Trigeorgis trees simulation

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.