Financial importance of the diffusion term in Black-Scholes partial differential equation

Question

Consider the Black-Scholes equation $$\begin{equation}\label{eq3} \frac{\partial{V}}{\partial{t}}+\frac{1}{2}\sigma^2S^2\frac{\partial^2{V}}{\partial{S}^2}+(r-D)S \frac{\partial{V}}{\partial{S}}-rV=0,~~~~S\in (0,\infty),~~~t\in(0,T) \end{equation}$$ where $D$ is the dividend yield, $\sigma$ is the market volatility, $r$ is the interest rate.

My Question:

What does the diffusion term $\displaystyle \frac{\partial^2{V}}{\partial{S}^2}$ mean financially in the model? I know that the diffusion term comes from the application of Ito's lemma to the stochastic differential equation $$ dS=(\mu-D) Sdt+\sigma S dW ,$$ where, $\mu$ is the drift rate , $dW$ is the increment of a standard Wiener process. But how can you explain the financial relevence of the diffusion term?

Answer 1

The Delta of the option is $\Delta = \dfrac{\partial V}{\partial S}$, which basically tells you the amount of shares you need to be short in order to perfectly hedge the option instantaneously. The other term is the Gamma, $\Gamma = \dfrac{\partial^2 V}{\partial S^2}$, which measures the sensitivity of $\Delta$ with respect to changes in $S$. It is a measure of how quickly the number of shares needed to hedge the position changes, as the underlying asset itself changes.

This is an important notion to have, since in practice trades are not executed infinitely often throughout a finite interval, so one must have an idea of whether the hedge will still be reasonable some finite time after the trade was made. As you may expect, the larger the $\Gamma$ in absolute value, the greater the discretisation error of the $\Delta$-hedging strategy.

The $\Gamma$ is also closely related to the Vega of an option, which is the sensitivity with respect to the change in volatility, $\nu = \dfrac{\partial V}{\partial S}$. You can check that for the Black-Scholes differential operator $L_{BS}$, one has: \begin{equation} L_{BS} \nu = -\sigma S^2 \dfrac{\partial^2 V}{\partial S^2} = - \sigma^2 S^2 \Gamma, \end{equation} so that $- \sigma^2 S^2 \Gamma$ acts as a source for the Vega.