Theta Says Minus 1.47 Cents and Ito Says Plus 0.47

A six-month at-the-money call on a 50 dollar share sheds a cent and a half a night to time decay, and its expected price tomorrow is higher anyway. The deterministic total differential gives minus 0.66 cents and predicts the opposite of the truth, while Ito's third term, half the gamma times the squared move, adds plus 1.13 and runs on variance rather than direction. Substituting the pricing equation for theta cancels that term exactly and leaves an expected return of the riskless rate plus elasticity times the premium, which is 35.54 percent a year here and turns negative below a real drift of 4.18.

A six-month call on a 50 dollar share, struck at 50, with rates at 5 percent and a swing of 30 percent a year, is worth 4.8174. Its theta is minus 5.357262 a year, which is a cent and a half a day. The clock is taking money out of it every night.

Its expected price tomorrow is higher than its price today. Both statements are exactly true at the same time, and the reason is not a trick of interpretation. It is the term that the calculus most people reach for does not have.

The three derivatives, evaluated

Everything below runs on three numbers, and each was recomputed by central finite differences of the pricing formula rather than taken from a table:

\Delta = 0.588589, \qquad \Gamma = 0.03668143, \qquad \Theta = -5.357262 \ \text{a year}

(1)

Theta here is the derivative with respect to calendar time, so a long vanilla position has it negative and one day of it is $-5.357262/365 = -0.014677$ dollars, the cent and a half. The share is expected to earn 10 percent a year in the real world. That is the drift the question about tomorrow runs on, and it is a different number from the 5 percent the pricing formula discounts at.

The reconciliation that makes it worse

The natural move is to write the total change as decay plus the move the share is expected to make. Time contributes minus 1.47 cents, the drift contributes $\Delta \mu S\,dt = 0.588589 \times 0.10 \times 50/365$ or plus 0.81 cents, and the sum is

-1.47 + 0.81 = -0.66 \ \text{cents a day}

(2)

Still negative. The reflex answer does not merely fail to resolve the contradiction, it predicts the opposite of the truth and it does so confidently, which is worse than being stuck. Whatever is missing is not a rounding.

A function of a random walk needs a third term

Equation (2) is the total differential of a smooth function of two smooth variables. The share price is not a smooth variable. Over an interval $dt$ its move has size $\sigma S\sqrt{dt}$ , so the square of the move is of order $dt$ itself rather than negligible against it, and a second-order term survives into the first-order expansion.

Ito's lemma, for a function of one diffusion

If $dS = \mu S\,dt + \sigma S\,dW$ and $c(S,t)$ is twice differentiable, then $dc = \Theta\,dt + \Delta\,dS + \tfrac{1}{2}\Gamma\,(dS)^2$ with $(dS)^2 = \sigma^2 S^2\,dt$ . The extra term is what ordinary calculus drops and what a nowhere-differentiable path forces back in.

Take expectations of that and the middle term picks up the drift while the last one picks up the variance:

\mathbb{E}[dc] = \underbrace{\Theta\,dt}_{-1.47} + \underbrace{\Delta\,\mu S\,dt}_{+0.81} + \underbrace{\tfrac{1}{2}\Gamma\,\sigma^2 S^2\,dt}_{+1.13} = +0.47 \ \text{cents}

(3)

The third term needs no view on direction at all. Gamma is positive for a vanilla call and a squared move is positive whatever its sign, so that contribution is positive up moves and down moves alike. It runs on variance, and variance does not care which way the share went.

Fig. 1 — The picture behind the third term. The drift term follows the dashed tangent; the curve sits above it in both directions, and the gap is what curvature pays.

Fig. 2 — Equation (3) as a column. Two positives against one negative, and the curvature term alone is nearly as large as the decay.

Half a cent a day on a 4.8174 price is 9.74 basis points, so the option is expected to appreciate by about ten basis points a night while its theta bleeds a cent and a half. Nothing contradicts anything.

Two independent checks on the half cent

Equation (3) is a first-order expansion, and a first-order expansion should be confronted with the exact quantity it approximates. Integrating the true one-day transition against the pricing formula with 4001-point Simpson quadrature gives an expected change of $+0.00469309$ dollars against the $+0.00469136$ of the three-term sum. A separate antithetic simulation of 600,000 paths returns $+0.00468429$ . The approximation is not carrying the result.

Why the sign was never a coincidence

Numbers this close to zero invite the suspicion that another parameter set would flip them. It would not, and the reason is that theta is not free. Any European claim satisfies the pricing equation, which can be read as a formula for theta:

\Theta = r c - r S \Delta - \tfrac{1}{2}\sigma^2 S^2 \Gamma

(4)

Substitute (4) into (3). The curvature term cancels exactly, and what is left has no gamma and no volatility in it at all:

\mathbb{E}[dc] = \bigl(r c + (\mu - r) S \Delta\bigr)dt \quad \Longrightarrow \quad \frac{\mathbb{E}[dc]}{c\,dt} = r + \Omega\,(\mu - r), \quad \Omega = \frac{S\Delta}{c}

(5)

The call earns the riskless rate plus its elasticity times the equity premium. Here $\Omega = 50 \times 0.588589/4.8174 = 6.1089$ , so the expected return is $0.05 + 6.1089 \times 0.05 = 35.54$ percent a year. Over 1200 parameter sets crossing five spots, five maturities, four rates, four volatilities and three drifts at or above the riskless rate, the expected one-day change was never negative.

Theta as rent rather than as loss

Equation (5) also settles an argument that people have on desks without noticing it is settled. Theta feels like a leak, something the position loses every night for nothing. Read the cancellation again and theta is the price of the curvature: the pricing equation sets it to exactly the level at which, under the pricing measure, the two effects net out to the riskless rate. Nobody is giving convexity away and nobody is paying too much for it.

What is left over in the real world is the premium, the second term in (5). A long option position earns the riskless rate on its capital plus its elasticity times whatever premium the underlying carries, and that is the whole of its expected return. The decay is already inside that number. Complaining about theta while holding a long call is complaining about the cost of an asset you have decided you want, which is a different conversation from asking whether the expected change is positive.

Where the answer flips

Read (5) as a condition rather than as a number and the exception falls out. The expected change is negative when $r + \Omega(\mu - r) < 0$ , that is when $\mu < r(\Omega - 1)/\Omega$ , which on this contract means a real drift below 4.18 percent a year. A share expected to earn less than that has a call expected to lose money overnight, decay included, and the leverage that magnifies a premium magnifies a shortfall just as faithfully.

Puts break the other half of the story. The delta of the put at these parameters is $-0.4114$ , so the drift term in (5) is negative rather than positive, and $\mathbb{E}[dc]$ comes to $-0.0023$ dollars a day. Its gamma is the same positive number as the call's, and its curvature term is the same positive 1.13 cents. The lesson is that gamma does not beat theta by itself. It beats theta together with a positive exposure to a share that is expected to rise.

One last caveat on (4). That cancellation is an identity of the model, not of the world. It assumes constant volatility and continuous trading, and it assumes the option is priced by the same formula whose derivatives are being substituted. If the market prices at a volatility other than the one the share realises, the curvature term and the decay term stop netting out the way (5) says, and the difference between them is precisely what a gamma trader is paid or charged.

Sources and further reading

The lemma the third term comes from — Ito's lemma
The equation rearranged in (4) — Black-Scholes equation
The three derivatives in (1), and the sign conventions around theta — Greeks (finance)
The property that makes the curvature term positive — Convex function
The quadratic variation that turns $(dS)^2$ into a $dt$ — Quadratic variation

The three derivatives in (1) were confirmed by central differences of the pricing formula, the half cent in (3) by Simpson quadrature and by an antithetic simulation, the cancellation in (5) symbolically, and the sign over a 1200-point parameter sweep.