A general formula for the multi-period case

The price for the option in an $n$ period model is given by $$C=\frac{1}{(1+r)^n}\sum_{k=0}^n {n \choose k} q^k (1-q)^{n-k}y_{u^k d^{n-k}}.$$ Here ${n \choose k}$ 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).

Black-Scholes in the limit

This section contains some undergraduate level mathematics.

Well, no-one knows exactly, but using stats you can make a good guess. This article tells you how and has an interactive life expectancy calculator.

Finding the Decision Numbers

As discussed in the main article, if the frogs are randomly numbered between 0 and 1, with the prince having the highest number, there is a sequence of decision numbers associated with the best strategy for choosing the top frog. Since we're meant to be doing some maths, we need some notation. We'll consider the general case with $N$ frogs, and let the number on the $j$th frog be $y_j$, $j = 1,2,\ldots,N$.

The 37% rule

The first question we need to answer is, why is the best strategy to throw back some number of frogs, and then kiss the next one that is better than all those you've seen so far? Well, what else could you do? All you know about the frogs that you've seen is how good they are relative to each other, which you can see from the numbers on their backs. When the first frog hops out, you have absolutely no information (the number on its back tells you nothing).

We are proud to present the winners of the Plus New Writers Award 2008.

Inverting and calculating the Radon transform using the Fourier transform

You might also want to listen to our podcast on the Fourier transform.