顶一下，金融工程的基础

Solutions to problems listed above:
1) aa' has two eigenvalues: zero and a'a.  the eigenvector of a'a is a, and the eigenvector of zero is the vectors orthogonal to vector a.
2) w^3 is not a martingale, but is a semi-martingale.
3) We note the price of a call with strike K as C(K). if we buy two calls with strike K_1 and K_2, and short two calls with strike K_3=(K_1+K_2)/2, then pay C(K_1)+C(K_2)-2*C(K_3) at first. At maturity, we will receive an non-negative payoff. So the initial value of this portfolio is non-negative. So we get the convexity.
4) I don't fully understand this question. It is easy to get the distribution of distance. And we know that the probability that dart falls inside the radius of R/2 is  p=1/4. So the expectation of gain is
     \sum_{n=0}^{n=100} C_100^n p^n (1-p)^{100-n} ([n/10]-(100-n)*0.1), where [x] means the integer part of x. And i think this expectation is negative, because we get 1 dollar for tens times,
     which means that every time we gain 10c, and we lost 10c if the dart is out of radius of R/2. So if the probability changes, so the result.
5)  Multiplication: x*y = {(x+y)^2-(x-y)^2}/4
     Division:  We just have to figure out 1/N, where N is an integer. We take the integer n satisfying 2^(n-1)<N<2^n.
                     So we have 1/N=(1/2^n)*(2^n/N), and addition can be applied to get the first number of 2^n/N, so we go on to repeat this process.
6) We take two positive integer x and y, and present x as x=2^N+x', where x' is positive and smaller than x, y as y=2^M+y', where y' is positive and smaller than y.
    So x*y = (2^N+x')*(2^M+y')=2^(N+M)+2^N*y'+2^M*x'+x'*y', and we go to to compute x'*y' until x' and y' become 0 or 1.
7) const const const, find a C++ book......
8) On my point, i will use a pure virtual class to present any function that we might to optimisation. And we can implement algorithms of optimisation. Once we want to optimise a function, we
    just present the function as a subclass of pure virtual class. If this, we don't have to rewrite algorithms of optimisation. That's what i think.
9) Find a book and check it.
10) Change variable as y=logx
11) By the proof of the derivative of compounded function. We can find it at any mathematics analyses book.
12) A is definitive, so use the Cholesky decomposition.
13) the expectation of max is N/(N+1), so the min is 1/(N+1).
14) An interesting question. I find one wich is not so convictive. Offer a better one!
15) Price of digital is the derivative of call w.r.t strike. So with imp vol, we can present it as vega* d vol / d strike
16) PDE
17) I am still working on it. And i know other method.
18)
19) Wall street Journal
20) We note the expected number of trials of N consecutive heads as T(N).
      So we has T(N)=0.5*(1+T(N))+0.5*(1+T(N-1)).
21) I have no idea to solve this problem, maybe Monte Carlo?????

Yes, this is my answer. I think it will be an easy work to find lots of grammer mistakes in my answer. So just ignore them.
And I am looking for an quantiative position for several months. But unfortunately i don't find a favorable one  even now.
Do you guys know somebody who work in this domain and could he do me a favor to recommand me?
I really need help now.......