When Can a European Option Have Positive Theta?
Theta, $\Theta = \partial V/\partial t$, measures how an option's value changes as calendar time passes with everything else fixed. For most options theta is negative: the option loses value as expiry approaches.
Under Black-Scholes-Merton (stock with continuous dividend yield $y$, risk-free rate $r$, volatility $\sigma$, time to maturity $\tau$), when can a *European* option have positive theta? Identify the cases, and justify each one from the theta formulas $$\Theta_{\text{call}} = -\frac{S e^{-y\tau} N'(d_1)\,\sigma}{2\sqrt{\tau}} - rKe^{-r\tau}N(d_2) + yS e^{-y\tau}N(d_1), \qquad \Theta_{\text{put}} = -\frac{S e^{-y\tau} N'(d_1)\,\sigma}{2\sqrt{\tau}} + rKe^{-r\tau}N(-d_2) - yS e^{-y\tau}N(-d_1).$$
Hints
- Each theta has a term $-Se^{-y\tau}N'(d_1)\sigma/(2\sqrt{\tau})$ that is always negative (time value decay) plus terms coming from the drift of the discounted strike and the dividend leakage. Positive theta needs the positive terms to dominate.
- Look at deep in-the-money puts: $d_1$ and $d_2$ are very negative, so $N'(d_1) \approx 0$ and $N(-d_2) \approx 1$, leaving $\Theta_{\text{put}} \approx rKe^{-r\tau} > 0$.
- For calls, the only positive term is $ySe^{-y\tau}N(d_1)$; deep in the money with a large dividend yield it can exceed $rKe^{-r\tau}N(d_2)$ plus the (tiny) decay term.
Worked Solution
How to Think About It: Split theta into a decay term that is always negative and "carry" terms driven by the drift of $Ke^{-r\tau}$ and $Se^{-y\tau}$. Theta is positive only where the option has almost no time value, so the decay term vanishes and the carry term has the right sign: a deep in-the-money put (carry $+rK$) or a deep in-the-money call on a heavy dividend payer (carry $+yS$ beating $rK$).
Quick Estimate: Deep in-the-money put with $S = 50$, $K = 100$, $r = 5\%$, $\sigma = 20\%$, $\tau = 1$, $y = 0$: $d_1 = -3.12$, so $N'(d_1) \approx 0.003$ and $N(-d_2) \approx 0.9996$. Then $\Theta_{\text{put}} \approx 0.05 \times 100 \times e^{-0.05} = +4.76$ per year, and the exact value is $+4.74$. The put is worth $Ke^{-r\tau} - S$ plus almost nothing, and that quantity grows by about $rKe^{-r\tau}$ per year.
Formal Solution:
*Step 1 -- Structure of theta.* In both formulas the first term $-Se^{-y\tau}N'(d_1)\sigma/(2\sqrt{\tau})$ is nonpositive; it is the erosion of optionality (time value). The remaining terms are the interest and dividend carry of the replicating portfolio. Positive theta requires the carry to be positive and larger than the decay.
*Step 2 -- Deep in-the-money European put.* As $S/K \to 0$ (or more generally $d_1, d_2 \to -\infty$): $N'(d_1) \to 0$, $N(-d_2) \to 1$, $N(-d_1) \to 1$, so $$\Theta_{\text{put}} \to rKe^{-r\tau} - ySe^{-y\tau}.$$ With no or small dividends this is $\approx rKe^{-r\tau} > 0$. Interpretation: the put is essentially worth $Ke^{-r\tau} - Se^{-y\tau}$ and the discounted strike accretes toward $K$ as time passes.
*Step 3 -- Deep in-the-money European call on a high-dividend stock.* As $S/K \to \infty$: $N'(d_1) \to 0$, $N(d_1), N(d_2) \to 1$, so $$\Theta_{\text{call}} \to ySe^{-y\tau} - rKe^{-r\tau},$$ which is positive when $ySe^{-y\tau} > rKe^{-r\tau}$, i.e. when the dividend yield on the stock leg exceeds the interest on the strike leg. Example: $S = 150$, $K = 100$, $r = 1\%$, $y = 8\%$, $\sigma = 20\%$, $\tau = 1$ gives $\Theta_{\text{call}} = +8.59$ per year. Without dividends ($y = 0$) a European call's theta is always negative.
*Step 4 -- Everything else.* Near the money, $N'(d_1)$ is at its largest and the decay term $\propto S\sigma/(2\sqrt{\tau})$ dominates (it blows up as $\tau \to 0$), so theta is negative: for example the at-the-money put above has $\Theta = -1.66$ and the at-the-money call $\Theta = -6.41$ per year. Deep out-of-the-money options have all terms near zero with the decay term slightly negative.
*Step 5 -- Connection to early exercise.* The two positive-theta regions are exactly where the corresponding American option may be exercised early: a deep in-the-money American put (collect $K$ now rather than $Ke^{-r\tau}$ later) and a deep in-the-money American call on a dividend payer (capture the dividends). The European holder cannot exercise, so the value simply drifts up toward the intrinsic value as expiry nears.
Answer: A European put that is deep in the money has $\Theta \approx rKe^{-r\tau} - ySe^{-y\tau} > 0$ (positive whenever dividends are small), and a European call that is deep in the money on a stock with a high dividend yield has $\Theta \approx ySe^{-y\tau} - rKe^{-r\tau} > 0$ when $yS e^{-y\tau} > rKe^{-r\tau}$. In both cases the decay term $Se^{-y\tau}N'(d_1)\sigma/(2\sqrt{\tau})$ is negligible because $N'(d_1) \approx 0$.
Intuition
A deep in-the-money European put is essentially a promise to receive $K$ at expiry in exchange for a stock that is almost surely below $K$: its value is close to $Ke^{-r\tau} - Se^{-y\tau}$, and as time passes the discount on $K$ unwinds, so the put *gains* value. The holder would love to exercise now and collect $K$ today, which is precisely why American puts carry an early exercise premium. The mirror case is a deep in-the-money call on a high-dividend stock, which is worth about $Se^{-y\tau} - Ke^{-r\tau}$ and rises with time because the dividend leakage $Se^{-y\tau}$ shrinks as $\tau$ falls. These are the same two situations where early exercise of American options can be optimal.