Binomial Option Valuation Model: Arbitrage Opportunities

CFA Level II Derivatives
#1

Hi,

I’m working through the binomial option valuation model (Chapter 41), and I understand how to calculate the optimal hedge ratio, risk-neutral probability, and value of call/put options. I end up having trouble reading the model, and understanding how to take advantage of the arbitrage opportunity. Could someone walk through the best way to analyze replicating the options?

Thank you.

#2

In arbitrage, an investor will always buy low and sell high

If your option pricing model gives you a $2 option price:

  • The investor would sell the option if it were priced at $3
  • The investor would buy the option if it were priced at $2

And you would take the opposite position in the underlying asset.

#3

Thanks, I’m following what you’re saying; however, in application I’m not quite seeing it.

Would you mind sharing the rationale behind chatper 41 CFAI EOC question #4? I don’t see where the $4.50 overpriced call option was determined.

#4

So whenever you see a problem with over/underpriced options, this is always determined based on the no-arbitrage values calculated when you use a binomial tree.

Problem #3 calculated the no-arbitrage value as 3.714, and problem #4 asks “To write the call, and do what?”

The problem is telling you to write the call, which automatically means that the call is overpriced relative to the no-arbitrage value (Remember, buy low and sell high). Since you sold the call, you have to take the opposite position in the underlying, which would mean buying shares of the stock. In arbitrage, you never put your own money on the line, which is why you have to borrow at the risk-free rate.

This problem doesnt explictly tell you that the call is overpriced, but it is implicitly stated when it says “write a call and do what”

The $4.50 is just a randomly generated example to illustrate the point: “_ For the example in this case _, the value of the call option is 3.714. If the option is overpriced _ at, say, 4.50 _, you short the option and have a cash flow at Time 0 of +4.50”

If it were under priced you would do the opposite. Short sell the underlying stock, INVEST at the risk-free rate with proceeds from the short sale, and buy the call. You would then close out the short position by exercising the call option, and keep the proceeds from the investment.

Hope that helped.

#5

Thank you!