Deriving the Black-Scholes PDE by Replication

From Replication to a Pricing Equation

The Black–Scholes PDE is a consequence of matching two descriptions of the same wealth. Itô’s formula gives the local dynamics of a smooth claim as a function of the stock price. A self-financing portfolio holding the right number of shares has the same random fluctuation; its remaining wealth is invested in the bank account. Equating the drift terms gives the PDE.

Work on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})$ carrying a standard Brownian motion $B_t$. Under the physical measure $\mathbb{P}$, assume the stock and bank account satisfy $dS_t=\mu S_t\,dt+\sigma S_t\,dB_t$ and $dA_t=rA_t\,dt$, where $S_t>0$, the constants $\mu$ and $r$ are the stock’s expected return and the interest rate, and $\sigma>0$ is the volatility. The drift coefficient is $\mu S_t$; the diffusion coefficient is $\sigma S_t$. Assume no dividends and allow trading in the stock and bank account.

Let $C(t,S)$ be the value of a claim, with $C$ continuously differentiable in $t$ and twice continuously differentiable in $S>0$. The calculation below applies on a time interval where the relevant stochastic integrands are progressively measurable and square-integrable: in particular, assume $\mathbb{E}\int_0^T (C_S(t,S_t)\sigma S_t)^2\,dt<\infty$ and the corresponding square-integrability for the portfolio’s stock holding. If these conditions hold only after stopping, the calculation applies to the stopped processes on those intervals. Every Itô integral below is with respect to $B_t$ and an integrand in this class.

Claim Dynamics and Portfolio Wealth

Itô’s formula, including its second-derivative correction, gives the claim dynamics. With $f(t,S)=C(t,S)$, the needed derivatives are $f_t=C_t$, $f_S=C_S$, and $f_{SS}=C_{SS}$. Since $dS_t\,dS_t=\sigma^2S_t^2\,dt$, while $dB_t\,dt=0$ and $dt\,dt=0$, we obtain

dC(t,S_t)= \left(C_t+\mu S_tC_S+\frac12\sigma^2S_t^2C_{SS}\right)dt +\sigma S_tC_S\,dB_t.

Here and below, derivatives of $C$ are evaluated at $(t,S_t)$. The correction term $\tfrac12\sigma^2S_t^2C_{SS}\,dt$ is essential: omitting it would amount to applying the ordinary chain rule to a process with nonzero quadratic variation.

A self-financing strategy holds $\Delta_t$ shares and invests the rest of its wealth $V_t$ in the bank account. Self-financing means that changes in wealth come only from gains on these holdings, not from deposits or withdrawals. Thus its wealth equation is $dV_t=\Delta_t\,dS_t+r(V_t-\Delta_tS_t)\,dt$. The strategy is self-financing when the corresponding stock integrand is admissible, for example progressively measurable with $\mathbb{E}\int_0^T(\Delta_t\sigma S_t)^2\,dt<\infty$. Substitution of the stock dynamics yields

dV_t= \left(\Delta_t\mu S_t+r(V_t-\Delta_tS_t)\right)dt +\Delta_t\sigma S_t\,dB_t.

To replicate the claim locally, set the portfolio value equal to the claim value, $V_t=C(t,S_t)$, and choose $\Delta_t=C_S(t,S_t)$. The portfolio and claim then have the same Brownian coefficient. Equating their drift coefficients gives

C_t+\mu S_tC_S+\frac12\sigma^2S_t^2C_{SS} =\mu S_tC_S+r(C-S_tC_S).

The terms $\mu S_tC_S$ on the two sides cancel. Rearranging the remaining terms produces the Black–Scholes PDE.

Theorem: Under the assumptions above, if a claim value $C(t,S_t)$ is replicated by a self-financing portfolio with $\Delta_t=C_S(t,S_t)$ and $V_t=C(t,S_t)$, then its value satisfies $$ C_t+\frac12\sigma^2S^2C_{SS}+rSC_S-rC=0. $$ Conversely, if a $C^{1,2}$ function $C$ satisfies this PDE and the delta strategy is admissible, then a self-financing portfolio with value $C(t,S_t)$ and stock holding $C_S(t,S_t)$ has the same dynamics as the claim.

Proof. For necessity, replication means $dV_t=dC(t,S_t)$. The two expressions have diffusion coefficients $\Delta_t\sigma S_t$ and $C_S\sigma S_t$, which agree for $\Delta_t=C_S$. Equality of the drift coefficients then gives the displayed matching equation. Subtracting $\mu S_tC_S$ from both sides and collecting terms gives $C_t+\tfrac12\sigma^2S_t^2C_{SS}+rS_tC_S-rC=0$. For sufficiency, start with that PDE and substitute $\Delta_t=C_S(t,S_t)$ and $V_t=C(t,S_t)$ into the portfolio wealth equation. Its diffusion coefficient is $\sigma S_tC_S$, and its drift is $\mu S_tC_S+r(C-S_tC_S)$. The PDE makes this drift equal to $C_t+\mu S_tC_S+\tfrac12\sigma^2S_t^2C_{SS}$, precisely the Itô drift of the claim. Thus the dynamics agree. $\square$

This is the local replication argument behind the pricing equation. A price process that admits this self-financing replication cannot have a different value without violating the no-arbitrage principle that two portfolios with the same cash flows must have the same price. The PDE alone does not specify which claim is being priced: the terminal payoff and boundary conditions are also needed, and are treated separately.

Why the Stock’s Expected Return Disappears

The cancellation is not an assumption that the stock earns the risk-free rate. Under $\mathbb{P}$, its drift remains $\mu S_t$. Rather, the claim’s stock exposure is $\Delta_t=C_S(t,S_t)$, so the stock-related expected gain $\mu S_tC_S$ appears both in the claim’s Itô drift and in the hedging portfolio’s gain. Their difference is determined by the bank-account financing cost $r(C-SC_S)$, together with the curvature correction from Itô’s formula.

The PDE is therefore the same for different physical expected returns $\mu$, provided the other assumptions and parameters are unchanged. This derivation uses $\mathbb{P}$ and the stock dynamics under that measure; it does not assume a risk-neutral measure $\mathbb{Q}$ or change measures. In particular, removing $\mu$ from the PDE does not mean that $\mu$ is zero, nor that the stock’s expected return has been replaced by $r$.

Proposition: For fixed $r$, $\sigma$, and a smooth replicable price function, the replication equation does not depend on the physical drift parameter $\mu$.

Proof. For any value of $\mu$, the claim drift contains the term $\mu SC_S$, and the delta portfolio drift contains the same term because its stock holding is $C_S$. Subtracting the portfolio drift from the claim drift cancels these identical terms. The remaining equation is $C_t+\tfrac12\sigma^2S^2C_{SS}+rSC_S-rC=0$, which contains no $\mu$. This calculation applies for every $\mu$ for which the stated dynamics and admissibility assumptions hold. $\square$

Worked Example: Checking the Stock Value

Take $C(t,S)=S$. Then $C_t=0$, $C_S=1$, and $C_{SS}=0$. Substitution into the PDE gives

C_t+\frac12\sigma^2S^2C_{SS}+rSC_S-rC =0+\frac12\sigma^2S^2\cdot0+rS\cdot1-rS=0.

The delta strategy holds one share, so its wealth is $V_t=S_t$ with no funds in the bank account. Its self-financing dynamics are $dV_t=dS_t=\mu S_t\,dt+\sigma S_t\,dB_t$, matching the claim dynamics exactly. Check: the PDE residual is zero for every $t$ and every $S>0$, and the replication uses the stock itself.

Worked Example: Checking a Bank-Account Value

Let $K$ be a fixed amount payable at time $T$, and consider the candidate value $C(t,S)=Ke^{-r(T-t)}$, independent of $S$. Its derivatives are $C_t=rKe^{-r(T-t)}=rC$, $C_S=0$, and $C_{SS}=0$. Hence

C_t+\frac12\sigma^2S^2C_{SS}+rSC_S-rC =rC+0+0-rC=0.

The replicating strategy holds $\Delta_t=0$ shares and invests $C(t,S_t)$ in the bank account. Its wealth satisfies $dV_t=rV_t\,dt$, while differentiating $C(t,S_t)$ in time gives $dC=rC\,dt$. At maturity, $C(T,S)=Ke^0=K$. Check: the terminal value is the stated fixed payment, and the PDE residual vanishes.

Worked Example: Matching the Drift at a Particular State

Suppose at a particular time and stock value, $r=\tfrac1{10}$, $\sigma=\tfrac12$, $\mu=\tfrac3{10}$, $S=20$, $C=30$, $C_S=2$, and $C_{SS}=\tfrac1{20}$. The PDE determines the time derivative at this point:

\begin{aligned} C_t &=rC-rSC_S-\frac12\sigma^2S^2C_{SS}\\ &=\frac1{10}\cdot30-\frac1{10}\cdot20\cdot2 -\frac12\left(\frac12\right)^2\cdot20^2\cdot\frac1{20}\\ &=3-4-\frac52=-\frac72. \end{aligned}

Now compute the claim’s Itô drift using the physical drift $\mu$:

C_t+\mu SC_S+\frac12\sigma^2S^2C_{SS} =-\frac72+\frac3{10}\cdot20\cdot2+\frac52 =-\frac72+12+\frac52=11.

The replicating portfolio’s drift is also $\mu SC_S+r(C-SC_S)=\tfrac3{10}\cdot20\cdot2+\tfrac1{10}(30-2\cdot20)=12-1=11$. Check: both drift calculations give $11$, and both diffusion coefficients are $\sigma SC_S=\tfrac12\cdot20\cdot2=20$. Thus the local claim and portfolio dynamics agree at the stated point.

A Common Mistake

A frequent error is to keep the claim’s stock-drift term but omit the stock gain from the replicating portfolio. That leads to the incorrect equation $C_t+\tfrac12\sigma^2S^2C_{SS}+\mu SC_S-rC=0$. In the numerical example, its left side would be $-\tfrac72+\tfrac52+12-3=8$, not zero. The correction is to write the portfolio drift before matching: it includes $\mu S\Delta$. With $\Delta=C_S$, that term cancels the identical term in the claim drift, and the correct PDE residual is $-\tfrac72+\tfrac52+4-3=0$.

Another important distinction is that the PDE is a local condition on a sufficiently smooth price function, while replication of a particular claim also requires an appropriate terminal payoff and admissible strategy. The equation does not by itself select a unique solution without those additional data. Finally, the result depends on the model assumptions: for example, dividends or additional traded sources of risk would change the wealth equation and therefore the matching calculation.

Check Your Understanding

For each question, identify which terms come from Itô’s formula and which come from self-financing portfolio dynamics.

Write the claim drift and diffusion when $C=C(t,S)$ and $dS_t=\mu S_t\,dt+\sigma S_t\,dB_t$.
For a self-financing strategy with wealth $V_t$ and stock holding $\Delta_t$, derive its drift and diffusion coefficients.
Explain why choosing $\Delta_t=C_S(t,S_t)$ eliminates the difference between the claim and portfolio diffusion coefficients.
Substitute $C(t,S)=S$ into the Black–Scholes PDE and verify the residual.
Identify the specific matching step that removes $\mu$ from the PDE. Why does this not assert that the stock’s expected return equals $r$?

Deriving the Black-Scholes PDE by Replication

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