 # Equity options and risk free rate

“The risk-neutral probability of an up- and down-move in the interest rate tree is always 50%, unlike with equity options, where the probabilities depend on the risk-free rate and the size of the up- and down-moves”

Source: Derivatives, Schweser pg. 70

Is this because pricing options on interest rates and bond prices are valued using binomial trees only? I don’t understand the statement. Can someone please clarify? An example would also help. Thanks!

That part is poorly written, in my opinion. I think it has to do with some assumption that the drift for rates is non-biased. Or something. You can definitely construct a rates model with non-50% probabilities.

For now… just accept and don’t question, I guess.

I’ve looked into this further and I think I’ve figured out what the statement means.

Both the interest rate/bond prices and equity options use binomial trees. The difference is that for interest rates/bond prices there is only up or down move, thus ‘always 50%’. In contrast, equity options probabilities rely on this risk-neutral probability formula:

up move = (1+Rf-D) / (U-D)

down move = 1 - up move

So, the probability of non-50% is most definite, depending on the risk-free rate and the up/down move size.

Also, each node in the ‘interest rate/bond prices’ binomial tree is valued using backward induction methodology. In contrast, ‘equity options’ are valued using payoff at maturity and discounting them back using a risk-free rate.

Hello Trekker, The main reason is your last sentence “equity options’ are valued using payoff at maturity and discounting them back using a risk-free rate.”