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I know that the value of a collar option on a stock (buy stock, buy put at $K_1$ and sell call at $K_2$) is given by

$$Collar\ Value = K_1d(t,T)+Put\ Value-Call\ Value$$

My question is, why do we have the $K_1$ term and why do we need to discount it?

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The collar strategy combines one unit of stock with a (long) put option with strike $K_1$ and a (short) call option with strike $K_2$. The payoff of this strategy is exactly $K_1$ if $S_T\leq K_1$, $S_T$ if $K_1<S_T\leq K_2$ and $K_2$ if $S_T>K_2$. The easiest way to see that the statement is false is by comparing the payoff profiles of the collar and that of your statement. Below $K_1$, the payoff is correct $(K_1)$, but for $S_T>K_1$, the payoff diverges.


Start again from the Collar present value $$ Collar=S+P(K_1)-C(K_2) $$ and make use of the Put-Call-Parity, $S+P=C+K$ to arrive at

$$ Collar=\mathbf{C(K_1)}+K_1D(t,T)-C(K_2) $$

Please note the emphasis. The payoff profile of a collar is thus achieved by buying / selling calls and entering a bond position with face value $K_1$.

HTH?

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  • $\begingroup$ this is old but I'm wondering why just discount K1? $\endgroup$ – habdie Nov 19 at 10:43
  • $\begingroup$ That’s due to PCP. Only one bond needs to be put out. $\endgroup$ – Kermittfrog Nov 19 at 11:59
  • $\begingroup$ At expiry an option pays out 𝐾𝑒𝑝 if 𝑆>𝐾𝑒𝑝, πΎπ‘‘π‘œπ‘€π‘› if 𝑆<πΎπ‘‘π‘œπ‘€π‘› and 𝑆 otherwise, where πΎπ‘‘π‘œπ‘€π‘›<𝐾𝑒𝑝 and 𝑆 is the underlying. will this be the same ? @kermittfrog $\endgroup$ – habdie Nov 19 at 12:07
  • $\begingroup$ do you know in which book i can find more details about this please? $\endgroup$ – habdie Nov 19 at 12:43
  • $\begingroup$ You could give Hull Options, Futures, and other Derivatives a go for this. $\endgroup$ – Kermittfrog Nov 19 at 13:17

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