Time to Expiration for European Calls

Why Time to Expiration Matters

A European call gives its holder the right, but not the obligation, to buy an underlying asset for a specified exercise price at a specified expiration date. A longer-lived call gives the stock more time to move above that exercise price. For a non-dividend-paying stock and nonnegative interest rates, that additional opportunity cannot reduce the call’s value when the other inputs are held fixed.

That conclusion is conditional, not a rule to apply to any two options with different expiration dates. The stock price, exercise price, interest-rate environment, volatility, and cash distributions must be compared consistently. Changing one of those inputs at the same time can obscure the effect of time itself.

Definition: The time to expiration $T$ is the time, measured in years, from the valuation date until a European option’s expiration. A clean time comparison holds the underlying asset, current stock price $S_0$, exercise price $K$, and market assumptions fixed, and compares call values $C(T_1)$ and $C(T_2)$ for $T_2>T_1$.

The call’s expiration payoff is $\max(S_T-K,0)$, where $S_T$ is the stock price at expiration. This payoff cannot be negative, and it increases when the expiration stock price increases. However, the option’s value before expiration is not simply its expected payoff: future cash flows must be valued consistently with the interest-rate and stock-price model.

The Usual Monotonicity Result

A broad result applies under risk-neutral pricing. Assume the stock pays no dividends, the risk-free rate $r$ is constant and nonnegative, trading is frictionless, and the stock’s discounted price $e^{-rt}S_t$ is a martingale under the risk-neutral probability measure. These assumptions include the standard non-dividend-paying stock model. The value of a European call expiring at $T$ is $C(T)=e^{-rT}E[(S_T-K)^+]$, where $x^+=\max(x,0)$.

For any $T_2>T_1$, conditional Jensen’s inequality and the martingale property show why the longer-dated call cannot be worth less. The complete argument is important: discounting the stock and discounting the exercise price both matter.

\begin{aligned} C(T_2) &=E\!\left[(e^{-rT_2}S_{T_2}-Ke^{-rT_2})^+\right]\\ &\ge E\!\left[(e^{-rT_1}S_{T_1}-Ke^{-rT_2})^+\right]\\ &\ge E\!\left[(e^{-rT_1}S_{T_1}-Ke^{-rT_1})^+\right]\\ &=C(T_1). \end{aligned}

For the first inequality, $e^{-rt}S_t$ is a martingale, so its conditional expectation at $T_2$, given information at $T_1$, equals $e^{-rT_1}S_{T_1}$. The function $x\mapsto(x-a)^+$ is convex. Conditional Jensen’s inequality therefore says that the conditional expected positive part is at least the positive part of the conditional expectation. Taking expectations gives the first inequality. For the second, $r\ge0$ and $T_2>T_1$ imply $e^{-rT_2}\le e^{-rT_1}$. Thus the longer-dated call’s discounted exercise price is no higher, and its positive-part payoff in the displayed comparison is no smaller. Together, the inequalities establish $C(T_2)\ge C(T_1)$.

This argument does not require a particular volatility distribution. It does require the stated assumptions, including no dividends and nonnegative rates. If those conditions change, this proof does not establish the same conclusion.

Black–Scholes View

Under the Black–Scholes model, the stock pays no dividends, follows geometric Brownian motion with constant volatility $\sigma>0$, and the continuously compounded risk-free rate $r$ is constant. The model also assumes frictionless trading and European exercise. Its call formula is $C(T)=S_0\Phi(d_1)-Ke^{-rT}\Phi(d_2)$, where $\Phi$ is the standard normal cumulative distribution function and $d_1=[\ln(S_0/K)+(r+\sigma^2/2)T]/(\sigma\sqrt{T})$, $d_2=d_1-\sigma\sqrt{T}$.

With these assumptions, the sensitivity of call value to expiration is $\partial C/\partial T=S_0\sigma\phi(d_1)/(2\sqrt{T})+rKe^{-rT}\Phi(d_2)$, where $\phi$ is the standard normal density. If $r\ge0$, both terms are nonnegative, so the call value is nondecreasing in $T$. If $r<0$, the second term is negative and can outweigh the first. The derivative then need not be positive.

Worked Example: Comparing Two Expirations in a Binomial Model

A hypothetical non-dividend-paying stock is worth $100. A European call has exercise price $100. In each one-year period, the stock either rises by a factor of 1.20 or falls by a factor of 0.90. The annual risk-free rate is 5%, so the one-year accumulation factor is $R=1.05$. Find the call values for expirations in one and two years using risk-neutral binomial pricing, then compare them.

Method and assumptions. The risk-neutral probability of an up move is $p=(R-d)/(u-d)$, with $u=1.20$ and $d=0.90$. This gives $p=(1.05-0.90)/(1.20-0.90)=0.15/0.30=0.50$; the down probability is also 0.50. The risk-free rate is positive, and there are no dividends.

One-year call. The stock prices at expiration are $100(1.20)=120$ and $100(0.90)=90$. The respective call payoffs are $\max(120-100,0)=20$ and $\max(90-100,0)=0$. The expected risk-neutral payoff is $0.50(20)+0.50(0)=10$. Discounting one year gives $C(1)=10/1.05=\$9.5238$, or $9.52 to the nearest cent.

Two-year call. The possible terminal stock prices are $100(1.20)^2=144$, $100(1.20)(0.90)=108$, and $100(0.90)^2=81$. Their probabilities are $0.25$, $0.50$, and $0.25$. The corresponding call payoffs are 44, 8, and 0. The expected payoff is $0.25(44)+0.50(8)+0.25(0)=11+4=15$. The two-year discount factor is $1/(1.05)^2=1/1.1025$, so $C(2)=15/1.1025=\$13.6054$, or $13.61 to the nearest cent.

Check. The longer-dated call is worth $13.61 versus $9.52 for the one-year call, consistent with the nonnegative-rate result. The common mistake is to compare the undiscounted expected payoffs alone; option values require discounting each expected payoff to the valuation date.

Worked Example: More Time for an Out-of-the-Money Call

A hypothetical non-dividend-paying stock is worth $80, and the exercise price is $90. In a binomial model, each year the stock moves up by a factor of 1.25 or down by a factor of 0.75. The annual risk-free rate is 0%, so the one-year accumulation factor is 1. Calculate the one- and two-year European call values.

Method. The risk-neutral up probability is $p=(1-0.75)/(1.25-0.75)=0.25/0.50=0.50$. The down probability is also 0.50. The risk-free rate is zero, so discounting does not change the expected payoffs.

One year. The stock is either $80(1.25)=100$ or $80(0.75)=60$. The call payoffs are $\max(100-90,0)=10$ and 0. The expected payoff is $0.50(10)+0.50(0)=5$, so $C(1)=\$5.00$.

Two years. The terminal stock prices are $80(1.25)^2=125$, $80(1.25)(0.75)=75$, and $80(0.75)^2=45$, with probabilities 0.25, 0.50, and 0.25. The call payoffs are 35, 0, and 0. Thus the expected payoff, and the call value at the zero rate, is $0.25(35)+0.50(0)+0.25(0)=\$8.75$.

Check. The longer expiration raises the value from $5.00 to $8.75, even though the call is out of the money initially. The trap is to focus only on the current stock price being below the exercise price. Expiration value depends on possible future stock prices, not just today’s moneyness.

When the Usual Result Can Fail

The nonnegative-rate condition is substantive. With negative rates, the discounted exercise price can increase as expiration gets longer. That change can offset the benefit of additional time and, in some models, reverse the usual comparison.

Worked Example: A Negative-Rate Counterexample

Consider a hypothetical non-dividend-paying stock worth $100 and a European call with exercise price $100. In each year, the stock moves up by a factor of 1.20 or down by a factor of 0.80. The annual risk-free accumulation factor is 0.90, corresponding to a negative effective annual rate of $-10\%$. Compare one- and two-year call values in this binomial model.

Method and risk-neutral probabilities. The risk-neutral up probability is $p=(0.90-0.80)/(1.20-0.80)=0.10/0.40=0.25$; the down probability is 0.75. The accumulation factor 0.90 lies between the down and up factors, so these probabilities are between zero and one.

One year. The stock prices are $120 and $80, and the call payoffs are $20 and $0. The expected payoff is $0.25(20)+0.75(0)=5$. Discounting by 0.90 gives $C(1)=5/0.90=\$5.5556$, or $5.56 to the nearest cent.

Two years. The terminal stock prices are $144, $96, and $64, with probabilities $0.25^2=0.0625$, $2(0.25)(0.75)=0.3750$, and $0.75^2=0.5625$. Only the $144 outcome has a positive call payoff, equal to $44. The expected payoff is $0.0625(44)=2.75$. The two-year accumulation factor is $0.90^2=0.81$, so $C(2)=2.75/0.81=\$3.3951$, or $3.40 to the nearest cent.

Check. In this model the two-year call is worth less than the one-year call: $3.40 versus $5.56. The common mistake is to apply the usual monotonicity result without checking the interest-rate assumption. Here the rate is negative, so the nonnegative-rate result does not apply.

A second important boundary is the no-dividend assumption. When a stock pays distributions, its discounted stock price alone is not the martingale used in the proof above. The relationship between distributions, stock growth, and the option’s discounted exercise price must be modeled before making a maturity comparison. In general, do not infer monotonicity from the no-dividend result if the underlying assumptions have changed.

Exam Traps and Practical Interpretation

A statement that “more time increases a call’s value” is useful only when the comparison is properly controlled. In particular, distinguish a change in expiration from a change in the underlying market inputs.

Different contracts are not automatically a time comparison. If the exercise prices differ, or one stock price is measured on a different valuation date, other effects are present.
Do not confuse payoff with value. The payoff is determined at expiration; the premium is a current value and includes discounting.
Check the rate assumption. Nonnegative rates are part of the broad monotonicity result. Negative rates can invalidate the conclusion.
Check for distributions. The proof given here assumes a non-dividend-paying stock; a distribution-paying stock requires a different analysis.
Hold volatility fixed in a model comparison. If a longer-dated option is also assigned a different volatility, its price change cannot be attributed to expiration alone.

The central result is therefore precise: for a European call on a non-dividend-paying stock, under risk-neutral pricing with a nonnegative constant risk-free rate and otherwise consistent assumptions, increasing time to expiration cannot lower the call’s value. This conclusion is aligned with the SOA VEE Accounting and Finance syllabus.

Check Your Understanding

For each problem, keep the stated assumptions fixed and distinguish an expiration effect from a change in another input.

A non-dividend-paying stock is worth $50. A one-period binomial model has up factor 1.20, down factor 0.80, and a 0% annual risk-free rate. A European call has exercise price $50. Find the one- and two-year values. (Answer: $C(1)=\$5.00$; $C(2)=\$5.50$.)
Under Black–Scholes, a non-dividend-paying stock is worth $80, the exercise price is $80, volatility is 30% per year, and the risk-free rate is 0%. Compare expirations of 0.25 and 1 year. Use $\Phi(0.075)\approx0.5299$ and $\Phi(0.15)\approx0.5596$. (Answer: approximately $C(0.25)=\$4.78$ and $C(1)=\$9.54$.)
In a one-period binomial model, the up and down factors are 1.30 and 0.85, while the annual accumulation factor is 1.00. Calculate the risk-neutral up probability and determine whether it is valid. (Answer: $p=(1.00-0.85)/(1.30-0.85)=1/3$, which is valid because it lies between 0 and 1.)
A call is on a non-dividend-paying stock, but the risk-free rate is negative. Is the nondecreasing-in-expiration result established by the theorem in this tutorial? State the reason. (Answer: No. The theorem requires a nonnegative rate; negative rates can make the discounted exercise price rise with expiration.)
Two calls have the same stock price and expiration, but different exercise prices and different volatilities. Can their price difference be attributed solely to time to expiration? (Answer: No. The exercise prices and volatilities also differ, so this is not a controlled time comparison.)

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