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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?

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$Z$ 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.

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  • $\begingroup$ What is $Z$? What is the other information provided? $\endgroup$
    – Gordon
    Commented Dec 15, 2017 at 15:23
  • $\begingroup$ 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. $\endgroup$
    – nbbo2
    Commented Dec 15, 2017 at 18:45
  • $\begingroup$ Thanks for the reply. Sure for the BBM formula but i was wondering if there had been an other way to pricing that value. $\endgroup$ Commented Dec 15, 2017 at 19:14
  • $\begingroup$ 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? $\endgroup$ Commented Dec 15, 2017 at 19:24

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Conceptually , the value of a calendar spread at T1 equals the "time value" of the remaining option. The time value equals the total value minus the intrinsic value. This time value is obviously a function of S(T1) and looks like a bell shaped curve centred on K. The expected value of the calendar spread at T1 could be found by taking the value at t < T1 (obtained from a model such as BS) and multiplying by exp (T1-t).

The calendar spread "looks cheap" if the volatility surface is inverted (long dated options cheaper than short dated). However you will only win if the stock price is near K at T1 .

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