Arbitrage profit using options

CFA Level II
#1

If an option is underpriced/overpriced in the marketplace, how come we have to take a short/long position on the underlying asset as well.

Wouldn’t taking a short/long position on just the option (depending on what the option is being valued at in the marketplace) be enough to generate arbitrage profit?

Thanks,

#2

Suppose that the spot price of the stock is $50 and you’re considering a 6-month call option with a strike of $60. BSM says that it should be priced at $1.35, but the market price is $3.00.

How, exactly, are you going to make an arbitrage profit by taking a long or short position only in the option?

Be specific.

#3

If you short the call option (since the market price of this option is 3.00) then writing the calll option (aka taking a short position) you would generate profit on the option.

Not sure what the role of taking an position on the underlying asset would produce in this scenario.

Thanks,

#4

You’ve received a $3.00 premium, but is that really a profit?

Is the transaction over yet?

#5

Yes? Sorry I am lost in what you are saying.

#6

You sold someone a call option that expires in 6 months.

What’s going to happen to you in 6 months?

#7

You would buy back the call option

#8

And if the price of the underlying goes to $80 you’ll pay at least $20 to buy it back.

So . . . where’s your arbitrage profit?

Or any profit, for that matter?

#9

Are you saying we need to buy the underlying stock in order to secure a profit, no matter which way the stock moves?

#10

No.

I’m saying that we need to buy the underlying stock to protect the profit if the stock price _ increases _.

However, if the stock price decreases, we’re going to lose money on the stock, and there goes our profit.

So . . . after you buy the stock, how are you going to protect your profit if the price of the stock decreases?

#11

The profit is protected with selling the call option

#12

Fundamental rules of arbitrage which are both violated with your current strategy:

Rule #1 Do not use your own money.

Rule #2 Do not take any price risk.

#13

So how would we create arbitrage profit in this situation?

#14

Because the option is overpriced, we will short the call options and purchase a number of shares of stock determined by the hedge ratio or delta factor.

The net amount of outlay is the cost of the stock minus the premium received from the call short.

To fund the outlay, you would borrow at the risk free rate.

In a binomial model, the # of stocks to buy is hS and the amount to borrow is PV(–hS– + c–).

In the BSM describes above, the # stocks is N(d1)S and the amount to borrow is –N(d2)^(e–rT)X

something like that…

now or you have eliminated price risk (assuming your model is correct, if the stick moves up or down you still have hedged away the risk with the stock) and you’re not using your own money.

#15

You can never − _ never! _ − protect yourself by giving someone else the opportunity to take your money.

#16

The same way you create an arbitrage profit in any situation:

  • Identify an asset in one market that is mispriced
  • Find an identical (or equivalent) asset in the same or another market with a different price
  • Buy the cheap one; sell the dear one

So, the question is: what’s equivalent to a call option? (Hint: you learned this at Level I. Or, at least, should have.)