The General Formula to Back Out The Risk-Neutral Probability
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The General Formula to Back Out The Risk-Neutral Probability
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Iāll be repetitive because itās good practice.
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Hereās a simple game.
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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
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Assume the game is fair (ie has an expectancy of zero).
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Whatās the probability of pulling a green ball?
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Answer
25%
Iām offering you 3-1 odds so the implied probability is 25%
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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.
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In this case, you are āgettingā 3:
There are 3+1 or 4 total possibilities.
3/4 = 75%
But since you are āgetting oddsā 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.
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Try anotherā¦
If you are ālayingā or āgivingā 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.
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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.
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In the above example, the bet is settled immediately so the risk-free rate can be ignored.
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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.
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More practice
Suppose a stock can return only 2 outcomes over the next year:
Up 20%
Down 40%
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We will denote the probability it goes up as p*
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Assume the risk-free rate is 0%
What is p*?
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Answer
You are risking 40% to make 20% or ālayingā 2-1 odds.
Itās 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 itās fairly priced
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Assume the risk-free rate is 10%
What is p*?
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Again, itās 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.
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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.
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We can compute that and we will below, but letās just step back for a moment. This lines up with our real-world intuition:
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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.
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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
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The reality will be complex ā the price will change and it will be difficult to decompose the change since stocks usually donāt have 2 possible payoffs and changes in interest rates have differing effects on businesses.
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The point is, that changing the RFR has a mechanical effect on whatās implied ā prices will change and p* in our simple models will also change.
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The General Formula to Back Out The Risk-Neutral Probability
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The formula rests on 2 principles we have established:
The probability is set by the odds to make a fair bet
A fair bet is benchmarked to the RFR
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This outstanding video I found on YT derives the formula:
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Iāll walk you through the algebra
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Definitions:
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:
āFuture value of the stockā = 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:
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If the down move represents a large proportion of the total range then the probability of the up move must be large
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The example from earlier with RFR = 10%
r = 10%
u = 20%
d = 40%
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p* = [10% - (-40%)] / [20% - (-40%)]
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p* = 83.3%
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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%