Equivalent Measures in Binomial Risk-Neutral Pricing
Summary
The document resolves an apparent contradiction in a binomial stock model: if one next-period outcome has zero probability in the real-world model, why should its price affect today’s stock price under risk-neutral valuation? The answer is that standard no-arbitrage pricing requires the physical probability measure and the risk-neutral measure to be equivalent. Equivalence means they assign zero probability to the same events. A branch that is impossible under the physical measure therefore cannot receive positive risk-neutral probability.
If one branch has probability zero, the setup must be treated as a one-outcome case for pricing, with the current price equal to the discounted value of the certain future stock price. The corresponding risk-neutral probability of that outcome is one. The explanation is concise and relies on the equivalence assumption used in the standard framework; it does not develop a general option-pricing derivation or discuss models where that assumption is relaxed.
Key ideas
- Equivalent physical and risk-neutral measures assign zero probability to the same events.
- A zero-probability branch under the real-world measure cannot have positive risk-neutral probability in the standard framework.
- With only one possible future stock price, the current price is its discounted value.
- The result depends on the equivalence assumption in no-arbitrage pricing.
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Full text
# What happens in the binomial model if the real-world probability is $0$
# What happens in the binomial model if the real-world probability is $0$
Consider a binomial model.
Suppose we know that the price of a stock will become a certain value at the next timestep. That is, one of the two outcomes has $0$ real-world probability.
Then it should not matter what the price of the stock is in that outcome, but indeed that does affect the price of today, through the risk neutral pricing formula.
How to resolve this (apparent) paradox?
## Answer by AdB (score 3, accepted)
https://quant.stackexchange.com/a/45001
If I understand your question correctly, another way to word it is: if an event that has probability 0 under the physical measure $\mathbb{P}$, how can it have a positive probability under the risk-neutral measure $\mathbb{Q}$?
The answer is simply: it cannot! According to the theory of risk-neutral pricing through no arbitrage arguments, we require that $\mathbb{Q}$ and $\mathbb{P}$ are equivalent measures. Simply put, if an event happens with probability 0 under one measure, it must also happen with probability 0 under the other.
Introducing this restriction, it is clear that the situation you describe violates this property (unless the risk-neutral probability is also 0 in one of the events, in which case the measures are equivalent and the result is correct!)
Edit: When it comes to the impact of the price of the stock, there is now only one price it can have, i.e. the disounted price of whatever its price will be with probability 1. Note that this is exactly the price that results in a risk-neutral probability of 1.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.