Derivation of the Black-Scholes PDE

As we will see, the derived partial differential equation is independent of the choice of $p$, $H:=\sqrt{\frac{1-p}{p}}$ and of $G:=H+H^{-1}$.
Let $V(S,t)$ be a smooth function and is defined over $0 \leq S_t< \infty$ and $t \leq T$. Using Taylor expansion, and recalling the values of $u$ and $l$, we can write $$V^+=V(S_t u, t+\Delta t)=V(S_t+ S_t(u-1), t+\Delta t)=V(S_t+ S_t(\mu \Delta t + \sigma \sqrt{\Delta t} H), t+\Delta t)$$ $$=V(S_t,t)+\frac{\partial V}{\partial S} S_t(\mu \Delta t + \sigma \sqrt{\Delta t} H)+ \frac{\partial V}{\partial t} \Delta t + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} S_t^2 \sigma^2 H^2 \Delta t+ \mathcal{O}{(\Delta t)^{3/2}}$$ and $$V^-=V(S_t l, t+\Delta t)=V(S_t+ S_t(l-1), t+\Delta t)=V(S_t+ S_t(\mu \Delta t - \sigma \sqrt{\Delta t} H), t+\Delta t)$$ $$=V(S_t,t)+\frac{\partial V}{\partial S} S_t(\mu \Delta t - \sigma \sqrt{\Delta t} H)+ \frac{\partial V}{\partial t} \Delta t + \frac{1}{2}\frac{\partial^2 V}{\partial S^2} S_t^2 \sigma^2 H^2 \Delta t+ \mathcal{O}{(\Delta t)^{3/2}}$$ Subtracting $V^-$ from $V^+$ gives $$V^+-V^-=2\frac{\partial V}{\partial S} S_t\sigma \sqrt{\Delta t} G+ \frac{1}{2}\frac{\partial^2 V}{\partial S^2} S_t^2 \sigma^2 (H^2-H^{-2}) \Delta t + \mathcal{O}{(\Delta t)^{3/2}} \label{Diff_V}$$ Substituting $V+$, $V^-$ and ($V^+-V^-$) into $V_t$, gives $$V_t (1+r\Delta t)=(1+r\Delta t)\frac{V^+-V^-}{u-l} +\frac{u V^-- l V^+ }{u-l} +\mathcal{O}{(\Delta t)^{2}}.$$ This gives $$- r V_t + \frac{\partial V}{\partial t} + r S \frac{\partial V}{\partial S} + \frac{1}{2} S_t^2 \sigma^2 \frac{\partial^2 V}{\partial S^2}= 0,$$ which is called the Black-Scholes partial differential equation. The Black-Scholes PDE can be transformed into the heat equation, which can be solved analytically.


Another derivation of the Black-Scholes partial differential equation
Assume that the security price follows a Geometric Brownian motion with a constant drift $\mu$ and a constant variance of $\sigma^2$. Use Ito's lemma on $V(t,S)$ $$\mathrm{d} V(t,S)= \mathrm{d} S \frac{\partial V}{\partial S} + \mathrm{d} t \frac{\partial V}{\partial t} + (\mathrm{d} S)^2 ~\frac{1}{2} \frac{\partial^2 V}{\partial S^2}$$ Substitute $\mathrm{d} S$ into $\mathrm{d} V(t,S)$ and use the Heuristic rules to get $$\mathrm{d} V(t,S)= (\mu~ S ~\mathrm{d} t+ \sigma~ \mathrm{d} \mathbf{B}_t) \frac{\partial V}{\partial S} + \mathrm{d} t \frac{\partial V}{\partial t} + (\sigma^2~ S^2 ~\mathrm{d} t) \frac{1}{2} \frac{\partial^2 V}{\partial S^2}$$ Consider the portfolio of $$\Pi= V + \Theta~ S$$ $$\mathrm{d} \Pi= \mathrm{d} V + \Theta~ \mathrm{d} S= \mathrm{d} V + \Theta (\mu~ S ~\mathrm{d} t+ \sigma ~\mathrm{d} \mathbf{B}_t)$$ $\Pi$ growth with risk-free interest rate such that to $\mathrm{d} \Pi=r \Pi \mathrm{d} t$ $$\mathrm{d}\Pi=(r-1)~ (V + \Theta~ S)= \mathrm{d}V + \Theta ~(\mu ~S ~\mathrm{d}t+ \sigma ~\mathrm{d}\mathbf{B}_t).$$ This gives $$(r~\mathrm{d}t)~(V~+~\Theta~S)=(\mu~S~\mathrm{d}t+~\sigma~\mathrm{d}\mathbf{B}_t)~\frac{\partial~V}{\partial~S}~+~\mathrm{d}t~\frac{\partial~V}{\partial~t}~+~(\sigma^2~S^2~\mathrm{d}t)~\frac{1}{2}~\frac{\partial^2~V}{\partial~S^2}+~\Theta~(\mu~S~\mathrm{d}t+~\sigma~\mathrm{d}\mathbf{B}_t).$$ Simplify to get $$-r~\mathrm{d}t~V~-~r~\mathrm{d}t~\Theta~S~+~(\mu~S~\mathrm{d}t+~\sigma~\mathrm{d}\mathbf{B}_t)~\frac{\partial~V}{\partial~S}~+~\mathrm{d}t~\frac{\partial~V}{\partial~t}~+~(\sigma^2~S^2~\mathrm{d}t)~\frac{1}{2}~\frac{\partial^2~V}{\partial~S^2}+~\Theta~(\mu~S~\mathrm{d}t+~\sigma~\mathrm{d}\mathbf{B}_t)=0.$$ Re-arrange the above equation to get $$-r~\mathrm{d}t~V~-~r~\mathrm{d}t~\Theta~S~+~\mu~S~\mathrm{d}t~\frac{\partial~V}{\partial~S}~+~\mathrm{d}t~\frac{\partial~V}{\partial~t}~+~(\sigma^2~S^2~\mathrm{d}t)~\frac{1}{2}\frac{\partial^2~V}{\partial~S^2}+~\Theta~\mu~S~\mathrm{d}t=-\sigma~\mathrm{d}\mathbf{B}_t~\frac{\partial~V}{\partial~S}-\Theta~\sigma~\mathrm{d}\mathbf{B}_t .$$ We then Hedge and choose $\Theta=-\frac{\partial V}{\partial S}$ (short selling) to eliminate the stochastic term such that $$%-r ~\mathrm{d} t~ V + r ~\mathrm{d} t ~S \frac{\partial V}{\partial S} + \mu~ S~ \mathrm{d} t~ \frac{\partial V}{\partial S} + \mathrm{d} t~ \frac{\partial V}{\partial t} + (\sigma^2 ~S^2~ \mathrm{d} t) \frac{1}{2} \frac{\partial^2 V}{\partial S^2}-\frac{\partial V}{\partial S} ~\mu ~S ~\mathrm{d} t~=0 -r ~\mathrm{d} t ~V + r~ \mathrm{d} t~ S \frac{\partial V}{\partial S} + \mathrm{d} t~ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 ~S^2~ \mathrm{d} t ~\frac{\partial^2 V}{\partial S^2}=0$$ Dividing the above equation by $\mathrm{d} t$ gives the Black-Scholes PDE.

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References

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4. Tang, Q. Mathematical Models in Industry and Finance, Lecture Notes University of Sussex (2008).
   
5. Wilmott P, Howison S and Dewunne J, The Mathematics of Financial Derivatives: A Student Introduction, (1995).