A risky business: how to price derivatives

A general formula for the multi-period case

The price for the option in an $n$ period model is given by $$C=\frac{1}{(1+r{)}^n}\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})q^k(1-q{)}^{n-k}{y}_{u^k{d}^{n-k}}.$$ Here $(\genfrac{}{}{0ex}{}{n}{k})$ denotes the number of ways in which one can choose $k$ objects from a selection of $n$ objects (called the binomial coefficient — you can read more in the Plus article Making the grade: Part II). Explicitly it is given by $$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-k)!},$$ where $$n!=n\times (n-1)\times (n-2)\times ...\times 2\times 1.$$ The symbol $y_{u^k{d}^{n-k}}$ stands for $y$ with a subscript consisting of $k$ $u^{\prime }s$ and $n-k$ $d^{\prime }s$ - these stand for the payoffs corresponding to the various combinations of good and bad periods.\ The expression $$\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})q^k(1-q{)}^{n-k}{y}_{u^k{d}^{n-k}}$$ means that you should sum the terms of the form $$(\genfrac{}{}{0ex}{}{n}{k})q^k(1-q{)}^{n-k}{y}_{u^k{d}^{n-k}}$$ in turn with $k$ substituted by 0, 1, 2, etc, up to $n$.