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.