LOS 56b. calculating interest rate options binomial model

LOS 56b. states that we must know how to calculate and interpret prices of interest rate options and options on assets using one- and two-period binomial models ;

There are a lot of formulas for this calculation such as C0 formula and the pi formula = (1+r-d)/(u-d)

Then the hedge ratio (C+ - C-)/(S+ - S-)

What’s the logic behind these formulas so that I can understand rather than rote memorization.

I believe for better and in depth understanding of the topics as the exam is not only focused for us to pass but also to develop the required set of skills and knowledge necessary for this profession.

Starting with the first one. That’s how I understand it. The formula tell you the probability of the up move. Do you remember continious distribution from the L1? Suppose we are in Time0. In T1 we are 100% sure that the stock will move to either to “d” or “u”. The interest rate is r, thus we might assume that without any market shocks the price will follow the interest rate path. The price in T1 will be 1+r. What is the probability of this move? It is the length of the segment relative to the lenght of all possible outcomes (100%), in other words 1+r-d / (u-d)

The second one the delta neutral hedging. The idea is that at any time period the value of the original portfolio stays constant. Assume you construct a portfolio in the following manner. You short the call option, and use the proceeds to buy stocks. Your risk is zero, your investment is zero. Your portfolio will look like this: P=n*S - C, where P=0. Assume that at T1 the price of the stock is S1 and the price of the call is C1. Thus, P1=n*S1 - C1. You want that the changes in prices do not impact the value of the initial portfolio. The question is, how many stocks there must be in your portfolio? Remember, that the objective is to keep the value of the protfolio constant, in consequence P=P1, in consequence n*S - C=nS1 - C1, resolving the equation for n, gives you n=(C1-C)/(S1-S).

With the put option it will be the same, just the DELTA (n, in my example) will be negative.