Simple Binomial Question - given chance of up move

[cwright]

I think this is a simple question but just wanted to verify... If a question says, "John thinks the probability of an up move at any point is 50%", does this ever go into the solution? or should we always calculate the risk neutral probability and use it instead?

 

[David Harper CFA FRM]

Hi cwright,

A great point. Without getting into the nitty gritty, GARP basically follows correct theory and ignores something like "John thinks the probability of an up move at any point is 50%" it is assumed to mean "John thinks the real-world probability of an up move at any point is 50%." And GARP almost always asks instead for the risk-neutral probability which is the (p) in our formulas, consider the question in our FR7 and from 2012:

GARP 2012.P1.7 (blue add and emphasis is mine) "A risk manager for Bank XYZ, Mark, is considering writing a 6-month American put option on a non-dividend-paying stock ABC. The current stock price is USD 50 and the strike price of the option is USD 52. In order to find the no-arbitrage price of the option Mark uses a two-step binomial tree model. The stock price can go up or down by 20% each period. Mark’s view is that the stock price has an 80% [real-world] probability of going up each period and a 20% probability of going down. The annual risk-free rate is 12% with continuous compounding. What is the risk-neutral probability of the stock price going up in a single step?"

It is maybe hard to decipher given the redundancy of the 20%, but this entire sentence is an unused red herring: "Mark’s view is that the stock price has an 80% probability of going up each period and a 20% probability of going down"

because:

• Mark's 80% is presumed to be a real-world probability
• but, as GARP almost always does, the question specifically asks for (p), the risk-neutral probability, which is given by (exp[12%*0.25]-0.8)/(1.2-0.8) = 57.61%

i hope that helps,