Do *all* non-dividend paying assets have the risk-free instantaneous return rate under the risk-neutral...












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$begingroup$


For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). I know that, by virtue of Girsanov's theorem, the Brownian motion under the risk-neutral measure is defined by
$$dB_t^{Bbb Q} = lambda dt + dB_t$$
where $lambda$ is the unique market price of risk, or the so-called Sharpe ratio.



Under the risk-neutral measure, any non-dividend paying stock price process $S_t$ thus follows
$$frac{dS_t}{S_t} = rdt + sigma_SdB_t^{Bbb Q}.$$



However, in Kerry Back's A Course in Derivative Securities page 220, the author claimed without a proof that the instantaneous rate of return for a call option on the stock price $C_t$ is also $r$, i.e.
$$frac{dC_t}{C_t} = rdt + sigma_C d B_t^{Bbb Q}$$
where $sigma_C$ is some stochastic process that we're not interested in. The author make crucial use of the above formula (i.e. the drift of $C_t$ is $rC_tdt$) to derive the BS PDE.



Question: is it true that under the risk neutral measure, any non-dividend paying asset price $X_t$ must have its instantaneous rate of return equal to $r$? If so, what would be a rigorous explanation for this?





Edit: Antoine is spot on. Under the risk neutral measure, any discounted asset price $Y_t=e^{-rt}X_t$ must be a martingale or equivalently an Ito integral without drift. Hence
$$frac{dY_t}{Y_t}=sigma_Y dB_t^{Bbb Q}.$$
where $sigma_Y$ can be a quite general stochastic process. On the other hand, by the compounding rule of Ito processes,
$$frac{dY_t}{Y_t}=-rdt+frac{dX_t}{X_t}$$
Therefore it follows
$$frac{dX_t}{X_t}=rdt+sigma_Y dB_t^{Bbb Q}.$$




option-pricing black-scholes arbitrage risk-neutral-measure







share|improve this question











$endgroup$

















    3












    $begingroup$


    For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). I know that, by virtue of Girsanov's theorem, the Brownian motion under the risk-neutral measure is defined by
    $$dB_t^{Bbb Q} = lambda dt + dB_t$$
    where $lambda$ is the unique market price of risk, or the so-called Sharpe ratio.



    Under the risk-neutral measure, any non-dividend paying stock price process $S_t$ thus follows
    $$frac{dS_t}{S_t} = rdt + sigma_SdB_t^{Bbb Q}.$$



    However, in Kerry Back's A Course in Derivative Securities page 220, the author claimed without a proof that the instantaneous rate of return for a call option on the stock price $C_t$ is also $r$, i.e.
    $$frac{dC_t}{C_t} = rdt + sigma_C d B_t^{Bbb Q}$$
    where $sigma_C$ is some stochastic process that we're not interested in. The author make crucial use of the above formula (i.e. the drift of $C_t$ is $rC_tdt$) to derive the BS PDE.



    Question: is it true that under the risk neutral measure, any non-dividend paying asset price $X_t$ must have its instantaneous rate of return equal to $r$? If so, what would be a rigorous explanation for this?





    Edit: Antoine is spot on. Under the risk neutral measure, any discounted asset price $Y_t=e^{-rt}X_t$ must be a martingale or equivalently an Ito integral without drift. Hence
    $$frac{dY_t}{Y_t}=sigma_Y dB_t^{Bbb Q}.$$
    where $sigma_Y$ can be a quite general stochastic process. On the other hand, by the compounding rule of Ito processes,
    $$frac{dY_t}{Y_t}=-rdt+frac{dX_t}{X_t}$$
    Therefore it follows
    $$frac{dX_t}{X_t}=rdt+sigma_Y dB_t^{Bbb Q}.$$




    option-pricing black-scholes arbitrage risk-neutral-measure







    share|improve this question











    $endgroup$















      3












      3








      3





      $begingroup$


      For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). I know that, by virtue of Girsanov's theorem, the Brownian motion under the risk-neutral measure is defined by
      $$dB_t^{Bbb Q} = lambda dt + dB_t$$
      where $lambda$ is the unique market price of risk, or the so-called Sharpe ratio.



      Under the risk-neutral measure, any non-dividend paying stock price process $S_t$ thus follows
      $$frac{dS_t}{S_t} = rdt + sigma_SdB_t^{Bbb Q}.$$



      However, in Kerry Back's A Course in Derivative Securities page 220, the author claimed without a proof that the instantaneous rate of return for a call option on the stock price $C_t$ is also $r$, i.e.
      $$frac{dC_t}{C_t} = rdt + sigma_C d B_t^{Bbb Q}$$
      where $sigma_C$ is some stochastic process that we're not interested in. The author make crucial use of the above formula (i.e. the drift of $C_t$ is $rC_tdt$) to derive the BS PDE.



      Question: is it true that under the risk neutral measure, any non-dividend paying asset price $X_t$ must have its instantaneous rate of return equal to $r$? If so, what would be a rigorous explanation for this?





      Edit: Antoine is spot on. Under the risk neutral measure, any discounted asset price $Y_t=e^{-rt}X_t$ must be a martingale or equivalently an Ito integral without drift. Hence
      $$frac{dY_t}{Y_t}=sigma_Y dB_t^{Bbb Q}.$$
      where $sigma_Y$ can be a quite general stochastic process. On the other hand, by the compounding rule of Ito processes,
      $$frac{dY_t}{Y_t}=-rdt+frac{dX_t}{X_t}$$
      Therefore it follows
      $$frac{dX_t}{X_t}=rdt+sigma_Y dB_t^{Bbb Q}.$$




      option-pricing black-scholes arbitrage risk-neutral-measure







      share|improve this question











      $endgroup$




      For simplicity let's consider a 1D BS world. The only source of randomness comes from the Brownian motion dynamics $dB_t$. The risk-free rate is $r$ (one may assume it as constant for the time being). I know that, by virtue of Girsanov's theorem, the Brownian motion under the risk-neutral measure is defined by
      $$dB_t^{Bbb Q} = lambda dt + dB_t$$
      where $lambda$ is the unique market price of risk, or the so-called Sharpe ratio.



      Under the risk-neutral measure, any non-dividend paying stock price process $S_t$ thus follows
      $$frac{dS_t}{S_t} = rdt + sigma_SdB_t^{Bbb Q}.$$



      However, in Kerry Back's A Course in Derivative Securities page 220, the author claimed without a proof that the instantaneous rate of return for a call option on the stock price $C_t$ is also $r$, i.e.
      $$frac{dC_t}{C_t} = rdt + sigma_C d B_t^{Bbb Q}$$
      where $sigma_C$ is some stochastic process that we're not interested in. The author make crucial use of the above formula (i.e. the drift of $C_t$ is $rC_tdt$) to derive the BS PDE.



      Question: is it true that under the risk neutral measure, any non-dividend paying asset price $X_t$ must have its instantaneous rate of return equal to $r$? If so, what would be a rigorous explanation for this?





      Edit: Antoine is spot on. Under the risk neutral measure, any discounted asset price $Y_t=e^{-rt}X_t$ must be a martingale or equivalently an Ito integral without drift. Hence
      $$frac{dY_t}{Y_t}=sigma_Y dB_t^{Bbb Q}.$$
      where $sigma_Y$ can be a quite general stochastic process. On the other hand, by the compounding rule of Ito processes,
      $$frac{dY_t}{Y_t}=-rdt+frac{dX_t}{X_t}$$
      Therefore it follows
      $$frac{dX_t}{X_t}=rdt+sigma_Y dB_t^{Bbb Q}.$$




      option-pricing black-scholes arbitrage risk-neutral-measure




      option-pricing black-scholes arbitrage risk-neutral-measure






      share|improve this question















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      edited Feb 7 at 11:46







      Vim

















      asked Feb 7 at 10:24









      VimVim

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          Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.






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            $begingroup$
            Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
            $endgroup$
            – Vim
            Feb 7 at 11:35











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          $begingroup$

          Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.






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          • 1




            $begingroup$
            Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
            $endgroup$
            – Vim
            Feb 7 at 11:35
















          4












          $begingroup$

          Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.






          share|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
            $endgroup$
            – Vim
            Feb 7 at 11:35














          4












          4








          4





          $begingroup$

          Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.






          share|improve this answer









          $endgroup$



          Under the assumption that the market is complete, any discounted contingent claim can be replicated as a stochastic integral against the discounted stock price, therefore the discounted contingent claim price is a martingale under the risk neutral measure, or said otherwise the contingent claim price instantaneous rate of return under the risk neutral measure is the risk free rate.







          share|improve this answer












          share|improve this answer



          share|improve this answer










          answered Feb 7 at 11:16









          Antoine ConzeAntoine Conze

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          3,770149








          • 1




            $begingroup$
            Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
            $endgroup$
            – Vim
            Feb 7 at 11:35














          • 1




            $begingroup$
            Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
            $endgroup$
            – Vim
            Feb 7 at 11:35








          1




          1




          $begingroup$
          Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
          $endgroup$
          – Vim
          Feb 7 at 11:35




          $begingroup$
          Oh yes. The discounted asset price must be a m.g. how could i forget this. Thanks.
          $endgroup$
          – Vim
          Feb 7 at 11:35


















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