Binomial trees simulation

The binomial pricing model

We aim at evaluating the price of an option with underlying \(S_t\), maturity \(T\), strike price \(K\) and payoff \(\varphi\), which can be path dependant or not.

The binomial pricing model traces the evolution of the option's price in discrete-time, under the risk-neutral measure, which is the measure under which the discounted price process is a martingale.

First of all we divide the time interval [0,T] into \(N\) discrete periods of lenght \(\Delta t\). At each time step, we assume that the underlying price will move up or down by a specific factor (\(u\) or \(d\), by definition, \(u>1\) and \(d<1\) ). If \(S_0\) is the spot value at time \(0 = t_0\), then at time \(t_1\) the price will either be \(S_{up}=S_0u\) or \(S_{down}=S_0d\).

If we compute all the possible values of the underlying \(S_t\) moving forward from time \(0\) to expiration time \(T\), we obtain a tree where each node represents a possible value of \(S_t\).

We compute the payoff at the final nodes, which corresponds to compute the possible values of \(C_T\), the option price at time \(T\).

We can now move backward through the tree, computing at each time step the discounted expected value of \(C_t\) with the formulas :

  • For an European Option :
    \(C_{t-\Delta t,i}=e^{-r\Delta t}(pC_{t,i+1}+(1-p)C_{t,i-1})\)
  • For an American Option :
    \(C_{t-\Delta t,i}=\max( {\varphi(S_{t-\Delta t})} , {e^{-r\Delta t}\mathbb{E}(C_{t,i} | C_{t-\Delta t})} )\)

A model will be determined by the choice of \(p\), \(u\) and \(d\), which are usually realized in order to match the expectation and the variance of the asset's price.

There are many different approaches for the computation of these values, here we present some of them.

Cox-Ross-Rubinstein model

Jarrow-Rudd model

Tian model

Trigeorgis model

We can compare these models in an unique graph.

Model comparison