Risk-Neutral Probabilities in Binomial Option Pricing

Summary

The document corrects the claim that a European option’s binomial price does not depend on probability. It distinguishes physical probabilities, which describe actual beliefs about market moves, from risk-neutral probabilities, which are implied by the model’s no-arbitrage pricing setup. In a one-period binomial model, the risk-neutral up probability is determined by the risk-free return and the up and down factors.

The option value is the discounted expected payoff under that risk-neutral probability. Although the payoff-weighted pricing expression can be rearranged so the probability is less visually obvious, risk-neutral probability remains embedded in the price. The example names a particular stock value and move size, but the response does not work through the numerical calculation or state assumptions beyond the one-period European model. It also does not explore extensions such as dividends, multiple periods, or early exercise.

Key ideas

• Physical probabilities describe real-world move likelihoods, while risk-neutral probabilities are used for no-arbitrage pricing.
• In a one-period binomial model, the risk-neutral up probability depends on the risk-free return and the up and down factors.
• The option value is the discounted expected payoff using risk-neutral probabilities.
• Rearranging the pricing formula can obscure the probability dependence without removing it.

Tags

# Why does option pricing not depend on probabilities in a binomial tree style valuation


# Why does option pricing not depend on probabilities in a binomial tree style valuation












I am new into learning option pricing and read that option pricing using binomial valuation does not depend on probabilities (real or risk neutral).

Example:

A 1 period binomial tree with $u = 1/d = 1.07$ and $S_0$ = $100.

If the up-move probability $u$ is 0.99 or 0.01, I read that the call option prices is the same and is equal to $4.8. They assumed European style options.

Can someone please help and guide me? Maybe dinner tutorial that explains this? Thanks.

## Answer by fni (score 0)

https://quant.stackexchange.com/a/34771

What you say is false. In general, the price of an option does not depend on physical probabilities but does depend on risk neutral probabilities. In your case, in a one-period binomial tree model, the probability of going up is $q=\frac{(1+r)-d}{u-d}$ therefore the price of the option paying $C_u$ and $C_d$ is $$C=\frac{1}{1+r}\left(qC_u+(1-q)C_d\right)=\frac{1}{1+r}\left(\frac{(1+r)-d}{u-d}C_u+\frac{u-(1+r)}{u-d}C_d\right)$$ If you focus on the second part of the equation it looks like the price does not depend on probability, but as the first part shows it does indeed depend on risk-neutral probabilities!

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.