# Collars

The collar is best thought of as combination of protective put and covered call.

An investor who is long the underlying could buy a put (most likely OTM) to hedge the stock’s downside, while at the same time selling a call (also most likely OTM) to sell off the upside and subsidize the cost of the put.

Usually the put strike is set, then an appropriate call strike is determined such that the call and put have the same premium. If the options are over-the-counter, rather than exchange-traded, this will be easy to do.

In this case there will be no net inflow or outflow at initiation and the investor will have constructed a zero-cost collar.

Collar P&L Diagram: Stock Purchased at 12, NOV 15 Put Purchased at 1.46, NOV 17 Call Written at 1.44

The value of the collar at expiration is the sum of the value of the underlying asset, the value of the long put (struck at X1), and the value of the short call (struck at X2):

• VT = ST + Max(0,X1 – ST) – Max(0,ST – X2), where X2 > X1.

The profit is the profit on the underlying share plus the profit on the long put and the short call so that:

• Π = ST + Max(0,X1 – ST) – Max(0,ST – X2) – S0 – p0 + c0.

Broken down into ranges, the total profit on the portfolio is as follows:

• Π = X1 − S0 – p0 + c0 if ST ≤ X1
• Π = ST − S0 – p0 + c0 if X1 < ST < X2
• Π = X2 − S0 – p0 + c0 if ST ≥ X2

A collar sacrifices the positive part of the return distribution in exchange for the removal of the adverse portion. With the short call option, the option writer sold the right side of the return distribution, which includes the most desirable outcomes. With the long put, the investor is protected against the left side of the distribution and the associated losses. The option premium paid for the put is largely and, often precisely, offset by the option premium received from writing the call. The collar dramatically narrows the distribution of possible investment outcomes, which is risk reducing. In exchange for the risk reduction, the return potential is limited.