Deriving the Collar Value with Put-Call Parity

Summary

The explanation derives a collar’s value from its components: one share of stock, a long put at the lower strike, and a short call at the upper strike. Its terminal payoff is capped at the lower strike below that level, follows the stock price between the strikes, and is capped at the upper strike above it. This payoff shows why a proposed formula consisting of the discounted lower strike plus the put value minus the call value is incorrect as written.

Starting with stock plus put minus call, the answer applies put-call parity to rewrite the position as a long call at the lower strike, a discounted bond payment with face value equal to that strike, and a short call at the upper strike. The discount factor converts the future bond payment to present value. This is an algebraic payoff decomposition; the note does not discuss dividends, early exercise, or practical pricing assumptions.

Key ideas

• A collar combines stock, a put at the lower strike, and a short call at the upper strike.
• Its expiration payoff is bounded below and above by the two strikes.
• Put-call parity rewrites the collar as a lower-strike call, a discounted bond position, and a short upper-strike call.
• The discounted strike represents the present value of a future bond payment.

Tags

# Collar Option K Term


# 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?

## Answer by Kermittfrog (score 3)

https://quant.stackexchange.com/a/58792

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.