## Stock Price Dynamics We assume a stock price $S_t$ follows an [[Itô Processes and Itô's Lemma|Itô Process]] of the following form, where $B_t$ is a standard [[Brownian Motion]] and $a$ and $b$ are smooth functions. $ dS_t = a(t,S_t)\, dt +b(t,S_t) \, dB$ ## Value of the Financial Derivative Also, consider a function $V(t, S_t)$, which represents the payoff from a financial derivative (e.g. stock option), whose underlying is the stock price $S_t$. We apply [[Itô Processes and Itô's Lemma]] to express the differential of that function $dV$ in terms of its dependencies in $t$ and $S_t$. $ dV= \frac{\partial V}{\partial t} dt+ \frac{\partial V}{\partial S_t}dS_t+\frac{b^2}{2}\frac{\partial^2V}{\partial S_t^2} dt$ ## Self-Financing Hedge Portfolio We form a portfolio by being *long* $1$ unit of the *financial derivative* $(V)$ and *short* $q$ units of the *underlying stock* $(-qS_t)$. Hence the portfolio value $\pi$ and its differential $d\pi$ are: $ \begin{align} \pi &= V -q S_t \\[2pt] d\pi&= dV- q \,dS_t \end{align}$ Substitute the Itô's expansion of $dV$ into $d\pi$: $ d\pi = \overbrace{\frac{\partial V}{\partial t} dt+ \frac{\partial V}{\partial S_t}dS_t+\frac{b^2}{2}\frac{\partial^2V}{\partial S_t^2} dt}^{dV} - q \,dS_t \\[4pt] $ Regrouping by $dt$ and $dS_t$. $ d\pi = \left(\frac{\partial V}{\partial t} +\frac{b^2}{2}\frac{\partial^2V}{\partial S_t^2} \right)dt + \left(\frac{\partial V}{\partial S_t} - q \right)dS_t$ ## Delta Hedging To ensure that the constructed portfolio is self-financing it needs to be *perfectly hedged*. This way no cash inflows or outflows are required. To achieve that, we set our number of stocks $q$ equal to $\frac{\partial V}{\partial S_t}$. This choice cancels the random $dS_t$ term. $ d\pi = \left(\frac{\partial V}{\partial t} +\frac{b^2}{2}\frac{\partial^2V}{\partial S_t^2} \right)dt$ ## No-Arbitrage Argument Because the portfolio is perfectly hedged (no exposure to $dS_t$​), its return must be the risk‐free rate $r$. Thus we can equate: $ d\pi = \left(\frac{\partial V}{\partial t} +\frac{b^2}{2}\frac{\partial^2V}{\partial S_t^2} \right)dt = r\pi \, dt$ where: - $r$ is the risk free interest rate - $\pi$ is the nominal value of the portfolio - $r \pi \,dt$ is the risk free profit of the portfolio over infinitesimal time $dt$. We can substitute again for $\pi$ and instead of $q$ we use $\frac{\partial V}{\partial S_t}$ which we set equal. $ \left(\frac{\partial V}{\partial t} +\frac{b^2}{2}\frac{\partial^2V}{\partial S_t^2} \right)dt = r \left(V-\frac{\partial V}{\partial S_t}S_t \right) \, dt$ We can cancel out the $dt$ terms and move everything to the left side, to obtain the standard PDE. $ \boxed{\frac{\partial V}{\partial t} +\frac{b^2}{2}\frac{\partial^2V}{\partial S_t^2}+rS_t\frac{\partial V}{\partial S_t} -rV = 0}$ ## Classical Black-Scholes PDE In the standard Black–Scholes model, we specialize $b(t,S_t)= \sigma S_t$​. Substituting $b^2 = \sigma^2 S_t^2$ into the PDE gives the familiar Black–Scholes partial differential equation. $ \boxed{\frac{\partial V}{\partial t} +\frac{1}{2} (\sigma S_t)^2 \,\frac{\partial^2V}{\partial S_t^2}+rS_t\frac{\partial V}{\partial S_t} -rV = 0}$ >[!note:] >This derivation is only an intuitive one, because it omits the fact that the hedge quantity $q$ cannot be a constant under constantly changing $S_t$. However in [[Black-Scholes PDE with Dynamic Hedging]] we will see that result is correct and remains unchanged.