# Calendar spread pricing: how find the final value of call long

Consider a calendar spread: short a call expiring at $T_1$ and long a call expiring at $T_2$ ($T_2>T_1$).

I didn't understand how estimate the price of the long call option at $T_1$. With a payoff like this $\left( c_1 - \max \left\{ S_{T_1} - K, 0 \right\} \right) + \left( -c_2 + Z \right)$, how can i find $Z$ as I don't know the future spot price?

Updated

$Z$ should represent the value of the call long but i don't understand how determine it if at $T_1$ (expiry of the short call) we don't know the future stock price yet.

From Hull-Treepongkaruna: "To understand the profile pattern from a calendar spread, first consider what happens if the stock price is very low when the short-maturity option expires. The short-maturity option in worthless and the value of the long-maturity option is close to zero". Again: "Consider next what happens is the stock price $St$ is very high when the short-maturity option expires. The short-maturity option costs the investore $St-K$ and the long maturity option is Worth a little more than $St-K$, where $K$ is the strike price of the options." Or: "If $St$ is close to $K$, the short-maturity option costs the investor either a small amount or nothing at all."

How can Hull say this? Maybe writing $Z=F(t_2)=S_{t1}e^{(r(t2-t1))]}$ so that payoff in $t_1$ can be interpreted $[c1-\max(S_{t1}-K,0)]+[-c_2+\max(S_{t1}e^{(r(T2-T1))}-K,0)]$ or in $t_0$ $(C_1-F(T_1)-K)+(-c_2+F(T_2)-K)$?

Excuse for the formules: $r(T_2-T_1)$ is the exponentation.

• What is $Z$? What is the other information provided? – Gordon Dec 15 '17 at 15:23
• A formula to compute the value of a call when you don't yet have the stock price at maturity does exist and is called the Black Scholes formula. – noob2 Dec 15 '17 at 18:45
• Thanks for the reply. Sure for the BBM formula but i was wondering if there had been an other way to pricing that value. – Francesco Totti Dec 15 '17 at 19:14
• After all if using BB it was the only way to determinate the value of the call and compute the payoff, talking about calendar, the author would at least have to mention it. So is it the only way? – Francesco Totti Dec 15 '17 at 19:24