The choice of portfolio in the proof of the Black-Scholes formula












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Consider a stock whose price $S$ satisfies $$dS_t=mu S_tdt+sigma S_tdW_t$$ for constants $mu,sigma$ and where $W$ is a $mathbb{P}$-Brownian motion. Further assume that the stock pays out dividends continuously at a rate of $d$ proportional to the current stock price.



Let $p_t$ denote the price at time $t$ of a European-style derivative which has a payoff of $f(S_T)$ at time $T$. In order to determine a formula for $p_t$ we essentially carry out the following steps:




  1. Use Girsanov's theorem to determine the risk-neutral probability measure $mathbb{Q}$ such that $widetilde{W}_t=left(frac{mu+d-r}{sigma}right)t+W_t$ is a $mathbb{Q}$-Brownian motion.

  2. Define $P_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$. Show that both $hat{S}_t=e^{-(r-d)t}S_t$ and $hat{P_t}=e^{-rt}P_t$ are $mathbb{Q}$-martingales.

  3. Use the Martingale Representation Theorem to conclude the existence of a predictable process $A$ such that $hat{P}_t=hat{P}_0+int_0^tA_sdhat{S}_s$ under $mathbb{Q}$.

  4. Construct the portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ which consists of $hat{P}_t-A_that{S}_t$ units of cash and $A_te^{dt}$ units of the stock at time $t$. The value of this portfolio is $P_t$.

  5. Since $P_T=p_T$ we conclude from the Law of One Price that $P_t=p_t$ for all $0leq tleq T$. In other words, $p_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$.


After going through the above steps I am wondering why the portfolio needs to be $(hat{P}_t-A_that{S}_t,A_te^{dt})$. It seems like we could simply choose $(hat{P}_t,0)$ as our portfolio and this would still have a value of $P_t$ at time $t$.






black-scholes stochastic-processes risk-neutral-measure







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    Consider a stock whose price $S$ satisfies $$dS_t=mu S_tdt+sigma S_tdW_t$$ for constants $mu,sigma$ and where $W$ is a $mathbb{P}$-Brownian motion. Further assume that the stock pays out dividends continuously at a rate of $d$ proportional to the current stock price.



    Let $p_t$ denote the price at time $t$ of a European-style derivative which has a payoff of $f(S_T)$ at time $T$. In order to determine a formula for $p_t$ we essentially carry out the following steps:




    1. Use Girsanov's theorem to determine the risk-neutral probability measure $mathbb{Q}$ such that $widetilde{W}_t=left(frac{mu+d-r}{sigma}right)t+W_t$ is a $mathbb{Q}$-Brownian motion.

    2. Define $P_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$. Show that both $hat{S}_t=e^{-(r-d)t}S_t$ and $hat{P_t}=e^{-rt}P_t$ are $mathbb{Q}$-martingales.

    3. Use the Martingale Representation Theorem to conclude the existence of a predictable process $A$ such that $hat{P}_t=hat{P}_0+int_0^tA_sdhat{S}_s$ under $mathbb{Q}$.

    4. Construct the portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ which consists of $hat{P}_t-A_that{S}_t$ units of cash and $A_te^{dt}$ units of the stock at time $t$. The value of this portfolio is $P_t$.

    5. Since $P_T=p_T$ we conclude from the Law of One Price that $P_t=p_t$ for all $0leq tleq T$. In other words, $p_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$.


    After going through the above steps I am wondering why the portfolio needs to be $(hat{P}_t-A_that{S}_t,A_te^{dt})$. It seems like we could simply choose $(hat{P}_t,0)$ as our portfolio and this would still have a value of $P_t$ at time $t$.






    black-scholes stochastic-processes risk-neutral-measure







    share|improve this question







    New contributor




    user375366 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      Consider a stock whose price $S$ satisfies $$dS_t=mu S_tdt+sigma S_tdW_t$$ for constants $mu,sigma$ and where $W$ is a $mathbb{P}$-Brownian motion. Further assume that the stock pays out dividends continuously at a rate of $d$ proportional to the current stock price.



      Let $p_t$ denote the price at time $t$ of a European-style derivative which has a payoff of $f(S_T)$ at time $T$. In order to determine a formula for $p_t$ we essentially carry out the following steps:




      1. Use Girsanov's theorem to determine the risk-neutral probability measure $mathbb{Q}$ such that $widetilde{W}_t=left(frac{mu+d-r}{sigma}right)t+W_t$ is a $mathbb{Q}$-Brownian motion.

      2. Define $P_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$. Show that both $hat{S}_t=e^{-(r-d)t}S_t$ and $hat{P_t}=e^{-rt}P_t$ are $mathbb{Q}$-martingales.

      3. Use the Martingale Representation Theorem to conclude the existence of a predictable process $A$ such that $hat{P}_t=hat{P}_0+int_0^tA_sdhat{S}_s$ under $mathbb{Q}$.

      4. Construct the portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ which consists of $hat{P}_t-A_that{S}_t$ units of cash and $A_te^{dt}$ units of the stock at time $t$. The value of this portfolio is $P_t$.

      5. Since $P_T=p_T$ we conclude from the Law of One Price that $P_t=p_t$ for all $0leq tleq T$. In other words, $p_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$.


      After going through the above steps I am wondering why the portfolio needs to be $(hat{P}_t-A_that{S}_t,A_te^{dt})$. It seems like we could simply choose $(hat{P}_t,0)$ as our portfolio and this would still have a value of $P_t$ at time $t$.






      black-scholes stochastic-processes risk-neutral-measure







      share|improve this question







      New contributor




      user375366 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.












      Consider a stock whose price $S$ satisfies $$dS_t=mu S_tdt+sigma S_tdW_t$$ for constants $mu,sigma$ and where $W$ is a $mathbb{P}$-Brownian motion. Further assume that the stock pays out dividends continuously at a rate of $d$ proportional to the current stock price.



      Let $p_t$ denote the price at time $t$ of a European-style derivative which has a payoff of $f(S_T)$ at time $T$. In order to determine a formula for $p_t$ we essentially carry out the following steps:




      1. Use Girsanov's theorem to determine the risk-neutral probability measure $mathbb{Q}$ such that $widetilde{W}_t=left(frac{mu+d-r}{sigma}right)t+W_t$ is a $mathbb{Q}$-Brownian motion.

      2. Define $P_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$. Show that both $hat{S}_t=e^{-(r-d)t}S_t$ and $hat{P_t}=e^{-rt}P_t$ are $mathbb{Q}$-martingales.

      3. Use the Martingale Representation Theorem to conclude the existence of a predictable process $A$ such that $hat{P}_t=hat{P}_0+int_0^tA_sdhat{S}_s$ under $mathbb{Q}$.

      4. Construct the portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ which consists of $hat{P}_t-A_that{S}_t$ units of cash and $A_te^{dt}$ units of the stock at time $t$. The value of this portfolio is $P_t$.

      5. Since $P_T=p_T$ we conclude from the Law of One Price that $P_t=p_t$ for all $0leq tleq T$. In other words, $p_t=e^{-r(T-t)}mathbb{E}_{mathbb{Q}}[f(S_T)midmathcal{F}_t]$.


      After going through the above steps I am wondering why the portfolio needs to be $(hat{P}_t-A_that{S}_t,A_te^{dt})$. It seems like we could simply choose $(hat{P}_t,0)$ as our portfolio and this would still have a value of $P_t$ at time $t$.






      black-scholes stochastic-processes risk-neutral-measure




      black-scholes stochastic-processes risk-neutral-measure






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          The portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ is chosen because it is a hedging portfolio. That is, unlike $(hat{P}_t,0)$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $(hat{P}_t,0)$.






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            The portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ is chosen because it is a hedging portfolio. That is, unlike $(hat{P}_t,0)$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $(hat{P}_t,0)$.






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              The portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ is chosen because it is a hedging portfolio. That is, unlike $(hat{P}_t,0)$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $(hat{P}_t,0)$.






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                The portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ is chosen because it is a hedging portfolio. That is, unlike $(hat{P}_t,0)$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $(hat{P}_t,0)$.






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                The portfolio $(hat{P}_t-A_that{S}_t,A_te^{dt})$ is chosen because it is a hedging portfolio. That is, unlike $(hat{P}_t,0)$ it will have the same value as the derivative an instant later. This is not generally the case for the portfolio $(hat{P}_t,0)$.







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                answered 2 days ago









                Bob Jansen♦Bob Jansen

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