Risk-neutral probability

CFA Level II Derivatives
#1

Hi everybody,

I am trying to grasp the logic/not the meaning behind Risk neutral probability equation. Especially regarding the numerator: 1+r-d. How can we substract a variation expressed in decimal form from an interest rate.

Thanks.

#2

If Rf = 5% and D = .87, it’s just 1.05 - .87 = .18 if U = 1.15, then divide by 1.15-.87 = .28 and Pi(U) = .64 or 64% (.18/.28).

It turns out that that’s what the answer has to be to avoid arbitrage opportunities. That’s the logic behind it.

Think of Rf as the up (and down) move on a risk free instrument. Because it’s risk free, it’s the same for the up and down moves. For the risky instrument, the U and D are different. (you could think of them as the “interest” earnings on the risky instrucment under different future scenarios; or more correctly, as the total return - and interest on the Rf is total return as well)

I’m not sure if that addresses your concern with “how can we subtract …”

#3

Thanks a lot Candidcam!

You are a master! Indeed you can consider a favorable up move (for a call) like an opportunity of earning money, the inverse being true for down move. Then it makes sense to add (substract here) with interest rate.

If I well understood…

Thanks again.

#4

I’m not sure that I understand the problem. An interest rate can be expressed as a percentage or as a decimal, and they’re equivalent.

Suppose that:

  • u = 1.2
  • d = 0.95
  • r = 4%

Then the upper probability is:

π = (1 + r − d) / (u − d) = ( 1 + 4% − 0.95) / (1.2 − 0.95) = (1.04 − 0.95) / 0.25 = 0.09 / 0.25 = 0.36.

The lower probability is:

1 − π = 1 – 0.36 = 0.64.

#5

Thanks S2000magician

I just didn’t get the fact that we substract from an interest rate a down move in the numerator. It was like mixing things not related with each other.

I finally grasped it I guess.

Have a nice day.

#6

This is nothing more than linear interpolation: you have the up factor and the down factor and you’re trying to find the weighted average that gives you the risk-free growth factor. What you’re calculating aren’t probabilities, their stupid name notwithstanding; what you’re calculating are weights.

By the way, there is no reason that the “down” factor need be less than 1, nor that the “up” factor be greater than 1. The calculations will work even if both the up and down factors are greater than 1 or if both are less than 1.

#7

Thanks a lot S2000magician.

#8

D’accord, Flo.