Deriving the Black–Scholes Limit from Binomial Probabilities
Summary
The document explains a step in deriving the Black–Scholes formula from a binomial tree. The number of upward moves is modeled as a random variable, and the derivation uses its binomial distribution, with mean equal to the number of trials times the up-move probability and standard deviation based on the corresponding binomial variance. Standardizing the variable maps it to a normal approximation.
The answer interprets the cumulative normal function as a probability below a threshold, then uses symmetry to express the probability of exceeding that threshold. This produces the stated normal-CDF term for the relevant binomial probability. The explanation addresses the probability transformation but does not show every step of the option-pricing derivation or define all tree parameters. It also notes that binomial prices approach the Black–Scholes price only as the number of trials becomes sufficiently large; the approximation’s accuracy therefore depends on tree resolution.
Key ideas
- The count of upward moves in a binomial tree is treated as a random variable.
- Its mean and standard deviation are used to standardize the move count for a normal approximation.
- The cumulative normal function gives probabilities below a threshold, while symmetry gives the corresponding upper-tail probability.
- The binomial model converges toward the Black–Scholes price as the number of trials grows sufficiently large.
- The probability argument explains one derivation step, not the entire option-pricing formula.
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# Black & Scholes formula derivation from a Binomial Tree - John C. Hull
# Black & Scholes formula derivation from a Binomial Tree - John C. Hull
I am reading "Option, Futures and other Derivatives" by John C. Hull, and on Appendix chapter 13, he derives BSM formula from a Binomial Tree.
When he builds U2, I just don't understood how to get equation 13A.5.
Which method is being used to get it, and why "a", that is equal to "J" that is defined as quantity of upward movements, is being probably used as a random variable?
Regards,
## Answer by David Duarte (score 3, accepted)
https://quant.stackexchange.com/a/54920
In the Black Scholes formula the $N(\alpha)$ gives you cumulative probability, i.e, the probability of a randomly selected occurence being below $\alpha$.
To transform the distribution of your variable into the standard normal you subtract the mean and divide by the standard deviation. It is said in the paragraph preceding formula 13A.5 that the mean is $np$ and the standard deviation is $\sqrt{np(1-p)}$.
So:
- $N(\alpha)$ gives you the cumulative probabily of $\alpha$ in the normal distribution, i.e, probability of a random selection being below $\alpha$
- $N\left(\frac{\alpha - np}{\sqrt{np(1-p)}}\right)$ gives you the cumulative probabily of $\alpha$ in the standard normal distribution
But because what you want is the probability of the random selection being above $\alpha$ (and not below), i.e. $1 - N(x)$, you can use the fact that the normal distribution is symetric and just use $N(-x)$.
Applying this logic to the case above would give you what you want:
$U_2 = N\left(-\frac{\alpha - np}{\sqrt{np(1-p)}}\right) = N\left(\frac{np - \alpha}{\sqrt{np(1-p)}}\right)$
Also, be aware that the price of the binomial model will only converge to the Black Scholes price for a sufficiently larget number of trials.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.