The General Formula to Back Out The Risk-Neutral Probability

I’ll be repetitive because it’s good practice.
Here’s a simple game.
I ask you to reach into a bag with 4 balls.
  • If you pull out a green ball, I pay you $3
  • If you pull any other color, you pay me $1
Assume the game is fair (ie has an expectancy of zero).
What’s the probability of pulling a green ball?
25% I’m offering you 3-1 odds so the implied probability is 25%
Common sense proof
You play the game 4x
  • 3x you pay me $1
  • 1x you get paid $3 You break even when green shows up 25% of the time
General tactic: converting odds to probability
Simply divide the odds you are getting by the total number of possibilities.
In this case, you are “getting” 3:
  • There are 3+1 or 4 total possibilities.
  • 3/4 = 75% But since you are “getting odds” meaning you are risking less than you are being offered, then you must be the underdog. So you have 25% (ie 1-75%) chance of winning.
Try another…
If you are “laying” or “giving” 5-3 odds then you are implying that you are a 62.5% favorite to win the bet.
  • There are 8 possibilities (5+3) and since you are risking more than you receive you must be the favorite. You must be winning 5/8 or 62.5% of the time for this bet to be fair.
The risk-neutral or arbitrage-free probability is the one that makes a proposition fair to just receiving the risk-free rate on your money.
In the above example, the bet is settled immediately so the risk-free rate can be ignored.
But for other random processes that occur over time, like the unfolding of a stock price path, we want to generalize the formula to account for the risk-free rate. We are effectively discounting the payoffs to present value where all comparisons are apples-to-apples and then implying the probabilities.
More practice
Suppose a stock can return only 2 outcomes over the next year:
  • Up 20%
  • Down 40%
We will denote the probability it goes up as p*
  1. Assume the risk-free rate is 0%
    1. What is p*?
      1. You are risking 40% to make 20% or “laying” 2-1 odds.
      1. It’s clear the stock must be a favorite to go up 2/(2+1) = 66.7% p* or the probability of the stock going up must be 66.7% if it’s fairly priced
  1. Assume the risk-free rate is 10%
    1. What is p*?
      Again, it’s clear the stock must be a favorite to go up.

      But this seems trickier.

      Nominally you are risking 40% to make 20% but if you just invest in T-bills, you earn 10% without risk.
      In this 10% risk-free rate regime, the probability of the stock going up (ie p*) needs to be higher and it was to justify the same stock price as the 0% world.
      We can compute that and we will below, but let’s just step back for a moment. This lines up with our real-world intuition:
      With interest rates at 10%, a stock price must be lower, ie offer a better risk/reward, to justify the same up and down probabilities.
In our complex reality, prices can react in several ways when rates change:
  • Payoffs change If the stock price falls, we might say the up/down probabilities are unchanged but the terminal prices possibilities are unchanged reflecting a more attractive payoff to account for the opportunity cost of 10% interest rates
  • The stock price stays the same P* must have increased
The reality will be complex — the price will change and it will be difficult to decompose the change since stocks usually don’t have 2 possible payoffs and changes in interest rates have differing effects on businesses.
The point is, that changing the RFR has a mechanical effect on what’s implied — prices will change and p* in our simple models will also change.
The General Formula to Back Out The Risk-Neutral Probability
The formula rests on 2 principles we have established:
  1. The probability is set by the odds to make a fair bet
  1. A fair bet is benchmarked to the RFR
This outstanding video I found on YT derives the formula:
Video preview
I’ll walk you through the algebra
u = size of the up move
d = size of the down move
p* = probability of up move
1-p* = probability of down move
r = risk-free rate
S = stock price
The identity relating fair expectancy to the risk-free rate:

“Future value of the stock” = Expected Value

Rearranging to solve for p*
p*+p*u +1 + d - P* - p*d = 1+r
P*u + d - P*d = r
The intuition in the formulas is familiar:
If the down move represents a large proportion of the total range then the probability of the up move must be large
The example from earlier with RFR = 10%
r = 10%
u = 20%
d = 40%
p* = [10% - (-40%)] / [20% - (-40%)]
p* = 83.3%
In a 10% risk-free rate world, if the stock stayed the same price while maintaining the same payoff profile, the implied probability of it increasing is now 83.3%