Pricing Equity and Interest Rate Collars from Their Components
Summary
The document distinguishes two structures that share the name collar. An equity collar combines ownership of a stock with a purchased out-of-the-money put and a written out-of-the-money call. The put limits downside below its strike, while the short call limits gains above its strike. The position can therefore be understood as stock plus a put minus a call.
An interest rate collar instead combines a purchased cap with a sold floor on the same rate and maturity. Its value is the cap value less the floor value, with the component instruments priced using Black’s formula. The discussion also describes the reverse collar as the opposite position. These are structural explanations rather than a full pricing derivation: the equity example does not provide a specific valuation formula, and the appropriate construction depends on whether the exposure is to equity or interest rates.
Key ideas
- An equity collar combines long stock, a long put, and a short call.
- The put and call strikes define the range within which the equity position participates in gains and losses.
- An interest rate collar can be formed by buying a cap and selling a floor on the same rate and maturity.
- The interest rate collar value is the cap value minus the floor value, using Black’s formula for each component.
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Full text
# Black Scholes Formula for Collar Option
# Black Scholes Formula for Collar Option
I am wondering if there exists a Black Scholes pricing formula for a collar option?
## Answer by ash (score 4)
https://quant.stackexchange.com/a/7874
SRXX has talked about Intrest Rate Collar. Since it is not clear if you are looking for IR or equity here is my explaination of equity Collar
Equity Collar :-
- Structure :- Buy Underlying Asset (e.g. Stock) and Buy an out of money put and write out of money call
- Payoff daigram
- Replication COLLAR = long stock + long put (K1) + short call (K2)
As you can see the Max gain / Loss is limited which the objective here
HTH
## Answer by SRKX (score 3)
https://quant.stackexchange.com/a/7867
Your question lacks a bit of background to make sure that you are using the right terminology.
In short, you buy an interest rate collar to hedge exposure in rates when they get out of a zone. As you can see on the wiki page when you buy a collar, you essentially:
- Buy an interest rate cap with strike price $K_c$
- Sell an interest rate floor with strike price $K_f$
with the same underlying rate and maturity.
As a result, you make a profit when $r>K_c$ and and loss when $r<K_f$. Hence, if you have a short exposure to rates (i.e. you are willing them to go down, for example if you own a bond) then you are giving away any profit beyond $K_f$ and getting insurance against any loss beyond $K_c$.
So, the value of your collar is:
$$v_\text{collar}= v_\text{cap} - v_\text{floor}$$
You can price both the cap and the floor using Black's formula, and you get the value of the collar.
Note that you can take the exact same opposite position which is then called a reverse collar.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.