# 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 :)

• When you say "at $T1$ we have", why is this actually true ? Oct 26 '19 at 13:04
• @baibo Consider the payoff at $T_2$ of the call vs $S-K$.
– htd
Oct 27 '19 at 14:40

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.

• Doh. Of course, thanks. If we denote the initial price difference $\Delta P$. Then if at $T_1$ $S < K (1-e^{-r(T_2-T_1)})-\Delta P e^{rT_1}$ the short position will be worth at least $K + \Delta P e^{rT_2}$ at maturity dominating the rest of the portfolio giving a loss.
– htd
Jan 3 '15 at 10:32
• Could either of you explain a bit more? So an European put option may increase and decrease in its time to maturity? Could you give some examples?
– tvk
Jan 10 '15 at 0:40
• @FangJing The above example shows that we can not conlude it to be increasing, I bet copying the argument using the natural lower bound of the put would yield the other conclusion. It is great exercise to go through it yourself as I just did :)
– htd
Jan 10 '15 at 9:13
• @Henrik Would you mind explaining what the sentence "Then if at $T_1$ $S<K(1−e^{−r(T2−T1)})−\Delta Pe^{rT_1}$ the short position will be worth at least $K+\Delta Pe^{rT_2}$ at maturity dominating the rest of the portfolio giving a loss" means? How is $\Delta P$ defined here? Is it the price of the later-maturity-date put minus that of the earlier-maturity-date put?
– tvk
Jan 10 '15 at 19:29
• @FangJing Exactly! At time 0 when the position is initialized.
– htd
Jan 11 '15 at 6:17