# Black Scholes Formula for Collar Option

I am wondering if there exists a Black Scholes pricing formula for a collar option?

• @edouard Collar is Long Put , short call , long underlying. – ash Apr 29 '13 at 8:38
• @Geraldine - You need to break the collar down into Calls + Puts + Underlying Asset and price each. BS will give you price for calls and puts. – ash Apr 29 '13 at 8:43
• @SRXX will do when I reach home. Not a good day at work so far – ash Apr 29 '13 at 9:39
• It looks like you already had an account here - if you'd like the two accounts merged, please use this form: quant.stackexchange.com/help/user-merge – Shog9 Apr 29 '13 at 20:09

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.

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

• Arf yeah of course I made that assumption... I get what you meant now... – SRKX Apr 29 '13 at 14:21