# 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?

• Intuitively when the option expires ITM (which happens with probability $p$) you will not receive stock worth S (the current stock price) but worth an amount $S' > S$ and you will be in a situation where the stock has appreciated more than expected by the factor $u/r$. Commented May 8, 2023 at 6:05
• You could handle the situation by using the enhanced price $S′$ in the call price formula, instead Cox uses $S$ and handles it by using an "enhanced probability" $p′$. What matters is the product of the two, so both methods give the same result. Commented May 8, 2023 at 12:03
• This makes sense, thanks. What made it click for me was computing the expected value after one period, so $E[S] = (puS + qdS) / r$.
– jds
Commented May 9, 2023 at 11:06