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Thread: Derivative of implicit function (risk & reward)

  1. #1
    Member Vinod's Avatar
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    Post Derivative of implicit function (risk & reward)

    Hi members,
    I want to differentiate w.r.t.$\sigma^2$ the following equation.
    $u'(Y)+\frac{u''(Y)}{2}(\sigma^2 + \mu^2) = 0$
    where we can consider $\mu$(reward) as an implicit function of $\sigma^2$(risk) of small bets
    $u'(Y)\frac{du}{d\sigma^2}+\frac{u''(Y)}{2}+\mu u''(Y)\frac{du}{d\sigma^2}=0$

    Hence,
    $u'(\sigma^2)=\frac{u''(Y)}{2(u'(Y)+\mu u''(Y)}$
    But answer given in the PDF downloaded from internet is $u'(\sigma^2)=\frac{u''(Y)}{2(u'(Y)+\mu(\sigma^2)u ''(Y)}$

    Which answer is correct?
    What does 1st and 2nd derivative of Utility function imply?
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  2. #2
    MHF Contributor
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    Re: Derivative of implicit function (risk & reward)

    Hey Vinod.

    Did you consider the chain rule? Also what is mu a function of specifically?
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  3. #3
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    Re: Derivative of implicit function (risk & reward)

    Quote Originally Posted by chiro View Post
    Hey Vinod. Also what is mu a function of specifically?
    $\displaystyle \mu$ is the mean representing the expected return (most likely expected value of lognormal distribution)
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  4. #4
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    Re: Derivative of implicit function (risk & reward)

    So is it a constant? The answer seems to suggest that it is at least a function of sigma^2.
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