The 50-50 rule

Remember that in the previous discussion, we noted that $q_{u}+q_{d}=b_{0}=%% \frac{1}{1+r}$. If the two state prices are equal, it is easy to confirm that they have to be equal to

$$q_{u}=q_{d}=\frac{0.5}{1+r/2}$$.

Making an assumption as the one above might be really close to or quite far from the truth. Nevertheless, it is widely used by practitioners, and it is known as the fifty-fifty rule. Is has the advantage that no prior knowledge is required other than the short interest rate, in order to calculate the state prices and therefore all contigent claims. This feature makes this assumption extremely valuable when one want to value fixed income securities. It also implies that the risk neutral probabilities are both equal to $Q_{u}=Q_{d}=\frac{1}{2}$, which is not such a bad idea, as we are going to see later.