Physical and Risk-Neutral Probabilities in a Binomial Model
Summary
The document clarifies why a binomial stock model can use different probabilities for the same up and down outcomes. The random variable’s possible values, represented by the up and down factors, remain fixed when moving between probability measures. What changes is the weight assigned to each outcome: physical probabilities describe beliefs or observed frequencies under the real-world measure, while risk-neutral probabilities are used for pricing under the martingale measure.
The answer notes that the up and down factors determine the risk-neutral weights under the model’s pricing conditions, while the physical weights need not match them. It also says the factors are supplied initially but may be calibrated to market prices. This is a conceptual explanation rather than a full derivation; it does not specify the calibration procedure or discuss the conditions required for an arbitrage-free model.
Key ideas
- A probability measure changes the likelihood assigned to outcomes, not the outcomes themselves.
- Physical and risk-neutral measures can assign different probabilities to the same up and down states.
- Risk-neutral probabilities support pricing and are constrained by the model’s market inputs.
- The document notes that binomial up and down factors may be calibrated to market prices.
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# Binomial model in Björk's Arbitrage Theory in Continuous Time
# Binomial model in Björk's Arbitrage Theory in Continuous Time
I am having some trouble with variable $Z$ introduced in chapter $2$ in Björk's text. In the beginning, it is the random variable that attains $u$ resp. $d$ with probabilities $p_{1}$ and $p_{2}$, i.e., the actual probabilities of the stock going up or down.
Later on, however, it seems that he assumes that $Z$ attains $u$ resp. $d$ with probabilities related to the martingale measure $\mathbb{Q}$.
From what I can see, these do not have to be the same. In fact, $u$ and $d$ determine $q_{1}$ and $q_{2}$, but $u$ and $d$ are fixed from the beginning and, thus, can not be chosen freely.
Is this observation correct? If it is correct, can someone explain why this is reasonable?
## Answer by Magic is in the chain (score 1)
https://quant.stackexchange.com/a/47571
It essentially boils down to: same random variable, different probability measures. So when you set u and d, you fix the values that the random variable can take. Probability Measure does not change that- it only re-weights the probability in a way.
The probability $p_1$ and $p_2$ are the probabilities of the two states under the P(physical) measure, and these same states have probabilities $q_1$ and $q_2$ under the Q(risk neutral) measure.
In the textbooks, when the binomial model is introduced, u and d are assumed to be given, but these will need to be calibrated based on the market prices, just as one would calibrate the volatility of the geometric brownian process as in the Black scholes model.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.