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How to compute the dynamic of stock using Geometric Brownian Motion?


A question on ItoGeometric Brownian motion - Volatility Interpretation (in the drift term)How to get Geometric Brownian Motion's closed-form solution in Black-Scholes model?Square of arithmetic brownian motion processSimulate drifted geometric brownian motion under new measureIto's Lemma, differentiating an integral with Brownian motionGiven $S$ is a Geometric Brownian Motion, how to show that $S^n$ is also a Geometric Brownian Motion?Stochastic process and brownian motionNormalized Gains Process is a Q-Martingale - Proof and IntuitionHow to derive the dynamic of the log forward price?













2












$begingroup$


I have been given the following question:




Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$




For the first part of the question, I have got this answer:
$$dS_t = mu S_tdt + sigma S_t dWt$$



Is it correct?



And for the second part, I know that the price $f(t,S_t)$ follows the process
$$df = (frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2} sigma ^2S_tfrac{partial^2f}{partial S_t^2})dt +sigma S_t dWt$$



I am having trouble finding the answer using this process and given the information.



Any help is appreciated.






black-scholes stochastic-processes stochastic-calculus







share|improve this question











$endgroup$

















    2












    $begingroup$


    I have been given the following question:




    Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$




    For the first part of the question, I have got this answer:
    $$dS_t = mu S_tdt + sigma S_t dWt$$



    Is it correct?



    And for the second part, I know that the price $f(t,S_t)$ follows the process
    $$df = (frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2} sigma ^2S_tfrac{partial^2f}{partial S_t^2})dt +sigma S_t dWt$$



    I am having trouble finding the answer using this process and given the information.



    Any help is appreciated.






    black-scholes stochastic-processes stochastic-calculus







    share|improve this question











    $endgroup$















      2












      2








      2





      $begingroup$


      I have been given the following question:




      Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$




      For the first part of the question, I have got this answer:
      $$dS_t = mu S_tdt + sigma S_t dWt$$



      Is it correct?



      And for the second part, I know that the price $f(t,S_t)$ follows the process
      $$df = (frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2} sigma ^2S_tfrac{partial^2f}{partial S_t^2})dt +sigma S_t dWt$$



      I am having trouble finding the answer using this process and given the information.



      Any help is appreciated.






      black-scholes stochastic-processes stochastic-calculus







      share|improve this question











      $endgroup$




      I have been given the following question:




      Given that $S_t$ follows Geometric Brownian Motion, write down the dynamic of $S_t$ and then compute the dynamic of $f(t,S_t) = e^{tS^{2}}$




      For the first part of the question, I have got this answer:
      $$dS_t = mu S_tdt + sigma S_t dWt$$



      Is it correct?



      And for the second part, I know that the price $f(t,S_t)$ follows the process
      $$df = (frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2} sigma ^2S_tfrac{partial^2f}{partial S_t^2})dt +sigma S_t dWt$$



      I am having trouble finding the answer using this process and given the information.



      Any help is appreciated.






      black-scholes stochastic-processes stochastic-calculus




      black-scholes stochastic-processes stochastic-calculus






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited 3 hours ago









      Emma

      27112




      27112










      asked 3 hours ago









      Rito LoweRito Lowe

      164




      164






















          1 Answer
          1






          active

          oldest

          votes


















          3












          $begingroup$

          The above equation should correctly read as follows:



          $df=big(frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2}sigma^2 S_t^2frac{partial^2 f}{partial S_t^2}big)+sigma S_t frac{partial f}{partial S_t}dW$



          Using:



          (a) $frac{partial f}{partial t}=S_t^2f$



          (b) $frac{partial f}{partial S_t}=2S_ttf$



          (c) $frac{partial^2 f}{partial S_t^2}=2tf+4S_t^2t^2f$



          The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:



          $frac{df}{f}=dt big(S_t^2+2 mu S_t^2t+sigma^2S_t^2t+2sigma^2S_t^4t^2 big)+2S_t^2tsigma dW$






          share|improve this answer











          $endgroup$













            Your Answer





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            1 Answer
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            active

            oldest

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            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            3












            $begingroup$

            The above equation should correctly read as follows:



            $df=big(frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2}sigma^2 S_t^2frac{partial^2 f}{partial S_t^2}big)+sigma S_t frac{partial f}{partial S_t}dW$



            Using:



            (a) $frac{partial f}{partial t}=S_t^2f$



            (b) $frac{partial f}{partial S_t}=2S_ttf$



            (c) $frac{partial^2 f}{partial S_t^2}=2tf+4S_t^2t^2f$



            The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:



            $frac{df}{f}=dt big(S_t^2+2 mu S_t^2t+sigma^2S_t^2t+2sigma^2S_t^4t^2 big)+2S_t^2tsigma dW$






            share|improve this answer











            $endgroup$


















              3












              $begingroup$

              The above equation should correctly read as follows:



              $df=big(frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2}sigma^2 S_t^2frac{partial^2 f}{partial S_t^2}big)+sigma S_t frac{partial f}{partial S_t}dW$



              Using:



              (a) $frac{partial f}{partial t}=S_t^2f$



              (b) $frac{partial f}{partial S_t}=2S_ttf$



              (c) $frac{partial^2 f}{partial S_t^2}=2tf+4S_t^2t^2f$



              The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:



              $frac{df}{f}=dt big(S_t^2+2 mu S_t^2t+sigma^2S_t^2t+2sigma^2S_t^4t^2 big)+2S_t^2tsigma dW$






              share|improve this answer











              $endgroup$
















                3












                3








                3





                $begingroup$

                The above equation should correctly read as follows:



                $df=big(frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2}sigma^2 S_t^2frac{partial^2 f}{partial S_t^2}big)+sigma S_t frac{partial f}{partial S_t}dW$



                Using:



                (a) $frac{partial f}{partial t}=S_t^2f$



                (b) $frac{partial f}{partial S_t}=2S_ttf$



                (c) $frac{partial^2 f}{partial S_t^2}=2tf+4S_t^2t^2f$



                The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:



                $frac{df}{f}=dt big(S_t^2+2 mu S_t^2t+sigma^2S_t^2t+2sigma^2S_t^4t^2 big)+2S_t^2tsigma dW$






                share|improve this answer











                $endgroup$



                The above equation should correctly read as follows:



                $df=big(frac{partial f}{partial t}+mu S_t frac{partial f}{partial S_t}+frac{1}{2}sigma^2 S_t^2frac{partial^2 f}{partial S_t^2}big)+sigma S_t frac{partial f}{partial S_t}dW$



                Using:



                (a) $frac{partial f}{partial t}=S_t^2f$



                (b) $frac{partial f}{partial S_t}=2S_ttf$



                (c) $frac{partial^2 f}{partial S_t^2}=2tf+4S_t^2t^2f$



                The Stochastic Differential Equation (SDF) governing the dynamics of $f$ becomes:



                $frac{df}{f}=dt big(S_t^2+2 mu S_t^2t+sigma^2S_t^2t+2sigma^2S_t^4t^2 big)+2S_t^2tsigma dW$







                share|improve this answer














                share|improve this answer



                share|improve this answer








                edited 1 hour ago









                Emma

                27112




                27112










                answered 2 hours ago









                ZRHZRH

                583112




                583112






























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