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Introduction

When valuing options using the binomial approach, we assume the underlying option price follows a process where at given times the price can jump either up or down.

$S$$uS$$dS$$uuS$$udS$$ddS$

One way to think about what is happening is that we are approximating the terminal distribution of the price of the underlying at the maturity date of the option. At each node of this tree, the value of the option can be found using simple arbitrage arguments, it is possible to construct a portfolio by combining riskless borrowing and the risky asset to duplicate the payoff of an option, which means the price of the option is easy to find.

To value an option using this approach, we specify the number $n$ of periods to split the time to maturity $(T-t)$ into, and then calculate the option using a binomial tree with that number of steps.

Given $S,X,r,\sigma,T$ and the number of periods $n$, calculate

\begin{displaymath}\Delta t = \frac{T-t}{n} \end{displaymath}


\begin{displaymath}u = e^{\sigma\sqrt{\Delta t}} \end{displaymath}


\begin{displaymath}d = e^{-\sigma\sqrt{\Delta t}} \end{displaymath}

We also define the ``risk neutral probabilities''

\begin{displaymath}R = e^{r\Delta t} \end{displaymath}


\begin{displaymath}p = \frac{R-d}{u-d} \end{displaymath}

To find the option price, will ``roll backwards:'' At node $t$, calculate the call price as a function of the two possible outcomes at time $t+1$. For example, if there is one step,

$C_0$ $Cu=\max(0,S_u-X)$ $Cd=\max(0,S_d-X)$

find the call price at time 0 as

\begin{displaymath}C_0 = e^{-r}(pC_u+(1-p)C_d) \end{displaymath}

The original source for binomial option pricing was the paper by Cox et al. (1979). Good textbook discussions are in Cox and Rubinstein (1985) and Hull (1993).


next up previous contents index
Next: European Options. Up: Binomial option pricing. Previous: Binomial option pricing.   Contents   Index
Bernt Arne Odegaard
1999-09-09