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

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

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

    I have started it, and so far got to:

    $\displaystyle 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 $

    $\displaystyle + \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 $

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

    Is this right?

    Then I need to simplify this, and I got:

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

    I don't know whether this is correct?

    Any suggestions would be much appreciated.

    Thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by signature View Post
    I have to perform a Taylor Expansion in both for $\displaystyle S_i$ where $\displaystyle i=1,2,......,n$ and $\displaystyle t$ for the function $\displaystyle V(S_1,S_2,.......,t)$

    I have started it, and so far got to:

    $\displaystyle 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 $

    $\displaystyle + \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 $

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

    Is this right?

    Then I need to simplify this, and I got:

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

    I don't know whether this is correct?

    Any suggestions would be much appreciated.

    Thanks
    Where are the:

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

    terms?
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  3. #3
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    Do I need still that since I have $\displaystyle O(t^2)$?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by signature View Post
    Do I need still that since I have $\displaystyle O(t^2)$?
    Yes (you in fact have terms $\displaystyle O(dt^2)$ you have missed the terms $\displaystyle O(dtdS_i)$)

    CB
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