Option Arbitrage With Hedge Ratio


With regards to the hedge Ratio I am trying to work out in the CFA EOCs some different solutions.

Generally i understand the principles.

  1. Calculate the tree and see which options are in the money

  2. Calculate the probabilities

  3. Apply probabilities and values to work out price of option

  4. Now say the solution gives the real price of the option, we match that against our calculated value to see whether it is under prices or over prices

  5. Using the Hedge Ratio calculate the number of options to shares ratio.

Where i am confused is on the buying of the shares. Let’s say for example:

A put is priced in the market at $14.

Our calculated value of the put is $12.78

Hedge Ratio = -0.3125

The put in the market is OVER PRICED compared to our calculated valuue. So we sell the puts right? But what about the shares? In this example it says Sell the puts and sell the shares. Dont we normally have to buy the shares and sell the puts?

In call examples sometimes it says to buy the calls and sell teh shares. Other times sell the calls sell the shares etc etc

How do I know which way round I do this?

Sell put = Go long on shares

Sell the shares = go short on shares

to mantain sterilized (hedge) position.


Buy put = short underlying (shares)

Sell put = long underlying (shares).

Buying a put is a bearish stance. We are hoping to make money if the market falls. Therefore do we not need to take the opposite side with shares and go long them to maintain the hedge.

That way if the market goes up you make on the shares, if the market goes down you make on the put?

You have put buying a put = shorting the shares.

Sorry If i’m missing something obvious

sell put- we get the premium from option and we believe the market will go up so the put will not be exercised by people we sold it to, say stock a is 20 now, we think it will be 25, so we tell person b hey you can sell the stock to me at 18 in future,which we believe will not happen but we will get the option premium and we buy stock because we believe market is going up

Rex, I think you understand the concept here.

1.) Selling a put option- you’re betting that the market is going to go UP (because if the market rises, the put that you sold won’t get excerised and you just collect your premium). In order to ‘hedge’ that position incase the market goes DOWN, you would ‘short’ the shares in the underlying (because you profit from a short position when the stock declines).

2.) Selling a call option- you’re betting that the market is going to go DOWN (because if the market declines, the call that you sold won’t get excercised and you just collect the premium.) In order to hedge that call that you sold incase the market does UP, you would buy the underlying stock so that if the market rises, you make money on the stock that you bought!

Does that make sense? (disclaimer: derivatives is my weak area so if anybody wants to correct me feel free)

Hi Hardj,

Thanks what you’re saying seems right. I am just getting confused from Flash’s comment above where he said if we buy the put we short the shares.

If we buy the put (we are hoping the market goes DOWN so we can sell higher). Therefore to hedge against the market going UP, dont we go LONG the shares?

If you buy the shares and sell the puts you have double long position so we cannot talk about hedge in this context.

I am talking about maintain some kind of sterile hedge position (this in each market direction your gain/losses will be 0.00).

Hey rexthedog,

Yes, you are absolutely correct in determining that hedge. If we buy the put, we hedge by going long the underlying.

I think sometimes the verbage in the curriculum (especially for derivatives) tends to overcomplicate things.

When I solve these problems, I try to think of them in a very simple and logical manner. When a case asks anything about the appropriate way to hedge a position, the first thing I ask is:

1.) “What is the current position and what does that say about my general view on the market?” Example, if I’m long a call option on a stock, I am clearly making a bet that the market is going to go UP.

Now, to determine the hedge I ask…

2.) “How do I protect myself if the market moves in the opposite direction of my intial view (i.e initial position)?” In this example, to protect myself from the market going DOWN, I can go ‘short’ the underlying stock.

At the end, I ask “am I making money if the market moves in either direction based off of my current positions?” If yes then you got your hedge!

This is an overly simplistic explanation but I use it as an ‘approach’ when seeing questions like this.

Perfect Hardj Cheers