Risk-Neutral Probabilities in the Binomial Call Pricing Formula

Summary

The document asks why the binomial call pricing formula uses two probabilities: the risk-neutral up-move probability p and an adjusted probability p′ equal to (u/r)p. In the formula, the term weighted by p′ contributes the stock component, while the strike payment is discounted and weighted by p. The question sets up the standard no-arbitrage condition that the gross risk-free return lies between the up and down factors, and defines p from those quantities.

The document does not include an answer, so it does not explain the derivation or interpretation of p′. In the usual binomial valuation, the adjustment arises when the stock-value contribution is expressed as a probability-weighted sum: multiplying the risk-neutral probability by the up factor and dividing by the gross risk-free return gives the adjusted weight. The note is therefore a useful prompt about option-pricing mechanics, but readers need the derivation to understand the full formula and assumptions.

Key ideas

• The binomial call formula uses p for the discounted strike term and p′ for the stock term.
• The risk-neutral up probability is determined by the up and down factors and the gross risk-free return.
• p′ adjusts p by the up factor and gross risk-free return.
• The document poses the interpretation question but does not provide its answer.

Tags

# Understanding the adjustment $(u/r) p$ in the binomial options pricing formula


# Understanding the adjustment $(u/r) p$ in the binomial options pricing formula












I'm reading Option Pricing: A Simplified Approach and have a question. Assume the binomial tree model for the stock. So

- $n$ discrete time periods

- $S$ is stock

- $C$ is call

- $K$ is strike

- $u$ is upward move

- $d$ is downward move

- $r$ is total return $1+R$ where $R$ is the interest rate

- no arb: $d < r < u$

- $q$ is probability of upward move

- $p = (r-d) / (u-d)$ is risk-neutral probability of upward move

Fine so far. But then Cox defines $p^{\prime}$ as $(u/r) p$. And I don't understand what this parameter represents. Concretely, the binomial options pricing formula for a call $C$ is

$$ C = S \Phi(a; n, p^{\prime}) - K r^{-n} \Phi(a; n, p) $$

where $a$ is the min moves for the call to be in-the-money and $\Phi$ is the complementary binomial distribution function, i.e. 1 minus the CDF. I can't make sense of $p^{\prime}$ though. Why does this adjustment factor fall out of the model? What does it represent?

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