I have a big problem in understanding and deriving partial derivatives. I would like to understand the logic behind deriving partial derivatives for some exemplary processes. I have a few examples which I would like to understand 'how to' derive.
E.g. Basing on Ito process:
dG = ( ∂G/∂S * µS + ∂G/∂t + 1/2* ∂^2G/∂S^2 * ^2 *S^2) dt + ∂G/∂S * S dz
I have some examples of processes that should follow Ito process. For The examples below I know the solutions but I would like to know how to get to them.
The Following processes should be expressed in terms of Ito:
a) y = S^2
b)y = e^S
c) y = e^(r(T-t)/S
d) the hard one: Process is given as dS = ( r B - rA ) S dt + S dz which is the process for a price of currency A expressed in terms of the price of currency B. We should determine process followed by the price of currency B expressed in terms of currency A.
Answers: I would like to understand how partial derivatives are derived in the form of:
∂G/ ∂S
∂G/ ∂t
∂^2 G / ∂S^2
The latter, how to insert derivatives in form of above to Ito process - I can fully perform but I have so many doubts how the derivatives should be derived.
For instance the asnwers for a) and d) are:
a) y = S^2
Partial derivatives would be:
∂y/∂S = 2S
∂^2 y / ∂S^2 = 2
∂y/∂t = 0
Final Ito process followed:
dy = (2µS^2 + ^2 S^2)dt + 2 S^2dz
d) X = 1/S
Partial derivatives would be:
∂X/ ∂S = (-1/S^2)
∂F/ ∂t = 0
∂^2 X / ∂S^2 = (2/S^3)
Final Ito process followed:
dX = ( (rB - rA)S * (-1/S^2) + 1/2 * (2/S^3) * ^2S^2) dt + (-1/S^2) S dz
Could somebody explain point by point how this has been achieved? If I have provided too many examples (a to d), it will be enough to show me one of the points from a - c and point d.
I will fully appreciate because I have no clue.