Thus we have the following

Now if we have only  and  movement then the equation reduces to

Thus we should have an unique value of , say  such that the following holds true: . In fact it can be easily proved for the case when we have only two sates, i.e.,  and , then .

Furthermore combining we can easily see that

Thus we have

   and as one easily obtains

In general  and it reflects the risk premium example. In case , then we operate in the risk free environment and in that case  is termed as the risk-neutral probability measure. In general using this value of  (whether risk free or not is immaterial) we can obtain the discounted expected value of the future claim.