Tian trees simulation

The Tian Model

The moments

The computation of the first three moments in the Black and Scholes model gives

\[\mathbb{E}(S_{\Delta t})=S_{0}e^{r\Delta t},\] \[\mathbb{E}(S_{\Delta t}^2)=S_{0}^2e^{2r\Delta t}e^{\sigma^2\Delta t},\] \[\mathbb{E}(S_{\Delta t}^3)= S_0^3 e^{3rt + 3\sigma^2 t}.\]

On the other hand, on the binomial tree we get : \[\mathbb{E}(S_{\Delta t})=S_{0}(pu+(1-p)d),\] \[\mathbb{E}(S_{\Delta t}^2)=S_{0}^2(pu^2+(1-p)d^2),\] \[\mathbb{E}(S_{\Delta t}^3)=S_{0}^3(pu^3+(1-p)d^3).\]

The equations

By matching these moments, we get the following equations :

\[\mathbb{E}(S_{\Delta t}) =pu+(1-p)d=e^{r \Delta t },\] \[\mathbb{E}(S_{\Delta t}^2) =pu^2+(1-p)d^2=( e^{ r \Delta t } )^2 e^{\sigma^2 \Delta t },\] \[\mathbb{E}(S_{\Delta t}^3) =pu^3+(1-p)d^3=( e^{ r \Delta t } )^3 (e^{\sigma^2 \Delta t })^3,\]

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 : \[p=\frac{e^{r \Delta t } - d }{ u - d },\] \[u=\frac{1}{2} e^{r \Delta t} v ( v + 1 + \sqrt{v^2 + 2v -3} ),\] \[d=\frac{1}{2} e^{r \Delta t} v ( v + 1 - \sqrt{v^2 + 2v -3} ).\] \[v=e^{\sigma^2 \Delta t}\]

These three parameters define the model.

 Parameters

Here below we show the convergence of the Tian 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.