Why European Put Prices Can Fall with Longer Maturity

Summary

The discussion contrasts European call and put prices as maturity changes. It gives an arbitrage argument that, under a positive interest rate, a call with a longer maturity cannot be cheaper than an otherwise identical shorter-dated call: if it were, buying the longer call and selling the shorter one would leave a nonnegative position at the shorter expiry, alongside the invested proceeds.

The answer notes that this monotonicity does not generally apply to puts. A European put can lose value as its expiry is extended, which helps explain why early exercise may be optimal for an American put in some settings. The stated conditions favoring early exercise are a deeply in-the-money put, a high interest rate, and no dividends. The exchange is brief and gives no derivation, pricing formula, or quantitative example, so it illustrates a qualitative distinction rather than a full analysis of maturity effects.

Key ideas

• A positive interest rate supports a no-arbitrage argument that call prices are nondecreasing with maturity.
• European put prices do not have the same general maturity monotonicity as call prices.
• A longer-dated put can be worth less than a shorter-dated put.
• Early exercise of an American put may be attractive when it is deeply in the money, rates are high, and dividends are absent.

Tags

# Effect of time to maturity on european put option


# Effect of time to maturity on european put option












Let $C(K,T,S_0)$ denote the price of an European call option with strike K and maturity T on underlying price $S_0$. Assume interest rate $r>0$. Then of course $C(K,T,S_0) \geq 0$ and $C(K,T,S_0) \geq S_0 - K e^{-rT}$ both to avoid arbitrage.

Consider now $C(K,T_1,S_0)$ and $C(K,T_2,S_0)$ with $T_2>T_1$ and assume

$$ C(K,T_1,S_0) > C(K,T_2,S_0) $$ Sell the expensive and buy the cheap, put money in the bank. At $T_1$ we have

$$ C(K,T_2,S_{T_1}) - \max\{S_{T_1}-K;0\} \geq 0 $$ with the money in the bank an arbitrage. We conclude the call price is increasing in maturity.

Can a similar argument be made for the put? To me the corresponding inequality is not good enough and using put call parity did not help either.

Thanks :)

## Answer by Mark Joshi (score 2, accepted)

https://quant.stackexchange.com/a/16069

puts can be decreasing in time to maturity. This is why you sometimes early exercise an American put. This tends to happen deeply in the money with large r and zero dividend rate.

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