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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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  • $\begingroup$ 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 $u/r$. $\endgroup$
    – nbbo2
    Commented May 8, 2023 at 6:05
  • $\begingroup$ 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. $\endgroup$
    – nbbo2
    Commented May 8, 2023 at 12:03
  • $\begingroup$ This makes sense, thanks. What made it click for me was computing the expected value after one period, so $E[S] = (puS + qdS) / r$. $\endgroup$
    – jds
    Commented May 9, 2023 at 11:06

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