This is a continuation of Black-Scholes-Merton Derivation (Part 2: Option Value Evolution).
A hedging portfolio starts with initial capital 
![t\in [0,T]](https://s0.wp.com/latex.php?latex=t%5Cin+%5B0%2CT%5D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
One way to impose this equality is to ensure that
Comparing the expressions 
Comparing the 
Next, comparing the 
Simplifying and rearranging, we get
Let 
The terminal condition to be satisfied is 
Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. New York: Springer, 2004.
This is a continuation of Black-Scholes-Merton Derivation (Part 1: Portfolio Value Evolution).
Let 
According to the Ito-Doeblin formula, the differential of
The second equality uses the definition of the stock modeled by geometric Brownian motion 
Next, we compute the differential of the discounted option price 
The third equality is due to 
Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. New York: Springer, 2004.
First, we define the following variables.
The remainder of the portfolio value 
The differential 
Mathematically, we have
The interpretation of the three terms in the last line above is as follows:
Next, we consider the discounted stock price 
Differential of the discounted stock price:
Differential of the discounted portfolio value:
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
Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. New York: Springer, 2004.
The Ito-Doeblin Formula is an important formula in stochastic calculus, analogous to the chain rule in ordinary calculus. It has many applications in finance, including the derivation of the Black-Scholes-Merton equation.
Let 
The Ito-Doeblin formula in differential form is:
The main difference (compared to ordinary calculus) is that there is an extra term due to the fact that 
Integrating this expression, we get the Ito-Doeblin formula in integral form:
There is also a slightly more generalized version that allows 
Let 
The Ito-Doeblin formula in differential form is:
Shreve, Steven E. Stochastic calculus for finance II: Continuous-time models. Vol. 11. New York: Springer, 2004.
Compiling a list of cryptocurrency/blockchain terminologies and their Chinese translations: https://wcymath.wordpress.com/crypto-terms-in-chinese/
Compiled a list of papers applying (algebraic) topology to finance at: https://blog.nus.edu.sg/wuchengyuan/topology-in-finance/
Interestingly, topology seems best at detecting financial crashes, but can also predict stock price movement, do clustering and classification of financial time series, and more.
This 3 minute video explains very well how algebraic topology (in particular persistent homology) can be applied to finance:
Mathematics Genealogy is an interesting website where one can trace the academic genealogy of a researcher (in the field of math/applied math), all the way to mathematicians of previous centuries.
WordPress Homepage (https://wcymath.wordpress.com/)
![\displaystyle\begin{aligned}&\Delta(t)(\alpha-r)S(t)\, dt+\Delta(t)\sigma S(t)\, dW(t)\\&=\left[-rc(t,S(t))+c_t(t,S(t))+\alpha S(t)c_x(t,S(t))+\frac{1}{2}\sigma^2 S^2(t)c_{xx}(t,S(t))\right]\, dt\\&\quad+\sigma S(t)c_x(t,S(t))\, dW(t)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7D%26%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%5C%5C%26%3D%5Cleft%5B-rc%28t%2CS%28t%29%29%2Bc_t%28t%2CS%28t%29%29%2B%5Calpha+S%28t%29c_x%28t%2CS%28t%29%29%2B%5Cfrac%7B1%7D%7B2%7D%5Csigma%5E2+S%5E2%28t%29c_%7Bxx%7D%28t%2CS%28t%29%29%5Cright%5D%5C%2C+dt%5C%5C%26%5Cquad%2B%5Csigma+S%28t%29c_x%28t%2CS%28t%29%29%5C%2C+dW%28t%29%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![\displaystyle\begin{aligned}dc(t,S(t))&=c_t(t,S(t))\, dt+c_x(t,S(t))\, dS(t)+\frac{1}{2}c_{xx}(t,S(t))\, dS(t)\, dS(t)\\&=c_t(t,S(t))\, dt+c_x(t,S(t))(\alpha S(t)\, dt+\sigma S(t)\, dW(t))+\frac{1}{2}c_{xx}(t,S(t))\sigma^2 S^2(t)\, dt\\&=\left[ c_t(t,S(t))+\alpha S(t) c_x(t,S(t))+\frac{1}{2}\sigma^2 S^2(t)c_{xx}(t,S(t))\right]\, dt\\&\quad+\sigma S(t)c_x(t,S(t))\, dW(t)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7Ddc%28t%2CS%28t%29%29%26%3Dc_t%28t%2CS%28t%29%29%5C%2C+dt%2Bc_x%28t%2CS%28t%29%29%5C%2C+dS%28t%29%2B%5Cfrac%7B1%7D%7B2%7Dc_%7Bxx%7D%28t%2CS%28t%29%29%5C%2C+dS%28t%29%5C%2C+dS%28t%29%5C%5C%26%3Dc_t%28t%2CS%28t%29%29%5C%2C+dt%2Bc_x%28t%2CS%28t%29%29%28%5Calpha+S%28t%29%5C%2C+dt%2B%5Csigma+S%28t%29%5C%2C+dW%28t%29%29%2B%5Cfrac%7B1%7D%7B2%7Dc_%7Bxx%7D%28t%2CS%28t%29%29%5Csigma%5E2+S%5E2%28t%29%5C%2C+dt%5C%5C%26%3D%5Cleft%5B+c_t%28t%2CS%28t%29%29%2B%5Calpha+S%28t%29+c_x%28t%2CS%28t%29%29%2B%5Cfrac%7B1%7D%7B2%7D%5Csigma%5E2+S%5E2%28t%29c_%7Bxx%7D%28t%2CS%28t%29%29%5Cright%5D%5C%2C+dt%5C%5C%26%5Cquad%2B%5Csigma+S%28t%29c_x%28t%2CS%28t%29%29%5C%2C+dW%28t%29%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![\displaystyle\begin{aligned}d(e^{-rt}c(t,S(t)))&=df(t,c(t,S(t)))\\&=f_t(t,c(t,S(t)))\, dt+f_x(t,c(t,S(t)))\, dc(t,S(t))\\&\quad+\frac{1}{2}f_{xx}(t,c(t,S(t)))\, dc(t,S(t))\, dc(t,S(t))\\&=-re^{-rt}c(t,S(t))\,dt+e^{-rt}\, dc(t,S(t))\\&=e^{-rt}[-rc(t,S(t))+c_t(t,S(t))+\alpha S(t)c_x(t,S(t))\\&\quad+\frac{1}{2}\sigma^2S^2(t)c_{xx}(t,S(t))]\, dt+e^{-rt}\sigma S(t)c_x(t,S(t))\, dW(t).\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5Cbegin%7Baligned%7Dd%28e%5E%7B-rt%7Dc%28t%2CS%28t%29%29%29%26%3Ddf%28t%2Cc%28t%2CS%28t%29%29%29%5C%5C%26%3Df_t%28t%2Cc%28t%2CS%28t%29%29%29%5C%2C+dt%2Bf_x%28t%2Cc%28t%2CS%28t%29%29%29%5C%2C+dc%28t%2CS%28t%29%29%5C%5C%26%5Cquad%2B%5Cfrac%7B1%7D%7B2%7Df_%7Bxx%7D%28t%2Cc%28t%2CS%28t%29%29%29%5C%2C+dc%28t%2CS%28t%29%29%5C%2C+dc%28t%2CS%28t%29%29%5C%5C%26%3D-re%5E%7B-rt%7Dc%28t%2CS%28t%29%29%5C%2Cdt%2Be%5E%7B-rt%7D%5C%2C+dc%28t%2CS%28t%29%29%5C%5C%26%3De%5E%7B-rt%7D%5B-rc%28t%2CS%28t%29%29%2Bc_t%28t%2CS%28t%29%29%2B%5Calpha+S%28t%29c_x%28t%2CS%28t%29%29%5C%5C%26%5Cquad%2B%5Cfrac%7B1%7D%7B2%7D%5Csigma%5E2S%5E2%28t%29c_%7Bxx%7D%28t%2CS%28t%29%29%5D%5C%2C+dt%2Be%5E%7B-rt%7D%5Csigma+S%28t%29c_x%28t%2CS%28t%29%29%5C%2C+dW%28t%29.%5Cend%7Baligned%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
![\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=1a1a1a&s=0&c=20201002)
![\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=1a1a1a&s=0&c=20201002)
Chengyuan Wu 