金融にフーリェ変換

最近英国のImperial CollegeのCernyが書いた論文を読んでいます。
オプションの計算方式を勉強していないから分からないところが多いです。主の課題は以下に書きます。
1. Complex numbers on the unit circle (exponential form).
2. Sequences of numbers on the unit circle and their reversal.
3. Writing the discrete Fourier transform in terms of a summation of multiples of complex numbers on the unit circle.
4. Convolutions: align elements of vector "a" on circle clockwise and elements of "b" on circle anticlockwise. Then 0th component of "a convolve b"is the sum of the product of the a and b coordinates at the same angular position. a0b0+a3b1+a2b2+a1b2. or the (j-k)th coord of a and the kth coord of b are multiplied over all k to produce the jth coord of the convolution.
It is interesting to compare this discrete definition of convolution with the continuous counterpart:
INTEGRAL(g(x-y)f(y)dy).
5. Writing the binomial model option value at present time in terms of repeated convolutions of risk-free probability distribution by itself and one convolution of this with final option value vector (all divided by power of risk-free return (e.g. 1.003)). In the binomial model, a stock can move two one of two positions in each trading period (up or down). The risk-free probabilities are found by first obtaining the return for up/down movement (by assuming an initial probability distribution, expected rate of return, and market volatility), then by setting the expected return to equal the risk free return.
6. Implementing above in software packages and measuring performance. Zero-padding of input vector can be used to make the size of the input optimal for fast execution (2^p3^q5^r)
7. moving to continuous time (e^(rT)) replaces (R^N). Also, something called E^Q[CT(ln ST)] replaces the F*1^N) used earlier. N means N(delta t) or T/(delta t), ie number of trading periods. I am not sure what E^Q is, but it seems to be some sort of expectation function based on the probability distribution of the stock's movement.
Fourier transform is not needed if characteristic function is known and there is only one strike point. Some points in this passage were unclear to me, i.e. how C0 formula was derived and how integrals were simplified. Also, I do not know why strike price would be e^k, but it seems that k is the ln(strike price). The reasoning behind the move to logarithmic values confuses me.
8. Why FFT is needed, if there are multiple strike points to be evaluated simultaneously. A general transformation known as a z-transform is specified. This is used to write the C0(ln S0) formula for multiple strike points. In some cases, this can be computed as a Fourier transform, and in some cases it is better to use the CHIRPZ z-transform algorithm. Here again some quantity called "v" is made use of, and there is a v(max) also. I am not sure what this is, but I saw a mention of something called "vega", which is d(option value)/volatility.
9. Appendix 1: Proof F(F^-1(a))=a
10. Appendix 2: Proof F(a convolve b)= sqrt(n)F(a)F(b)

*1:F^-1CT)*(sqrt(N+1)*F(q