# The Jarrow-Rudd model

### The moments

The Jarrow-Rudd model is based on the log-transformation of the binomial model.

We set $$Y_{\Delta t}=\log(S_{\Delta t})$$. Due to the Black and Scholes model, we know that $$Y_{\Delta t}$$~$$\mathcal{N}(Y_{0}+\gamma \Delta t, \sigma^2 \Delta t)$$ and if we set $$\gamma=r-\frac{\sigma ^2}{2}$$, then we have $\mathbb{E}(Y_{\Delta t})=Y_{0}+\gamma \Delta t,$ $\mathbb{Var}(Y_{\Delta t}^2)=\sigma^2 \Delta t.$ On the other hand, if we set $$p = \frac{1}{2}$$ on the binomial tree, we get : $\mathbb{E}(Y_{\Delta t})=Y_{0}+\frac{\ln(ud)}{2},$ $\mathbb{Var}(Y_{\Delta t})=\frac{1}{4}(\ln(u/d))^2.$

### The equations

By matching these first two moments, we get the following equations: $p = \frac{1}{2},$ $p\ln (u) +(1-p)\ln (d) = \left( r- \frac{\sigma^2}{2} \right) \Delta t,$ $p(1-p)\left( \ln(u/d) \right)^2 = \sigma^2 \Delta t,$ where $$r$$ and $$\sigma$$ are the riskless interest rate and the volatility in the Black and Scholes model.

### Our unknowns

By solving these three equations, we finally get : $p=\frac{1}{2},$ $u=e^{\left(r - \sigma^2/2 \right) \Delta t + \sigma \sqrt{\Delta t}},$ $d=e^{\left(r - \sigma^2/2 \right) \Delta t - \sigma \sqrt{\Delta t}}.$

These three parameters define the model.

### Parameters

Here below we show the convergence of the Jarrow-Rudd 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.