# The Trigeorgis Model :

### The moments

The Trigeorgis 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 $$\gamma=r-\frac{\sigma ^2}{2}$$, the first two moments read

$\mathbb{E}(Y_{\Delta t})=Y_{0}+\gamma \Delta t,$ $\mathbb{E}(Y_{\Delta t}^2)=Y_{0}^2+(\sigma ^2 + 2\gamma Y_{0} )\Delta t+ \gamma^2\Delta t^2,$

On the other hand, if we set $$u=1/d$$ (which gives $$\log d=- \log u$$) on our binomial tree, we get : $\mathbb{E}(Y_{\Delta t})=Y_{0}+(2p-1)(\log u),$ $\mathbb{E}(Y_{\Delta t}^2)=Y_{0}^2+2(2p-1)(\log u)Y_{0}+(\log u)^2.$

### The equations

By matching these first two moments, we get the following equations: $u=\frac{1}{d},$ $\gamma \Delta t=(2p-1)(\log u),$ $(\sigma ^2 + 2\gamma )\Delta t+ \gamma^2\Delta t^2=2(2p-1)(\log u)Y_{0}+(\log u)^2,$

where $$r$$ and $$\sigma$$ are the riskless interest rate and the volatility in the Black-Scholes model.

### Our unknowns

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

These three parameters define the model.

### Parameters

Here below we show the convergence of the Trigeorgis 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.