How Maturity Affects European Call and Put Values
Summary
The document explains why increasing time to maturity does not have a uniformly positive effect on European option values. More time generally adds time value for an at-the-money option because the underlying has more opportunity to move favorably while the option payoff cannot fall below zero. Away from at-the-money conditions, however, the balance between optionality and the present value of expected payments can produce different outcomes.
An in-the-money put on a non-dividend-paying stock may lose value as maturity extends, and a call on a currency with a high foreign interest rate is noted as another exception. The answer also distinguishes European from American exercise: absent dividends, calls on stocks have no early-exercise advantage, while puts may benefit from early exercise. For an at-the-money forward call on a zero-dividend stock, it gives an approximate value proportional to volatility, spot price, and the square root of maturity. The approximation is limited to that special case; other moneyness and dividend conditions are more complex.
Key ideas
- Longer maturity often raises at-the-money European option value by expanding favorable price-move opportunities.
- Maturity can reduce the value of some in-the-money European options as discounting changes their intrinsic-value component.
- For a non-dividend-paying stock, American calls do not gain value from early exercise, while American puts may.
- At-the-money forward call value is approximately proportional to the square root of maturity under the stated assumptions.
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Full text
# How does longer time to maturity affect standard European call and put option values?
# How does longer time to maturity affect standard European call and put option values?
Denote American call and put option values as $C$ and $P$ respectively. Similarly, denote European call and put options values as $c$ and $p$.
It is well known that time to maturity affects all $C,P,c,p$ given all else are fixed (such as stock price, strike price, risk-free interest rate, volatility, etc).
For American $P$ and $C,$ it is quite clear that longer time to maturity increases $P$ and $C.$ However, things are not so clear cut when it comes to European $p$ and $c.$
> Question: How does longer time to maturity affect $p$ and $c?$
## Answer by Dom (score 7, accepted)
https://quant.stackexchange.com/a/49988
In simple terms, more time to expiry $T$ increases the value of an at-the-money (ATM) option as it gives more time for the stock to rise further (or fall further in the case of a put option). This means that the potential upside of the option is greater (the downside is not as it is floored at zero). So the option is worth more. This effect has nothing to do with being able to exercise the option early.
The situation is more complicated when the option is not ATM. For example, an in-the-money put option with no dividend may see its value decrease with increasing time to expiry due to its value being driven by its intrinsic value whose present-value declines, rather than increases with increasing $T$. Another exception would be an in-the-money call option on a currency with a high interest rate.
Unless the stock is paying a dividend, the value of an American and European call option are the same - there is no advantage to being able to exercise early. For a put there can be an advantage to exercising early even if there is no dividend. So the price of an American put will be slightly higher than that of the European put option. But this is not typically a significant difference in value.
To return to the special case of an ATM forward call option where $K=Se^{rT}$ on a zero-dividend stock, the price dependence of the option price $C$ with $T$ years to expiry and a stock price $S$ is approximately proportional to the square root of the time to expiry. The approximation is:
$C(S,T) \simeq 0.4 S \sigma \sqrt{T}$
where $\sigma$ is the volatility of the stock price.
This $T$ dependency does not change very much even if the option pays a dividend. For options that are not ATM forward the $T$ dependency is more complex.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.