Black–Scholes PDE via Self-Financing Replication¶
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, 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.
The emphasis of this page is therefore on structure rather than algebra: semimartingale dynamics, the self-financing condition, the role of the martingale representation theorem (MRT), and completeness. The unifying view across the five derivations is set out in § Why So Many Derivations?.
Mechanism: Self-Financing on a Two-Step Tree¶
Before the continuous version, the structure is visible on a two-step binomial tree. With stock \(S_0 \to \{S_0 u, S_0 d\} \to \{S_0 u^2, S_0 ud, S_0 d^2\}\) and bond \(B_t = (1+r)^t\), a trading strategy is a pair \((\alpha_t, \beta_t)\) giving share and bond holdings. Self-financing means wealth changes only through asset moves: at every node,
Working backward from the terminal payoff \(\Phi(S_2)\), the replication condition \(\Pi_t = V(t, S_t)\) in both successor states pins down \(\alpha_t\) uniquely as a difference quotient — the discrete analog of \(V_S\) — and the wealth identity fixes \(\beta_t\). No external cash is needed at any node: rebalancing finances itself out of price moves. The continuous derivation below replaces difference quotients with Itô's formula and invokes the martingale representation theorem in place of the backward induction; the same self-financing structure carries over.
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).
Why completeness matters
Recall (see § Martingale Representation Theorem): in the one-dimensional Brownian filtration, every square-integrable martingale is a stochastic integral against \(W_t\). This dimensional match — one Brownian motion, one tradable risky asset — is what forces \(\alpha_t = V_S\) in Step 6 and pins down the price uniquely. When the match fails (Exercise 6), MRT no longer determines the integrand and the replicating strategy is non-unique.
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¶
Recall (see § Itô's Lemma): applying Itô's formula to \(V(t, S_t)\) under \(\mathbb{P}\) gives
5. Replication Strategy¶
We seek a self-financing strategy \((\alpha_t, \beta_t)\) such that:
and
This is the replication condition.
6. Matching Dynamics — MRT Forces \(\alpha_t = V_S\)¶
From self-financing, \(d\Pi_t = \alpha_t\,dS_t + \beta_t\,dB_t\). Matching the diffusion coefficient of \(d\Pi_t\) with that of \(dV\) from §4 — both written against the same Brownian motion \(W\) — and invoking MRT in the one-dimensional Brownian filtration to ensure uniqueness, we obtain
This is the rigorous counterpart of the heuristic choice \(\Delta = V_S\) on the delta-hedging page.
7. Substitution and the PDE¶
Substituting \(\alpha_t = V_S\) into the drift equality and using the wealth identity \(\beta_t B_t = V - V_S S\) yields the Black–Scholes PDE:
with terminal condition \(V(T,S) = \Phi(S)\). The drift \(\mu\) cancels on both sides of the matching equation — algebraically identical to the cancellation in § delta hedging, but here driven by MRT rather than an explicit hedge choice.
8. Hedging Interpretation¶
The replication strategy is:
- \(\alpha_t\): delta hedge
- \(\beta_t\): financing position
9. Why the Drift μ Disappears¶
Recall (see § Why So Many Derivations?): each derivation removes \(\mu\) by a distinct mechanism. Here it is the MRT-enforced choice \(\alpha_t = V_S\): locally, the replicating portfolio carries no \(W\)-exposure, so its drift must equal \(r\) — leaving the PDE in terms of \(r\) and \(\sigma\) only.
10. 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 \(\sigma\).
11. 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 \(\mathbb{Q}\). The same self-financing strategy \((\alpha_t, \beta_t)\) derived here reappears in risk-neutral pricing, where the discounted wealth \(\widetilde{V}_t = e^{-rt}V_t\) satisfies \(d\widetilde{V}_t = \alpha_t\,d\widetilde{S}_t\) — the martingale property is the conclusion of the replication argument and the starting point of the risk-neutral derivation. The same identification holds for the change-of-numéraire and equilibrium derivations: in each case the strategy is the same; only the lens through which it is characterized changes.
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.