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Math Help - [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 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
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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 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?
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  3. #3
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    Do I need still that since I have 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 O(t^2)?
    Yes (you in fact have terms O(dt^2) you have missed the terms O(dtdS_i))

    CB
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