8. Binomial Model for European Options

There exists a big zoo of options of which the boundary conditions of the Black-Scholes differential equation are too complex to solve analytically. An example is the American option. For this reason one has to rely on numerical price computation. The best known methods approximate the stock price process by a discrete time stochastic process, or, as in the approach followed by Cox, Ross, Rubinstein, model the stock price process as a discrete time process from the start. By doing this, the options time to maturity $ T$ is decomposed into $ n$ equidistant time steps of length

$\displaystyle \Delta t=\frac {T}n.$

We consider therefore the discrete time points

$\displaystyle t_j=j\Delta t , \; j=0,...,n .$

By $ S_j$ we denote the stock price at time $ t_j.$ At the same time, we discretize the set of values the stock price can take such that it takes on the finitely many values $ S_j^k ,
k=1,...,m_j ,$ with $ j$ denoting the point of time and $ k$ representing the value. If the stock price is in time $ t_j$ equal to $ S_j^k,$ then it can jump in the next time step to one of $ m_{j+1}$ new states $ S_{j+1}^l,~l=1,...,m_{j+1} .$ The probabilities associated to these movements are denoted by $ p_{kl}^j$:

$\displaystyle p_{kl}^j = \P( S_{j+1} = S_{j+1}^l \vert S_j = S_j^k ) ,$

with

$\displaystyle \sum_{l=1}^{m_{j+1}} p_{kl}^j = 1 , \quad 0\leq p_{kl}^j \leq 1 .$

If we know the stock price at the current time, we can build up a tree of possible stock prices up to a certain point of time, for example the maturity date $ T=t_n.$ Such a tree is also called stock price tree. Should the option price be known at the final point of time $ t_n$ of the stock price tree, for example by means of the options intrinsic value, the option value at time $ t_{n-1}$ can be computed (according to (6.24)) as the discounted conditional expectation of the corresponding option prices at time $ t_n$ given the stock price at time $ t_{n-1}:$
$\displaystyle V(S_{n-1}^k,t_{n-1})$ $\displaystyle =$ $\displaystyle e^{-r \Delta t} {\mathop{\text{\rm\sf E}}} \{ V(S_n,t_n) \, \vert \, S_{n-1} = S_{n-1}^k \}$  
  $\displaystyle =$ $\displaystyle e^{-r \Delta t} \sum_{l=1}^{m_n}p_{kl}^{n-1}V(S_n^l,t_n).$ (8.1)

$ V(S,t)$ again denotes the option value at time $ t$ if the underlying has a price of $ S.$ Repeating this step for the remaining time steps $ t_j,~j=n-2,n-3,...,0,$ the option prices up to time $ t=0$ can be approximated.