In the previous section, our compounding frequency could be annual (m=1), semiannual (2), quarterly (4), monthly (12), or daily (365). If the compounding frequency increases further and further, such as by the hour, minute, and second, the limit is called continuously compounded. The following is the conversion formula:

Here, Rc is the continuously compounded rate, ln() is a natural log function, APR is the annual percentage rate, and m is the compounding frequency per year. For the natural log function, refer to the following code:

>>>import math
>>>math.e
2.718281828459045
>>>math.log(math.e)
1.0

For example, if a given APR of 5 percent is compounded semiannually, its corresponding continuously compounded rate will be 4.9385225 percent, as shown in the following code:

>>>import math
>>>2*math.log(1+0.05/2)
0.04938522518074283

In the next chapter, for a call option, the risk-free rate used is continuously...