Why the Binomial Option Model Uses Discrete Delta

# Is the binomial model wrong?


# Is the binomial model wrong?












In the standard MBA one-period binomial model, the value of an option is

$v = \frac{1}{R}\bigl(\frac{u - R}{u - d}V(sd) + \frac{R - d}{u - d}V(su)\bigr)$

where $R$ is the realized return over the period and the stock goes from $s$ down to $sd$ or up to $su$, where $d\lt R\lt u$, and $V$ is the option payoff. Note

$\frac{dv}{ds} = \frac{1}{R}\bigl(\frac{u - R}{u - d}V'(sd)d + \frac{R - d}{u - d}V'(su)u\bigr)$

is the "delta" hedge. Suppose $V$ is a call spread consisting of long a call struck at slightly higher than $sd$ and short a call struck slightly lower than $su$, then $V'(sd) = V'(su) = 0$, hence $dv/ds = 0$.

Wat?! How can that be???

## Answer by mepuzza (score 1)

https://quant.stackexchange.com/a/3460

Binomial model is just a model, and a rather simplistic one. Think about it, anywhere you put the strikes of the call spread, provided that, as you did, they are between su and sd, you get the same value! Also, to be fair to the binomial model, the delta hedge in the binomial model is not defined as the derivative of the value with respect to s, but rather as

(V(su)-V(sd))/(su-sd)

## Answer by Vijay (score 0)

https://quant.stackexchange.com/a/4746

OK, I'm going to be slightly facetious but I hope marginally helpful, by quoting legendary Prof. George E. P. Box: "all models are wrong, but some are useful."

So, yes, certainly, the binomial model is wrong. It is also incredibly useful!

Where you are going wrong is using it as a continuous model (and using differential calculus) when it is intended to be a simplified discrete model of the world. The only changes that matter in this discrete model are discrete changes. If you look at everything as differences in value between t to t+1 then it all makes sense. Anything between t and t+1 (including at time t+$\delta$t) is undefined.