Deriving the Black–Scholes PDE With a Delta Hedge

Apply Itô’s lemma to the derivative price, hold the amount of stock needed to match its instantaneous Brownian exposure, and put the remaining value in a cash account. A self-financing strategy that replicates the option must then match its drift as well. With no dividends and a constant risk-free rate

r

, the result is

V_t+\tfrac12\sigma^2 S^2V_{SS}+rSV_S-rV=0

At one instant, a call and its stock are exposed to the same unpredictable shock. A call worth

10

can move by roughly

0.60

for each

1

stock move if its delta is

0.60

. Holding

0.60

shares reproduces that first-order movement. The stock position usually costs more than the call, so the replicating portfolio also holds a negative cash balance. The interest on that balance is part of the portfolio's return.

The drift of the real-world stock process appears on both sides of the replication equation. It cancels because the amount of stock exposed to that drift is the same on both sides. Volatility survives through curvature. A convex payoff reacts differently to two small moves in opposite directions, and the squared Brownian increment contributes at the order of elapsed time.

Match the exposures before matching the drift

Treat $V(t,S)$ as a price per option and $V_S$ as its value change per dollar of stock move. Holding $V_S$ shares matches the coefficient of $dW$ because the stock itself contributes $\sigma S\,dW$ per share. The cash position is whatever remains of the option value after buying those shares. The hedge is self-financing if changes in the number of shares are paid for from cash. That is why the portfolio gains contain $\Delta\,dS$ and the interest on cash, with no extra injection of money when delta changes.

Suppose $dS=\mu S\,dt+\sigma S\,dW$ . Before expiration, Itô’s lemma gives

dV=\left(V_t+\mu SV_S+\tfrac12\sigma^2S^2V_{SS}\right)dt+\sigma SV_S\,dW.

A portfolio worth $V$ holding $\Delta$ shares has cash $V-\Delta S$ . When it is self-financing, its gains are

dV=\Delta\,dS+r(V-\Delta S)\,dt.

Self-financing means trades between stock and cash are paid from within the portfolio. It is why this gains equation contains

\Delta\,dS

even when

\Delta

changes with time. Comparing diffusion terms gives

\Delta=V_S

. Comparing drifts and cancelling

\mu S V_S

yields the PDE. A European call supplies the terminal condition

V(T,S)=(S-K)^+

. Other payoff conditions can lead to other solutions of the same pricing equation.

Write the two drift coefficients side by side. Itô gives $V_t+\mu S V_S+\sigma^2 S^2 V_{SS}/2$ . The hedge earns $\mu S V_S+r(V-SV_S)$ from the stock and cash. The $\mu S V_S$ terms are identical because the two strategies hold the same instantaneous stock exposure. Subtracting them leaves $V_t+\sigma^2 S^2 V_{SS}/2=r(V-SV_S)$ . Rearranging moves the stock financing term to the left, producing $rSV_S-rV$ . This explains why the physical drift drops out even though the stock itself still has that drift.

What the equation says about theta

Take a moment when $S=100$ , $V=10$ , $V_S=0.60$ , and $V_{SS}=0.02$ . Use $r=5\%$ and $\sigma=20\%$ per year. The cash position is $10-60=-50$ , so it contributes $-2.50$ per year at that instant. The curvature term contributes $\tfrac12(0.20)^2(100)^2(0.02)=4.00$ per year. The PDE therefore gives $V_t=-2.50-4.00=-6.50$ per year.

Now let the assumed physical stock drift be

8\%

. The option’s local expected change is

-6.50+4.80+4.00=2.30

per year. The portfolio earns

4.80-2.50=2.30

per year in expectation, and both sides carry the same random term. Changing

\mu

changes both expected gains by the same amount. These figures describe an instantaneous calculation, so a one-year forecast cannot be read off this local rate.

The options pricer lets you vary spot while holding volatility fixed and see the curvature that enters the PDE.

Deriving the Black–Scholes PDE With a Delta Hedge

Match the exposures before matching the drift

What the equation says about theta

Move volatility and expiry in Black–Scholes

References and next steps