# Collar Option K Term

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?

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 $$K_1 and $$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?

• this is old but I'm wondering why just discount K1? Nov 19 '20 at 10:43
• That’s due to PCP. Only one bond needs to be put out. Nov 19 '20 at 11:59
• At expiry an option pays out 𝐾𝑢𝑝 if 𝑆>𝐾𝑢𝑝, 𝐾𝑑𝑜𝑤𝑛 if 𝑆<𝐾𝑑𝑜𝑤𝑛 and 𝑆 otherwise, where 𝐾𝑑𝑜𝑤𝑛<𝐾𝑢𝑝 and 𝑆 is the underlying. will this be the same ? @kermittfrog Nov 19 '20 at 12:07