Theta and Time Decay

Calendar Time and Time to Maturity

Theta measures how an option price changes as the valuation date moves forward, with the other market inputs held fixed. That qualification matters: a derivative with respect to calendar time is not the same derivative as a derivative with respect to the remaining time to maturity. The two have opposite signs, because advancing calendar time shortens the remaining life of the contract.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ carrying a Brownian motion $B_t$. In the Black–Scholes model, under the risk-neutral measure $\mathbb{Q}$, suppose the stock has constant volatility $\sigma>0$, pays no dividends, and satisfies $dS_t=rS_t\,dt+\sigma S_t\,dB_t$, where $r$ is a constant interest rate. Here $B_t$ is Brownian motion under $\mathbb{Q}$; $\mathbb{P}$ and $\mathbb{Q}$ are distinct measures in general. We use the Black–Scholes formulas below, not a stochastic-integral calculation. Let $S>0$ be the current stock price, $K>0$ the strike, $T$ the fixed maturity date, $t<T$ the calendar valuation time, and $\tau=T-t>0$ the time to maturity.

Definition: For a call price $C(t,S)$ or put price $P(t,S)$, calendar-time theta is $\Theta_C=\left.\frac{\partial C}{\partial t}\right|_{S,K,r,\sigma,T}$ or $\Theta_P=\left.\frac{\partial P}{\partial t}\right|_{S,K,r,\sigma,T}$. The derivative holds the displayed inputs, including the maturity date $T$, fixed. If instead the price is written as $c(\tau,S)$ or $p(\tau,S)$, its maturity sensitivity is the derivative with respect to $\tau$, holding $S,K,r,\sigma$ fixed.

Write $\Phi$ for the standard normal cumulative distribution function and $\phi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ for its density. The Black–Scholes quantities and prices, with no dividend yield, are

d_1=\frac{\ln(S/K)+(r+\tfrac12\sigma^2)\tau}{\sigma\sqrt{\tau}}, \qquad d_2=d_1-\sigma\sqrt{\tau},

C=S\Phi(d_1)-Ke^{-r\tau}\Phi(d_2), \qquad P=Ke^{-r\tau}\Phi(-d_2)-S\Phi(-d_1).

Deriving Call and Put Theta

The derivatives below are taken for $t<T$, so $\tau>0$, with $S,K,r,\sigma,T$ fixed. Since $d\tau/dt=-1$, differentiating $d_2=d_1-\sigma\sqrt{\tau}$ gives $d_{2,t}=d_{1,t}+\sigma/(2\sqrt{\tau})$. A useful identity for cancelling the terms involving $d_{1,t}$ is $S\phi(d_1)=Ke^{-r\tau}\phi(d_2)$. To verify it, $d_1^2-d_2^2=(d_1-d_2)(d_1+d_2)=2r\tau+2\ln(S/K)$, so $\phi(d_1)/\phi(d_2)=e^{-r\tau}(K/S)$.

Theorem: For $S,K,\sigma>0$, constant $r$, and $t<T$, calendar-time theta is $$ \Theta_C=-\frac{S\phi(d_1)\sigma}{2\sqrt{\tau}}-rKe^{-r\tau}\Phi(d_2), \qquad \Theta_P=-\frac{S\phi(d_1)\sigma}{2\sqrt{\tau}}+rKe^{-r\tau}\Phi(-d_2). $$ The maturity sensitivities are $\partial c/\partial\tau=-\Theta_C$ and $\partial p/\partial\tau=-\Theta_P$, with the same other inputs fixed.

Proof. The function being differentiated for the call is $C=S\Phi(d_1)-Ke^{-r\tau}\Phi(d_2)$. Using $\Phi'(x)=\phi(x)$ and $\frac{\partial}{\partial t}e^{-r\tau}=re^{-r\tau}$, its derivative is $$ C_t=S\phi(d_1)d_{1,t}-rKe^{-r\tau}\Phi(d_2)-Ke^{-r\tau}\phi(d_2)d_{2,t}. $$ Substitute $d_{2,t}=d_{1,t}+\sigma/(2\sqrt{\tau})$. The terms containing $d_{1,t}$ cancel by $S\phi(d_1)=Ke^{-r\tau}\phi(d_2)$, leaving the stated call expression. For the put, the differentiated function is $P=Ke^{-r\tau}\Phi(-d_2)-S\Phi(-d_1)$. Its derivative is $$ P_t=rKe^{-r\tau}\Phi(-d_2)-Ke^{-r\tau}\phi(d_2)d_{2,t}+S\phi(d_1)d_{1,t}. $$ The last two terms reduce, using the same identity and the relation between $d_{2,t}$ and $d_{1,t}$, to $-S\phi(d_1)\sigma/(2\sqrt{\tau})$. This gives the stated put expression. Finally, at fixed $S,K,r,\sigma,T$, the price written as a function of $t$ is its representation as a function of $\tau=T-t$; the chain rule gives $\partial_t=-\partial_\tau$. This proves both maturity-sensitivity identities. $\square$

The first term in each theta formula is negative: uncertainty decays as expiry approaches. The rate-dependent term differs between calls and puts. In particular, the signs of total theta are not universally fixed if rates can be negative or the rate term dominates. Theta is a local rate of price change per unit of calendar time; it is not the total expected price change over a finite interval unless the other inputs and the theta itself remain constant over that interval.

Worked Evaluations

Worked Example: One-Year At-the-Money Call and Put

Take $S=K=100$, $r=0.05$, $\sigma=0.2$, and $\tau=1$. Find call and put calendar-time theta.

First, $\sigma^2=0.04$, so $r+\tfrac12\sigma^2=0.07$. Since $\ln(S/K)=\ln(1)=0$ and $\sigma\sqrt{\tau}=0.2$, the inputs to the normal distribution are $d_1=0.07/0.2=0.35$ and $d_2=0.35-0.2=0.15$. The needed values are $\phi(0.35)\approx0.3752$, $e^{-0.05}\approx0.9512$, $\Phi(0.15)\approx0.5596$, and $\Phi(-0.15)\approx0.4404$.

The common volatility term is $-100(0.3752)(0.2)/(2\sqrt{1})\approx-3.752$. The call's rate term is $-0.05(100)(0.9512)(0.5596)\approx-2.662$, giving $\Theta_C\approx-6.414$. The put's rate term is $0.05(100)(0.9512)(0.4404)\approx2.095$, giving $\Theta_P\approx-1.658$. Both theta values are in price units per year when time is measured in years.

Check: put-call parity gives $C-P=S-Ke^{-r\tau}$. Differentiating with respect to calendar time at fixed $T$ gives $\Theta_C-\Theta_P=-rKe^{-r\tau}$. Here the right-hand side is $-0.05(100)(0.9512)\approx-4.756$; the computed difference is $-6.414-(-1.658)=-4.756$, up to rounding.

Worked Example: Maturity Sensitivity at Zero Interest Rate

Let $S=K=100$, $r=0$, $\sigma=0.2$, and $\tau=0.25$. Compute both theta values and both maturity sensitivities.

Here $\sqrt{\tau}=0.5$, so $\sigma\sqrt{\tau}=0.1$. The numerator of $d_1$ is $\ln(1)+(0+0.02)(0.25)=0.005$. Thus $d_1=0.005/0.1=0.05$, and $d_2=0.05-0.1=-0.05$. Since $r=0$, both rate terms in the theta formulas vanish. Also $\phi(0.05)\approx0.3984$, so the common term is $-100(0.3984)(0.2)/(2(0.5))\approx-7.968$. Therefore $\Theta_C=\Theta_P\approx-7.968$. The maturity sensitivities are the negatives of these values: $\partial c/\partial\tau=\partial p/\partial\tau\approx7.968$.

Check: at $r=0$, put-call parity gives $C-P=S-K=0$ for every $\tau$. Its maturity derivative is therefore zero, consistent with the equal call and put maturity sensitivities.

A Second Check Through the Black–Scholes PDE

In the no-dividend, constant-rate model, the Black–Scholes PDE for a European claim $V$ is $V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0$. It is useful as an independent theta check: writing $\Delta=V_S$ and $\Gamma=V_{SS}$, it yields $\Theta=rV-rS\Delta-\tfrac12\sigma^2S^2\Gamma$. The drift and diffusion roles are distinct here: $rS$ is the risk-neutral drift of the stock, while $\sigma S$ is its diffusion coefficient.

Proposition: If a Black–Scholes price $V(t,S)$ is twice differentiable in $S$ and satisfies the Black–Scholes PDE, then its calendar-time theta satisfies $\Theta=rV-rS\Delta-\tfrac12\sigma^2S^2\Gamma$.

Proof. The PDE states $V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0$. Isolating $V_t$, and using the definitions $\Theta=V_t$, $\Delta=V_S$, and $\Gamma=V_{SS}$, gives $\Theta=rV-rS\Delta-\tfrac12\sigma^2S^2\Gamma$. The conclusion uses the stated PDE and differentiability assumptions; it is an identity for calendar-time theta, not maturity sensitivity. $\square$

Worked Example: Verify Theta Using Delta and Gamma

Take $S=K=100$, $r=0$, $\sigma=0.2$, and $\tau=1$. Compute call theta from the formula, then check it with the PDE expression.

Since $\sigma\sqrt{\tau}=0.2$ and $r+\tfrac12\sigma^2=0.02$, $d_1=0.02/0.2=0.1$ and $d_2=0.1-0.2=-0.1$. The direct theta formula gives $\Theta_C=-100\phi(0.1)(0.2)/2$. Substituting $\phi(0.1)\approx0.3970$ gives $\Theta_C\approx-3.970$.

For the independent PDE check, $C=100[\Phi(0.1)-\Phi(-0.1)]\approx100(0.5398-0.4602)=7.966$. The call delta is $\Delta=\Phi(0.1)\approx0.5398$, and gamma is $\Gamma=\phi(0.1)/(S\sigma\sqrt{\tau})\approx0.3970/(100(0.2))=0.01985$. With $r=0$, the PDE expression gives $\Theta_C=-\tfrac12(0.2)^2(100)^2(0.01985)\approx-3.970$, matching the direct calculation up to rounding.

Common Pitfall: Reversing the Sign

A frequent error is to differentiate with respect to $\tau$ and label the result calendar-time theta. For the one-year call in the first example, the derivative with respect to maturity is approximately $+6.414$, while calendar-time theta is approximately $-6.414$. The mistake is not in the Black–Scholes formula; it is in forgetting that $\tau=T-t$, so $d\tau/dt=-1$. Always specify which time variable is held fixed and which is being differentiated.

The formulas also apply only before expiry, where $\tau>0$. At expiry the payoff has a kink at $S=K$, and the smooth formula for theta need not have a finite limit there. A numerical theta should therefore be interpreted at the stated positive time to maturity, not as a generic derivative at the terminal date.

Check Your Understanding

Use the definitions and formulas above, keeping calendar time distinct from maturity.

For $S=K=100$, $r=0$, $\sigma=0.2$, and $\tau=1$, calculate $d_1,d_2$, and express call theta in terms of $\phi(d_1)$.
Explain why $\partial c/\partial\tau=-\Theta_C$ when the maturity date is fixed.
For positive $r$, write the rate-dependent terms in call theta and put theta, and state their signs.
State the identity relating $S\phi(d_1)$ to $Ke^{-r\tau}\phi(d_2)$, and explain where it is used in the differentiation.
Use the Black–Scholes PDE to write calendar-time theta in terms of $V,\Delta,\Gamma,r,S,\sigma$.

What You'll Learn