# Thread: [SOLVED] Taylor Series Expansion help!!!

1. ## [SOLVED] Taylor Series Expansion help!!!

I have to perform a Taylor Expansion in both for $S_i$ where $i=1,2,......,n$ and $t$ for the function $V(S_1,S_2,.......,t)$

I have started it, and so far got to:

$dV = \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S_1} dS_1+ ........+\frac{\partial V}{\partial S_n} dS_n + \frac{1}{2}\frac{\partial^2 V}{\partial S_1^2}dS_1^2+........+ \frac{1}{2}\frac{\partial^2 V}{\partial S_n^2}dS_n^2$

$+ \frac{\partial^2 V}{\partial S_1 \partial S_2}dS_1dS_2 + \frac{\partial^2 V}{\partial S_1 \partial S_3}dS_1dS_3 +........+ \frac{\partial^2 V}{\partial S_{n-1} \partial S_n}dS_{n-1}dS_n$

$+ O(t^2) + O(t^3) + O(S_1^3)+........+O(S_n^3)$

Is this right?

Then I need to simplify this, and I got:

$dV = \frac{\partial V}{\partial t} dt + \sum_{i=1}^{n}\frac{\partial V}{\partial S_i}dS_i + \frac{1}{2}\sum_{i=1}^{n}\frac{\partial^2 V}{\partial S_i^2}dS_i^2 + \sum_{i=1}^{n} \sum_{j \neq i}^{n}\frac{\partial^2 V}{\partial S_i \partial S_j}dS_idS_j$

I don't know whether this is correct?

Any suggestions would be much appreciated.

Thanks

2. Originally Posted by signature
I have to perform a Taylor Expansion in both for $S_i$ where $i=1,2,......,n$ and $t$ for the function $V(S_1,S_2,.......,t)$

I have started it, and so far got to:

$dV = \frac{\partial V}{\partial t} dt + \frac{\partial V}{\partial S_1} dS_1+ ........+\frac{\partial V}{\partial S_n} dS_n + \frac{1}{2}\frac{\partial^2 V}{\partial S_1^2}dS_1^2+........+ \frac{1}{2}\frac{\partial^2 V}{\partial S_n^2}dS_n^2$

$+ \frac{\partial^2 V}{\partial S_1 \partial S_2}dS_1dS_2 + \frac{\partial^2 V}{\partial S_1 \partial S_3}dS_1dS_3 +........+ \frac{\partial^2 V}{\partial S_{n-1} \partial S_n}dS_{n-1}dS_n$

$+ O(t^2) + O(t^3) + O(S_1^3)+........+O(S_n^3)$

Is this right?

Then I need to simplify this, and I got:

$dV = \frac{\partial V}{\partial t} dt + \sum_{i=1}^{n}\frac{\partial V}{\partial S_i}dS_i + \frac{1}{2}\sum_{i=1}^{n}\frac{\partial^2 V}{\partial S_i^2}dS_i^2 + \sum_{i=1}^{n} \sum_{j \neq i}^{n}\frac{\partial^2 V}{\partial S_i \partial S_j}dS_idS_j$

I don't know whether this is correct?

Any suggestions would be much appreciated.

Thanks
Where are the:

$\frac{\partial^2V}{\partial t \partial S_i}$

terms?

3. Do I need still that since I have $O(t^2)$?

4. Originally Posted by signature
Do I need still that since I have $O(t^2)$?
Yes (you in fact have terms $O(dt^2)$ you have missed the terms $O(dtdS_i)$)

CB