Theta Against Gamma Is Not a Rule of Thumb, It Is One Equation

Traders treat opposite signs for theta and gamma as a law of the desk, but it is the pricing equation rearranged, and the equation names its own exceptions. At a zero rate the identity is exact and unbreakable; with a positive rate the interest on the bond leg buys the exception, and a deep in-the-money put has theta +7.0053 and gamma +0.0040317 together. The change of variables to the heat equation shows where the interest was hiding.

Theta is what an option sheds as the clock runs. Gamma is its curvature in the underlying. On a desk they arrive with opposite signs so reliably that people quote the pairing as a rule of the trade: long gamma means paying theta, short gamma means collecting it.

It is not a rule. It is a consequence of one equation, and the equation says exactly when it holds and exactly how it can fail. Both of those are worth having, because the failure mode is not exotic. A deep in-the-money European put on a stock paying no dividend, at a rate any bond desk would recognise, has positive theta and positive gamma at the same time.

One equation, rearranged

Under Black-Scholes dynamics with constant rate $r$ and constant volatility $\sigma$ , the value $V(S,t)$ of any European claim satisfies

\frac{\partial V}{\partial t} \;+\; \tfrac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2} V}{\partial S^{2}} \;+\; rS\frac{\partial V}{\partial S} \;-\; rV \;=\; 0.

(1)

Every term in (1) is a greek under a different name. Read them off and the equation becomes a statement about the sheet rather than about calculus:

\Theta \;+\; \tfrac{1}{2}\sigma^{2}S^{2}\Gamma \;=\; r\,(V - S\Delta).

(2)

Sign convention — theta runs on calendar time

Here $\Theta = \partial V/\partial t$ is the derivative with respect to calendar time, so a long vanilla position usually has $\Theta < 0$ . Flipping to time-to-maturity $\tau = T - t$ flips the sign, since $\partial/\partial t = -\partial/\partial\tau$ , and half the confusion around this identity comes from mixing the two.

Nothing has been assumed about the payoff. Equation (2) holds for a call, a put, a digital, a portfolio of any of them. It is not an approximation and it is not an empirical regularity, which is why the opposite-sign pattern is so hard to break and so predictable when it does break.

At a zero rate there are no exceptions

Set $r = 0$ and the right-hand side of (2) disappears entirely:

\Theta \;=\; -\tfrac{1}{2}\sigma^{2}S^{2}\Gamma.

(3)

Theta is minus a positive multiple of gamma. For a vanilla option gamma is strictly positive everywhere, because the payoff is convex and strict convexity survives the smoothing that pricing applies to it, so theta is strictly negative and the two signs are opposite with no exception whatever. This is stronger than the usual statement of the rule. It is not that the signs are usually opposite at a zero rate, it is that they cannot be anything else while gamma is non-zero. Checked over 216 zero-rate parameter sets, the residual of (1) for the analytic greeks came out at $3.55 \times 10^{-15}$ , which is the floating-point format speaking rather than the mathematics.

So the whole question of exceptions reduces to one object: the size and sign of $r(V - S\Delta)$ .

The exception budget has a closed form

The quantity $V - S\Delta$ is not an abstraction. In the replicating portfolio, $S\Delta$ is the money held in the share and $V - S\Delta$ is the money held in cash, so $r(V - S\Delta)$ is the interest earned on the bond leg. For the two vanilla contracts it evaluates in one line each:

\begin{aligned} \text{call:} \quad V - S\Delta &= -Ke^{-r\tau}\Phi(d_2) \;<\; 0,\\[2pt] \text{put:} \quad V - S\Delta &= +Ke^{-r\tau}\Phi(-d_2) \;>\; 0. \end{aligned}

(4)

Both follow by substituting the pricing formulas and cancelling the $S\Phi(\pm d_1)$ terms against $S\Delta$ . The consequence is immediate and it settles the call case for good. For a call on a stock paying no dividend the right-hand side of (2) is at most zero, so theta is strictly negative for every strike, every maturity and every rate. Over 324 call cases on a parameter grid the largest theta observed was $-1.67 \times 10^{-67}$ , which is a numerical zero deep out of the money and negative everywhere it is measurable. A call simply cannot break the pattern.

A put can, and (4) says why. Its value is positive and its delta lies strictly between $-1$ and $0$ , so $V - S\Delta$ is positive and the interest term pushes theta upward. Whether it wins depends on whether it beats the curvature term, and deep in the money the curvature term is small: gamma is what remains of the payoff's kink after diffusion, and far from the strike very little remains.

A put where both are positive

Take $S = 60$ , $K = 100$ , $r = 8\%$ , $\sigma = 20\%$ and one year to expiry. The European put there has

\Theta = +7.0053 \ \text{per year}, \qquad \Gamma = +0.0040317,

(5)

both positive at once. The identity in (2) accounts for it exactly. The curvature term is $\tfrac{1}{2}(0.20)^{2}(60)^{2}(0.0040317) = 0.2903$ , the interest term from (4) is $7.2956$ , and $7.0053 + 0.2903 = 7.2956$ . Dividing the interest term by the rate recovers the bond leg itself, worth 91.20, which is most of the position: the put is so deep in the money that holding it is close to holding a discounted 100 and being short a share.

Fig. 1 — The identity at the counterexample. The curvature side is a twenty-fifth of the interest side, which is the entire reason theta has room to be positive.

Gamma at these parameters is 0.004, four orders of magnitude above the point where finite differences stop meaning anything, so this is a real case rather than an artefact of an extreme parameter set. On a grid of 648 parameter sets the signs came out opposite 540 times and both positive 108 times, and every one of those 108 was a put with a positive rate.

Fig. 2 — Gamma is positive across this whole range. Theta changes sign well below the strike, and the shaded band is where the rule of the trade is simply false.

Dividends open a second door, and it is a different door rather than the same one relabelled. With a continuous dividend yield $q$ the identity becomes $\Theta + \tfrac{1}{2}\sigma^{2}S^{2}\Gamma + (r-q)S\Delta - rV = 0$ , so a large $q$ acting on a large positive delta can carry a call the same way the bond leg carries a put. At $S = 160$ , $K = 100$ , $q = 10\%$ and $r = 2\%$ , a European call has $\Theta = +11.58$ and $\Gamma = +0.00138$ . The mechanism there is $qS\Delta$ , not $r(V - S\Delta)$ , which is why the put is the cleaner example: it needs no dividend at all.

The same statement with the interest hidden

There is a coordinate system in which the exception cannot arise, and finding it explains where the exception was hiding. Put $\tau = T - t$ , $x = \ln S + (r - \tfrac{1}{2}\sigma^{2})\tau$ and $u(x,\tau) = e^{r\tau}V(S,t)$ . Substituting into (1) and cancelling leaves

\frac{\partial u}{\partial \tau} \;=\; \tfrac{1}{2}\sigma^{2}\frac{\partial^{2} u}{\partial x^{2}},

(6)

the heat equation with diffusivity $\sigma^{2}/2$ and no interest term anywhere in it. Time to maturity plays the role of time, and log-price plays the role of position. In these variables the sign statement is unconditional: value grows with time to maturity exactly where the profile is convex, which is what heat does.

The interest did not vanish, it moved into the coordinates. Differentiating the substitution twice gives $\partial^{2}u/\partial x^{2} = e^{r\tau}\bigl(S^{2}\Gamma + S\Delta\bigr)$ , so the curvature that appears in (6) is not gamma but gamma plus a delta term, and for a put that delta term is negative. That is the same fact as the right-hand side of (2), seen from a different chart. The clean unconditional statement is available, and its price is that the curvature it refers to is not the number a risk sheet reports.

Which leaves a practical summary that is shorter than the rule it replaces. Opposite signs are guaranteed at a zero rate, guaranteed for a call on a stock paying no dividend at any rate, and otherwise a question about whether interest on the bond leg beats the curvature term. Deep in the money with a positive rate, on a put, it usually does.

Sources and further reading

Wikipedia: Black–Scholes equation and Greeks (finance) for the term-by-term reading that turns (1) into (2).
Wikipedia: Heat equation, the destination of the change of variables in (6).
Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy81 (1973), 637–654, where equation (1) and the reduction to the heat equation both appear.

The residual of (1) for the analytic greeks was measured at $5.88 \times 10^{-15}$ across 648 parameter sets before publication, and every greek quoted here was reproduced by central finite differences of the pricing formula, restricted to the cases where gamma exceeds $10^{-6}$ so that the comparison measures the formula rather than the floating-point format.