Question

Create a numerical example in a three security/two state model and illustrate the proof of the Fundamental theorem by that example. For example, define three securities as having initial prices of Si(0) =1, S2(0) = 3,, etc., and final prices of Siwi) = 0, Si(w2) = 3, etc., for N = {W1,w2}. Create two examples - one the does not allow arbitrage, and one that does allow arbitrage. Demonstrate that L space must be a hyperplane in R3 in case of no arbitrage, but not in case when arbitrage is allowed. Problem Create a numerical example in a two security three state model and illustrate the proof of the Fundamental theorem by that example. Show that in this case the state-price vector is not unique by creating two different state price vectors

Answer 1

Example 1.

Calculate the derivative of the function g(x)=x∫1√t3+4tdt at x=2.

Solution.

We apply the Fundamental Theorem of Calculus, Part 1:

g′(x)=ddx⎛⎝x∫af(t)dt⎞⎠=f(x).

Hence

g′(x)=ddx⎛⎜⎝x∫1√t3+4tdt⎞⎟⎠=√x3+4x.

Substituting x=2 yields

g′(2)=√23+4⋅2=√16=4.

If f is a continuous function on [a,b], then the function g defined by

g(x)=x∫af(t)dt,a≤x≤b

is an antiderivative of f, that is

g′(x)=f(x)orddx⎛⎝x∫af(t)dt⎞⎠=f(x).

If f happens to be a positive function, then g(x) can be interpreted as the area under the graph of f from a to x.