# 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.