The book introduces the binomial option pricing model in which two outcomes are possible for each timestep, a move up to u or a move down to d, where u is (1+ru) and d is (1+rd) where ru and rd are the percent changes to the asset. So, if an asset can increase in price by 25% or dcrease by 20% then u and d are 1.25 and 0.8 respectively.
then pi is introduces as the ‘risk-neutral probability’ of the up move, (1+risk free - d)/(u-d)
I’m a little fuzzy on this ‘risk neutral probability’ concept. Lets assume we invest an asset into a risk free instrument, such that u = (1+risk free). I guess d could equal anything but there would be zero actual probability of d being the value at time 1. How do these actual probabilities tie into the risk-neutral probability?
That makes more sense - both from the derivation and qualitatively. Now, as the magnitude of the down move increases (so d gets closer to 0) the value of pi (the weight on the up-move option price) increases. Likewise, as the magnitude of the up move increases, the value of pi decreases.
So if pi is the weight on the up move it means that the weight of the up move is inversely proportionate to its magnitude. Likewise with the down move. I’m trying to digest that. The book does a little hocus pocus to go from one equation to the other to derive pi…
Is it an implied assumption of the binomial model that the “up move” and the “down move” occur with equal probability?
OR
Are the moves up and down weighted such that the probability of a percentage gain is the same as the probability of the same percentage loss? This would make sense for the weights to be inversely proportional to the magnitude of the move.
So it does in fact seem that pi=0.5 if u and d are equally spaced above and below the risk-free rate. So my assumption would be the term “risk-neutral probability” comes from teh fact that the weighting factors are set such that the probability of an X% spread above the risk-free rate is equivalent to an X% spread below the risk free rate.
Still spinning my wheels on the derivation of pi, though. Equations provided:
Admittedly, I only spent about 15-20 min on this before moving on. I apologize for being lazy but anyone who can shed light on the derivation, it would be much appreciated.
Also - thank you for your response yesterday Magic - as always I am extremely grateful for your assistance.
risk neutral probability is a pure mathematical concept used in derivative pricing. it is “risk neutral” because the expected return (based on risk neutral probability applied to up/down payoffs in the case of a binomial model and the price of the underlying) of the risky underlying asset will be the risk-free rate.