# binomial model help

supposdly when you use the correct hedge ratio you will have a risk free payoff and thus you discount it at risk free rate…based on this idea we solve for put or call But wait, is it realy risk free? What garantees the stock price will end up being one of the two values we expected? For all we know it can be some third value and we would be screwed…thus it is not a risk free payoff and the model falls apart! I hope someone can explain why it works

You probably meant ARBITRAGE free rates. Nothing is risk free, except Treasury bonds as they are theoretically default free. The rates that we calculate for option free forward spots or optioned binomial forward spots are arbitrage free, meaning if you calculate the value of asset or rate now it will be equal to current Treasury bond (remove option costs). The same argument is true for stock options. When you sell a call you borrow at risk free rate, buy the asset and hold until delivery. At the time of transaction time there no profit to make. The profit/loss happens as the time passes.

If someone can get their hands on Cox et al. (1979) that would be great! I am sure the guy who developed the model talked about its assumptions and weaknesses. CFAI seems to skip something here… It seems like built into the model is the assumption that you know for sure the Stock price in 1 period will be either S+ or S- Given that assumption, the valuation perfect sense… But in reality, does anyone know for sure what the stock price will be next? Here is an example: S0=65 X=70 S+=84.5 S-=50.7 c=8 risk free=8 It appears there is opportunity for arbitrage Buy 0.4290 shares, and sell 1 call Cost = (0.4290*65)-8= 19.985 Payoff= (0.4290*84.5)-(14.5)=21.75 or payoff= (0.4290*50.7)-0=21.75 so you have locked in 21.75 with no risk 21.75/19.985=8.8% which is bigger than the rf… Great, so arbitragers should eliminate this opportunity in theory, they will keep selling the call and bring its price down. But if you were an arbitrager, would you want in on this? You’re not hedged, it is not truly risk free, stock price could end up being worth 0, so you are taking on risk, and you should demand more than Rf, thus call may be correctly priced at 8… It seems it all goes back to the assumptions that the stock will take on one of only two values and we are sure it can’t take a third value, is that realistic ??? HELP

One of the assumptions is the random walk nature of the stock price which means that the expected return of the stock in the next period is 0. So in theory you can have “n” future nodes with different returns and their expected probabilities. But this is extremely difficult to use to price anything, so we just use a binomial model so we have an up and a down state with an expected return of 0 in the next period. How you determine the up and down returns is model specific, there are a lot of models where you can either end up with a 50/50 probability or a risk neutral probability that will always make the expected return 0. You don’t just randomly pick a S+ or an S-, any S+/S- combination will affect the probabilities in such a way that the expected return is still 0. Now the time period is 1 day or more than a day(like 3 months) in the usual CFA questions, but if you start reducing the time period to smaller periods then the binomial tree will approach a continuous distribution and this is where BSM model kicks in. The risk free rate that is used in the model is based on the no arbitrage theory where we invest in a synthetic portfolio of stock and bond that will replicate the payoff of the call. The risk free rate is not compensation for the risk, all else equal the risk of a call option is effectively the volatility of the stock and the premium of the call option reflects it. Yes, the model is simplified for understanding the basics of how to price things and there are a lot of assumptions (like constant volatility, constant risk free rate among others)… Hope this helps

thank you seeki. It is a great start. I will dwel more on ur answer in order to fully get it. So is this model used much in practice or there are so many adjustments before it is? Thanks again