Black–Scholes PDE via Delta Hedging (Rigorous Version)¶
This derivation removes the physical drift \(\mu\) by self-financing replication: the martingale representation theorem enforces \(\alpha_t = V_S\), and the self-financing condition does the rest. Where the heuristic delta hedge freezes \(\Delta\) over each infinitesimal interval and appeals to an informal rebalancing argument, this version constructs a genuinely self-financing strategy whose value process satisfies \(dV_t = \alpha_t\,dS_t + \beta_t\,dB_t\) with no external cash flows.
This section presents a fully rigorous derivation of the Black–Scholes partial differential equation using the framework of self-financing trading strategies, following the level of mathematical precision found in Shreve (2004) and Björk (2009).
We explicitly distinguish between informal heuristic arguments and the stochastic calculus formulation, and we formalize the notion of replication. This replication argument yields the same PDE obtained via risk-neutral valuation and equilibrium (SDF) pricing—the three approaches are equivalent, but differ in method and emphasis.
1. Market Model¶
We work on a filtered probability space \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \ge 0}, \mathbb{P})\) supporting a standard Brownian motion \(W_t\).
Assets¶
Stock¶
The stock price process \(S_t\) follows geometric Brownian motion:
where \(\mu \in \mathbb{R}\), \(\sigma > 0\).
Money Market Account¶
Hence \(B_t = e^{rt}\).
This market is complete: with one Brownian motion and one traded risky asset, every contingent claim can be replicated by a self-financing portfolio (see Exercise 6 for the failure of this condition when the dimensional matching breaks down).
2. Trading Strategies¶
A trading strategy is a pair of adapted processes \((\alpha_t, \beta_t)\), where:
- \(\alpha_t\): number of shares of stock
- \(\beta_t\): amount invested in the money market account
Wealth Process¶
Self-Financing Condition¶
A strategy is self-financing if:
That is, changes in wealth arise only from asset price movements, not external cash flows.
3. The Derivative Pricing Problem¶
Let \(V(t,S)\) denote the price of a European derivative with payoff
We assume:
- \(V \in C^{1,2}([0,T) \times (0,\infty))\)
- Growth conditions sufficient for Itô's formula
Note: The payoff \(\Phi\) need not be smooth; the solution becomes smooth for \(t < T\).
4. Itô Dynamics of the Derivative¶
By Itô's formula:
5. Replication Strategy¶
We seek a self-financing strategy \((\alpha_t, \beta_t)\) such that:
and
This is the replication condition.
6. Matching Dynamics¶
From self-financing, substitute the asset dynamics:
Since \(\Pi_t = V(t,S_t)\), the coefficients of \(d\Pi_t\) and \(dV\) must agree.
Diffusion term. \(\alpha_t \sigma S = \sigma S V_S\), so \(\alpha_t = V_S\) (since \(\sigma > 0\), \(S > 0\)).
Drift term. Substituting \(\alpha_t = V_S\) and cancelling \(V_S \mu S\) from both sides:
7. Elimination of beta¶
From the wealth identity:
So:
Substitute into drift equation:
8. Black–Scholes PDE¶
Rearranging:
with terminal condition:
9. Hedging Interpretation¶
The replication strategy is:
- \(\alpha_t\): delta hedge
- \(\beta_t\): financing position
10. Why the Drift mu Disappears¶
The drift \(\mu\) cancels because the pricing problem is one of replication, not prediction.
Once the diffusion term is eliminated by setting \(\alpha_t = V_S\), the portfolio becomes locally riskless. By the absence of arbitrage, it must earn the risk-free rate \(r\), which forces the drift in the PDE to adjust from \(\mu\) to \(r\). The cancellation appears algebraic—\(\mu\) enters both \(dV\) and \(\alpha_t\,dS_t\) through the same Itô expansion, and the matching \(\alpha_t = V_S\) cancels it from both the diffusion and the drift simultaneously—but it reflects the economic fact that a locally riskless portfolio must earn the risk-free rate.
This reflects a deeper principle:
Pricing depends only on the absence of arbitrage, not on investors' beliefs about expected returns.
11. Hedging Flow (Mermaid Diagram)¶
flowchart TD
A[Derivative V(t,S)] --> B[Apply Ito's Formula]
B --> C[Decompose into Drift + Diffusion]
C --> D[Construct Portfolio Π]
D --> E[Choose α = V_S]
E --> F[Diffusion Eliminated]
F --> G[Portfolio Becomes Riskless]
G --> H[Must Earn Risk-Free Rate r]
H --> I[Obtain PDE]
12. Replication Structure¶
flowchart LR
S[Stock S] -->|α = V_S| P[Portfolio Π]
B[Bond B] -->|β| P
P -->|Replicates| V[Derivative V]
13. Conceptual Summary¶
The derivation consists of three rigorous steps:
- Model specification (SDEs for assets)
- Self-financing replication (match dynamics)
- No-arbitrage principle (uniqueness of price)
The Black–Scholes PDE characterizes exactly those price processes that admit a self-financing replicating strategy. It is:
- independent of \(\mu\),
- determined entirely by \(r\) and \(\sigma\).
14. Remarks¶
- The PDE is a backward parabolic equation.
- The solution is unique under suitable growth conditions.
- By Feynman–Kac, the solution admits a probabilistic representation under the risk-neutral measure: \(V(t,S) = e^{-r(T-t)}\,\mathbb{E}^{\mathbb{Q}}[\Phi(S_T) \mid S_t = S]\). Equivalently, the replication argument implies that the discounted price process \(e^{-rt}V(t,S_t)\) is a martingale under \(\mathbb{Q}\). This connects the PDE formulation to the change-of-numéraire and equilibrium derivations, where the same martingale property is the starting point rather than the conclusion.
References¶
- Shreve, S. (2004). Stochastic Calculus for Finance II.
- Björk, T. (2009). Arbitrage Theory in Continuous Time.
Exercises¶
Exercise 1. Consider the Black–Scholes market with \(r = 0.05\), \(\sigma = 0.3\), \(S_0 = 100\), and \(T = 1\). Suppose a European derivative has price function \(V(t,S) = S\,e^{-q(T-t)}\) for some constant \(q > 0\). Verify that \(V\) satisfies the Black–Scholes PDE if and only if \(q = r\). Compute the replicating portfolio \((\alpha_t, \beta_t)\) and verify the self-financing condition.
Solution to Exercise 1
Compute the partial derivatives: \(V_t = qS\,e^{-q(T-t)}\), \(V_S = e^{-q(T-t)}\), \(V_{SS} = 0\). Substitute into the Black–Scholes PDE \(V_t + rSV_S + \frac{1}{2}\sigma^2 S^2 V_{SS} - rV = 0\):
This equals zero if and only if \(q = 0\). Wait — let us recompute. We have \(V = Se^{-q(T-t)}\), so \(V_t = qSe^{-q(T-t)} = qV\), but \(rV = rSe^{-q(T-t)}\). The PDE gives:
For this to vanish we need \(q = 0\), giving \(V = S\). Alternatively, if the derivative pays a continuous dividend yield \(q\), then the correct PDE is \(V_t + (r-q)SV_S + \frac{1}{2}\sigma^2 S^2 V_{SS} - rV = 0\), under which:
So \(V = Se^{-q(T-t)}\) satisfies the dividend-adjusted PDE for any \(q\). In the standard (no-dividend) PDE, only \(q = 0\) works, confirming \(V = S\) as a trivial solution.
The replicating portfolio has \(\alpha_t = V_S = e^{-q(T-t)}\) and \(\beta_t = (V - \alpha_t S)/B_t = (Se^{-q(T-t)} - Se^{-q(T-t)})/e^{rt} = 0\). The portfolio is fully invested in the stock with no bond position. Self-financing holds because \(d\Pi_t = \alpha_t\,dS_t + \beta_t\,dB_t = e^{-q(T-t)}\,dS_t\), and by Itô's formula \(dV = qVdt + e^{-q(T-t)}dS_t\), which matches \(d\Pi_t\) only when \(qV\,dt = 0\), i.e., \(q = 0\) in the standard model (or when dividend income \(q\alpha_t S_t\,dt\) is included in the self-financing condition for the dividend case).
Exercise 2. Starting from the self-financing condition \(d\Pi_t = \alpha_t\,dS_t + \beta_t\,dB_t\) and the replication requirement \(\Pi_t = V(t, S_t)\), prove that matching the diffusion coefficients of \(d\Pi_t\) and \(dV\) uniquely determines \(\alpha_t = V_S(t, S_t)\). Explain why this uniqueness is a consequence of the martingale representation theorem in the one-dimensional Brownian filtration.
Solution to Exercise 2
Apply Itô's formula to the replication condition \(\Pi_t = V(t, S_t)\):
From self-financing, substitute the asset dynamics:
Both expressions must agree pathwise. Matching the diffusion (i.e., \(dW\)) coefficients:
Since \(\sigma > 0\) and \(S > 0\), we can divide both sides to get \(\alpha_t = V_S(t, S_t)\). This is unique because there is only one Brownian motion driving the model.
The connection to the martingale representation theorem is as follows. In the filtration generated by a single Brownian motion \(W_t\), every square-integrable martingale \(M_t\) has a unique representation \(M_t = M_0 + \int_0^t \phi_s\,dW_s\) for a unique adapted process \(\phi_s\). After discounting, the replication condition \(e^{-rt}V(t,S_t) = e^{-rt}\Pi_t\) equates two martingales (under the risk-neutral measure). Their Itô integrands with respect to \(dW\) must therefore agree, which forces \(\alpha_t = V_S\). In a multi-dimensional setting with \(d\) Brownian motions but only one stock, the diffusion coefficient would still be uniquely determined along the stock's volatility direction, but hedging components orthogonal to the stock would be uncontrolled — reflecting market incompleteness.
Exercise 3. Show that the self-financing condition \(d\Pi_t = \alpha_t\,dS_t + \beta_t\,dB_t\) is equivalent to the integrated form \(\Pi_t - \Pi_0 = \int_0^t \alpha_u\,dS_u + \int_0^t \beta_u\,dB_u\), which in turn is equivalent to \(d(\alpha_t S_t) + d(\beta_t B_t) = \alpha_t\,dS_t + \beta_t\,dB_t + S_t\,d\alpha_t + B_t\,d\beta_t + d[\alpha, S]_t + d[\beta, B]_t = d\Pi_t\). From this, derive the condition \(S_t\,d\alpha_t + B_t\,d\beta_t + d[\alpha, S]_t = 0\).
Solution to Exercise 3
The wealth process is \(\Pi_t = \alpha_t S_t + \beta_t B_t\). By the product rule (Itô's product formula):
Since \(B_t = e^{rt}\) is a finite-variation process, \(d[\beta, B]_t = 0\). The self-financing condition requires \(d\Pi_t = \alpha_t\,dS_t + \beta_t\,dB_t\). Subtracting this from the product-rule expansion:
This is the differential form of the self-financing condition expressed in terms of the portfolio weights. It states that any rebalancing of positions must be self-financing: the cost of buying more stock (\(S_t\,d\alpha_t\)) must be funded by selling bonds (\(B_t\,d\beta_t\)), with an adjustment for the quadratic covariation \(d[\alpha, S]_t\) when the hedge ratio \(\alpha_t\) depends on \(S_t\).
If \(\alpha_t = f(t, S_t)\) for some smooth function \(f\), then \(d[\alpha, S]_t = f_S \sigma^2 S_t^2\,dt\), and the self-financing constraint becomes:
The integrated form \(\Pi_t - \Pi_0 = \int_0^t \alpha_u\,dS_u + \int_0^t \beta_u\,dB_u\) follows directly from integrating \(d\Pi_t = \alpha_t\,dS_t + \beta_t\,dB_t\), which is equivalent to the differential self-financing condition above.
Exercise 4. In the derivation, the drift \(\mu\) cancels when we match the drift coefficients after setting \(\alpha_t = V_S\). Suppose instead we worked with a more general stock model \(dS_t = \mu(t, S_t)S_t\,dt + \sigma(t, S_t)S_t\,dW_t\) where \(\mu\) and \(\sigma\) are functions of \((t, S)\). Show that the replication argument still eliminates \(\mu(t, S)\) and derive the resulting PDE.
Solution to Exercise 4
With general coefficients, Itô's formula gives:
The self-financing portfolio dynamics are:
Matching diffusion coefficients: \(\alpha_t \sigma(t,S)S = \sigma(t,S)S V_S\), so \(\alpha_t = V_S\) (since \(\sigma(t,S) > 0\) and \(S > 0\)).
Matching drift coefficients with \(\alpha_t = V_S\):
The terms \(V_S \mu(t,S)S\) cancel on both sides, giving:
Using \(\beta_t B_t = V - SV_S\):
Rearranging:
The drift \(\mu(t,S)\) cancels regardless of its functional form. The PDE involves only the volatility function \(\sigma(t,S)\) and the risk-free rate \(r\). This generalization is the foundation of local volatility models: even when volatility depends on \((t,S)\), the replication argument eliminates the drift and produces a PDE that can be solved for the option price.
Exercise 5. Prove that if two \(C^{1,2}\) functions \(V\) and \(U\) both satisfy the Black–Scholes PDE on \([0,T) \times (0,\infty)\) with the same terminal condition \(V(T,S) = U(T,S) = \Phi(S)\), then \(V \equiv U\). Use the probabilistic (Feynman–Kac) representation to establish uniqueness under appropriate growth conditions.
Solution to Exercise 5
Under the risk-neutral measure \(\mathbb{Q}\), the stock follows \(dS_t = rS_t\,dt + \sigma S_t\,d\widetilde{W}_t\). Define \(w = V - U\). Then \(w\) satisfies:
By the Feynman–Kac theorem, under suitable growth conditions (e.g., \(|w(t,S)| \leq C(1 + S^p)\) for some \(C, p > 0\)), the solution has the probabilistic representation:
Therefore \(V(t,S) = U(t,S)\) for all \((t,S) \in [0,T) \times (0,\infty)\).
The growth condition is essential: without it, uniqueness can fail. For example, the heat equation (to which the Black–Scholes PDE reduces after a change of variables) admits non-trivial solutions with \(u(x,0) = 0\) that grow faster than \(e^{cx^2}\) — the classical Tychonoff counterexample. In financial terms, the growth condition excludes "doubling strategies" that generate arbitrage through unbounded positions. The standard Black–Scholes call and put prices satisfy \(|V(t,S)| \leq CS\) for some constant \(C\), which is well within the required growth bound.
Exercise 6. Consider a market with two risky assets \(S_t^{(1)}\) and \(S_t^{(2)}\), each driven by the same Brownian motion: \(dS_t^{(i)} = \mu_i S_t^{(i)}\,dt + \sigma_i S_t^{(i)}\,dW_t\) for \(i = 1, 2\), plus a bond \(B_t = e^{rt}\). A trader forms a portfolio \(\Pi_t = \alpha_t^{(1)} S_t^{(1)} + \alpha_t^{(2)} S_t^{(2)} + \beta_t B_t\). Show that replication of a derivative \(V(t, S^{(1)}, S^{(2)})\) is over-determined: the single diffusion coefficient equation imposes a constraint relating \(\alpha_t^{(1)}\) and \(\alpha_t^{(2)}\), but does not uniquely determine both. What does this imply about the market?
Solution to Exercise 6
Apply Itô's formula to \(V(t, S_t^{(1)}, S_t^{(2)})\). Since both assets are driven by the same \(dW_t\):
The self-financing portfolio has:
Matching diffusion coefficients gives the single equation:
This is one equation in two unknowns \((\alpha^{(1)}, \alpha^{(2)})\). There is a one-parameter family of solutions: for any \(\lambda\), we can set \(\alpha^{(1)} = V_{S^{(1)}} + \lambda\) and \(\alpha^{(2)} = V_{S^{(2)}} - \lambda \sigma_1 S^{(1)} / (\sigma_2 S^{(2)})\).
This means the replicating portfolio is not unique — there are infinitely many self-financing portfolios that replicate \(V\). This is a consequence of redundancy: with two assets driven by the same single Brownian motion, one asset can be replicated by the other and the bond. Indeed, applying the standard one-asset replication argument to \(S^{(2)}\) viewed as a "derivative" of \(S^{(1)}\) (they share the same source of randomness), we find that \(S^{(2)}\) is redundant. The market effectively has only one source of risk and one independent risky asset. For a well-posed replication problem, the number of independent risky assets must equal the number of independent Brownian motions — this is the dimensional matching condition for market completeness.