8.6 The Black–Scholes PDE by delta hedging

Assumptions and the PDE#

Derivation

By Itô's lemma (8.14) with $a=\mu S$, $b=\sigma S$,

$$dV=\Big(V_t+\mu SV_S+\tfrac12\sigma^2S^2V_{SS}\Big)dt+\sigma SV_S\,dW_t .$$

Form a portfolio long one derivative and short $\Delta$ units of the underlying, $\Pi=V-\Delta S$, holding $\Delta$ fixed over $[t,t+dt]$. Over $dt$ the short position loses the price change $\Delta\,dS$ and also pays the dividends owed to the lender of the shares, $\Delta qS\,dt$. So

$$\begin{aligned} d\Pi&=dV-\Delta\,dS-\Delta qS\,dt\\ &=\Big(V_t+\mu SV_S+\tfrac12\sigma^2S^2V_{SS}-\Delta\mu S-\Delta qS\Big)dt+\sigma S\,(V_S-\Delta)\,dW_t . \end{aligned}$$

Choose $\Delta=V_S$. The $dW$ term vanishes, and so does every term containing $\mu$:

$$d\Pi=\Big(V_t+\tfrac12\sigma^2S^2V_{SS}-qSV_S\Big)dt .$$

The portfolio is now riskless over $dt$. By (A4) it must earn the risk-free rate, otherwise borrowing to buy it (or shorting it to lend) is an arbitrage: $d\Pi=r\Pi\,dt=r(V-SV_S)\,dt$. Equating the two expressions,

$$V_t+\tfrac12\sigma^2S^2V_{SS}-qSV_S=rV-rSV_S ,$$

which rearranges to (8.18).∎

Reading the PDE#

Remarks

1. $\mu$ has dropped out. As in the one-period model, the price depends on volatility and not on expected return. Two investors with different views on Nifty's drift agree on the option's value if they agree on $\sigma$.
2. The rigorous version. The step "hold $\Delta$ fixed over $dt$" hides a subtlety, since $\Delta=V_S$ changes as $S$ moves and $d(\Delta S)\neq\Delta\,dS$. The clean argument constructs a self-financing portfolio $X_t$ of $\Delta_t$ shares and the rest in the bond, with $dX=\Delta\,dS+\Delta qS\,dt+r(X-\Delta S)dt$, and asks for $X_t=V(t,S_t)$ for all $t$. Matching $dW$ terms forces $\Delta_t=V_S$; matching $dt$ terms gives (8.18). See Shreve II, 4.5.
3. Boundary behaviour. For a call, $V(t,0)=0$ and $V(t,S)\sim Se^{-q(T-t)}-Ke^{-r(T-t)}$ as $S\to\infty$; for a put, $V(t,0)=Ke^{-r(T-t)}$ and $V\to0$ as $S\to\infty$. With these, (8.18) has a unique solution of polynomial growth.
4. The trading reading. Rearranged with $\Delta=V_S$, $\Gamma=V_{SS}$, $\Theta=V_t$, (8.18) is (8.27) of 8.9: a delta-hedged option earns $\Theta\,dt$ from time decay and $\frac12\Gamma(dS)^2$ from convexity, and the PDE says these cancel on average when realised variance equals $\sigma^2$. M09 builds the volatility-trading P&L on this.