order coefficients in closed form by keeping the full Lipton-Lewis formula intact and using a dynamic cutoff that separates inner, core, and tail regions. They verify the coefficients numerically against existing closed-form expressions where those are available. The result sharpens short-maturity pricing in a model where small jumps drive the leading behavior and tempering shows up later.

An option with almost no time left does not fade in one smooth line. If you watch the price, you see two kinds of motion. In the CGMY model, the at-the-money, or ATM, call price splits into two parts as time shrinks. The first part scales like t^{1/Y}. The second part scales like t. Tiny jumps do not just make noise. They set the pace. The model keeps only jumps. It drops any smooth wiggle. That makes the short-time shape sharp enough to study. The work shows one more thing. The extra linear term carries the effect of the left and right jump tails. It is not just decoration. That is why the second term matters.

Two terms, two time scales

The short-time ATM call price has a two-step shape as time shrinks. The work writes it as c(t,0)=d1 t^{1/Y}+d2 t+o(t) as t goes to zero. Here c(t,0) is the normalized ATM call price. ATM means the strike matches the current price. The first coefficient d1 is the known stable-limit term. It comes from the domain of attraction of a symmetric Y-stable law. That phrase means the small jumps behave like a stable jump cloud when time is tiny. The second coefficient d2 is new in this setting. It comes from an explicit integral. The integral uses the model's characteristic exponent, which is the log of the characteristic function. It also uses the limiting stable exponent. The work then goes further. It keeps the full Lipton-Lewis integrand intact. It uses that to extract higher-order coefficients in closed form. It also checks the coefficients against known closed forms where those exist.

How the cutoff keeps the full formula alive

Instead of trimming the Lipton-Lewis formula, the method keeps it whole. The Lipton-Lewis formula turns an ATM call price into an integral from the model's characteristic function. A characteristic function is a compact way to encode how a random process moves. A dynamic cutoff then splits the integral into three parts. The cutoff moves with time. That keeps the balance right as t shrinks. The inner part handles very small values. The core part carries the main mass. The tail part catches the far edge. This split lets the method read off higher-order terms. It also keeps the remainder under control.

Why the split matters for pricing

Short-maturity pricing is where small jumps matter most. A first-order answer gives the main shape. A linear correction gives the next nudge. That helps when a contract has almost no time left. The CGMY model uses only jumps. It has no smooth wiggle from a Brownian term. So these extra coefficients sharpen the picture rather than blur it. They also let the full formula be checked against known closed forms where those forms exist. That makes the expansion more than a neat trick. It becomes a cleaner map for the last stretch before expiry. It also shows exactly where the second wave starts.

What the last stretch now reveals

The surprise is that the last-minute price has two clean parts. That means the last stretch before expiry is not just a guess from one term. It has a named correction you can point to. The work also gives a path to higher terms without throwing away the full Lipton-Lewis formula. So the model can do more than mark the main bend. It can also track the next step. That makes the short-time CGMY picture sharper, not just prettier. It is a better guide for the tiny time window where the price moves fastest.