Write a matlab or r function to compute the value of a

Question

This project contains 4 sections.

Section A)

Let X be a d-dimensional random vector expressed as column vector. Without loss of generality, assume X has zero empirical mean. We want to find a d-by-d orthonormal transformation matrix P such that

Y = P^TX

With the constraint that

Cov(Y) is a diagonal matrix and P^-1 = P^T

Section B)

Let S(t) be a positive stochastic process satisfying

dS(t) = μ(t)S(t)dt + σ(t)S(t)dW(t)

Where μ(t), σ(t) are processes adapted to filteration F(t) for t ≥ 0 associated with the Brownian motion W(t). Compute d {log(S(t))} to show that -

S(t) = S(0)exp{₀∫^t(μ(s) - (σ²(s)/2)ds + σ(s)dW(s)}

Section C)

Let S(t) be a positive stochastic process satisfying

dS(t) = μ(t)S(t)dt + σ(t)S(t)dW(t)

Where µ(t), σ(t) are processes adapted to filteration F(t) for t ≥ 0 associated with the Brownian motion W(t). Compute d{(s(t))^P} for p > 0

Section D)

Write a Matlab or R function to compute the value of a convertible bond. Please use two different models to price convertible bonds. Explain the difference and sources of optionality of the two models.