First, we define the following variables.
: The value of the portfolio at time 
. The portfolio invests in a money market account paying a constant interest rate 
and in a stock modeled by geometric Brownian motion
: At time , the portfolio holds shares of stock.
The remainder of the portfolio value 
is invested in the money market account.
The differential 
is due to two factors, the capital gain 
on the stock position and the interest earnings 
on the cash position.
Mathematically, we have
The interpretation of the three terms in the last line above is as follows:
: reflects an average underlying rate of return on the portfolio
: reflects a risk premium 
for investing in the stock
: a volatility term proportional to the size of the stock investment
Next, we consider the discounted stock price 
and the discounted portfolio value 
. We let 
, and apply the Ito-Doeblin formula (for an Ito process) to get the following.
Differential of the discounted stock price:
![\begin{aligned}d(e^{-rt}S(t))&=df(t,S(t))\\&=f_t(t,S(t))\, dt+f_x(t,S(t))\, dS(t)+\frac{1}{2}f_{xx}(t,S(t))\, dS(t)\, dS(t)\\&=-re^{-rt}S(t)\, dt+e^{-rt}\, dS(t)+0\\&=-re^{-rt}S(t)\, dt+e^{-rt}[\alpha S(t)\, dt+\sigma S(t)\, dW(t)]\\&=(\alpha-r)e^{-rt}S(t)\, dt+\sigma e^{-rt}S(t)\, dW(t)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7Dd%28e%5E%7B-rt%7DS%28t%29%29%26%3Ddf%28t%2CS%28t%29%29%5C%5C%26%3Df_t%28t%2CS%28t%29%29%5C%2C+dt%2Bf_x%28t%2CS%28t%29%29%5C%2C+dS%28t%29%2B%5Cfrac%7B1%7D%7B2%7Df_%7Bxx%7D%28t%2CS%28t%29%29%5C%2C+dS%28t%29%5C%2C+dS%28t%29%5C%5C%26%3D-re%5E%7B-rt%7DS%28t%29%5C%2C+dt%2Be%5E%7B-rt%7D%5C%2C+dS%28t%29%2B0%5C%5C%26%3D-re%5E%7B-rt%7DS%28t%29%5C%2C+dt%2Be%5E%7B-rt%7D%5B%5Calpha+S%28t%29%5C%2C+dt%2B%5Csigma+S%28t%29%5C%2C+dW%28t%29%5D%5C%5C%26%3D%28%5Calpha-r%29e%5E%7B-rt%7DS%28t%29%5C%2C+dt%2B%5Csigma+e%5E%7B-rt%7DS%28t%29%5C%2C+dW%28t%29%5Cend%7Baligned%7D&bg=ffffff&fg=000000&s=0 "\begin{aligned}d(e^{-rt}S(t))&=df(t,S(t))\\&=f_t(t,S(t))\, dt+f_x(t,S(t))\, dS(t)+\frac{1}{2}f_{xx}(t,S(t))\, dS(t)\, dS(t)\\&=-re^{-rt}S(t)\, dt+e^{-rt}\, dS(t)+0\\&=-re^{-rt}S(t)\, dt+e^{-rt}[\alpha S(t)\, dt+\sigma S(t)\, dW(t)]\\&=(\alpha-r)e^{-rt}S(t)\, dt+\sigma e^{-rt}S(t)\, dW(t)\end{aligned}")
Differential of the discounted portfolio value:
![\begin{aligned}d(e^{-rt}X(t))&=df(t,X(t))\\&=f_t(t,X(t))\, dt+f_x(t,X(t))\, dX(t)+\frac{1}{2}f_{xx}(t,X(t))\, dX(t)\, dX(t)\\&=-re^{-rt}X(t)\, dt+e^{-rt}\, dX(t)+0\\&=-re^{-rt}X(t)\, dt+e^{-rt}[rX(t)\, dt+\Delta(t)(\alpha-r)S(t)\, dt+\Delta(t)\sigma S(t)\, dW(t)]\\&=\Delta(t)(\alpha-r)e^{-rt}S(t)\, dt+\Delta(t)\sigma e^{-rt}S(t)\, dW(t)\\&=\Delta(t)\, d(e^{-rt}S(t))\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7Dd%28e%5E%7B-rt%7DX%28t%29%29%26%3Ddf%28t%2CX%28t%29%29%5C%5C%26%3Df_t%28t%2CX%28t%29%29%5C%2C+dt%2Bf_x%28t%2CX%28t%29%29%5C%2C+dX%28t%29%2B%5Cfrac%7B1%7D%7B2%7Df_%7Bxx%7D%28t%2CX%28t%29%29%5C%2C+dX%28t%29%5C%2C+dX%28t%29%5C%5C%26%3D-re%5E%7B-rt%7DX%28t%29%5C%2C+dt%2Be%5E%7B-rt%7D%5C%2C+dX%28t%29%2B0%5C%5C%26%3D-re%5E%7B-rt%7DX%28t%29%5C%2C+dt%2Be%5E%7B-rt%7D%5BrX%28t%29%5C%2C+dt%2B%5CDelta%28t%29%28%5Calpha-r%29S%28t%29%5C%2C+dt%2B%5CDelta%28t%29%5Csigma+S%28t%29%5C%2C+dW%28t%29%5D%5C%5C%26%3D%5CDelta%28t%29%28%5Calpha-r%29e%5E%7B-rt%7DS%28t%29%5C%2C+dt%2B%5CDelta%28t%29%5Csigma+e%5E%7B-rt%7DS%28t%29%5C%2C+dW%28t%29%5C%5C%26%3D%5CDelta%28t%29%5C%2C+d%28e%5E%7B-rt%7DS%28t%29%29%5Cend%7Baligned%7D&bg=ffffff&fg=000000&s=0 "\begin{aligned}d(e^{-rt}X(t))&=df(t,X(t))\\&=f_t(t,X(t))\, dt+f_x(t,X(t))\, dX(t)+\frac{1}{2}f_{xx}(t,X(t))\, dX(t)\, dX(t)\\&=-re^{-rt}X(t)\, dt+e^{-rt}\, dX(t)+0\\&=-re^{-rt}X(t)\, dt+e^{-rt}[rX(t)\, dt+\Delta(t)(\alpha-r)S(t)\, dt+\Delta(t)\sigma S(t)\, dW(t)]\\&=\Delta(t)(\alpha-r)e^{-rt}S(t)\, dt+\Delta(t)\sigma e^{-rt}S(t)\, dW(t)\\&=\Delta(t)\, d(e^{-rt}S(t))\end{aligned}")
The last line shows that change in the discounted portfolio value is solely due to change in the discounted stock price (note that the term reflecting underlying rate of return has vanished in the second-last line).
Reference
Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. New York: Springer, 2004.
Continued at: https://blog.nus.edu.sg/wuchengyuan/2022/08/13/black-scholes-merton-derivation-part-2-option-value-evolution/