The Cox-Rox-Rubinstein model

The moments

The computation of the first and second moment in the Black and Scholes model gives $\mathbb{E}(S_{\Delta t})=S_{0}e^{r\Delta t},$ $\mathbb{E}(S_{\Delta t}^2)=S_{0}^2e^{2r\Delta t}e^{\sigma^2\Delta t}.$

On the other hand, on the binomial tree we get : $\mathbb{E}(S_{\Delta t})=S_{0}(pu+(1-p)d),$ $\mathbb{E}(S_{\Delta t}^2)=S_{0}^2(pu^2+(1-p)d^2).$

The equations

By matching these moments and setting $$u = 1/d$$, we get the following equations, which caracterize the Cox-Rox-Rubinstein model :

$u=\frac{1}{d},$ $pu+(1-p)d=e^{r\Delta t},$ $pu^2+(1-p)d^2-e^{2r\Delta t}=\sigma ^2\Delta t,$ where $$r$$ and $$\sigma$$ are the riskless interest rate and the volatility of the Black and Scholes model.

Our unknowns

By solving this three equations, we finally get : $p=\frac{e^{r\Delta t}-d}{u-d},$ $u=e^{\sigma \sqrt{\Delta t}},$ $d=e^{-\sigma \sqrt{\Delta t}}.$

These three parameters define the model.

Parameters

Here below we show the convergence of the Cox-Ross-Rubinstein binomial model.

In the first resulting graph, we compute the price of the option with the binomial tree, with a time step size varying between $$N_{min}$$ and $$N_{max}$$. We compare this price to the analytical and semi-analytical solutions, computed with Quantlib library.

We search also for the first time steps size $$N \in [N_{min}, N_{max}]$$ such that the error between the analytical and the computed solution is smaller than a fixed $$\varepsilon$$. $$N$$ can be smaller of $$N_{min}$$ or greater than $$N_{max}$$.

In the second graph, we show the optimal exercise time, with respect to the number of steps of the tree.

The last graph shows the construction of a reference tree of size $$N_{ref}$$. The nodes represent the possible option values at each time.