3.3.2 The BSM PDE, Its Derivation & the Feynman–Kac Bridge

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

Two pictures produce the same price: the PDE picture (a parabolic partial differential equation with boundary conditions, solved analytically for European options) and the expectation picture (a discounted conditional expectation under the risk-neutral measure $\mathbb{Q}$). The Feynman–Kac theorem is the bridge that declares these two pictures identical. The practical objective: once you see the bridge, every option becomes the same kind of object - “integrate the payoff against a probability density under $\mathbb{Q}$” - and the PDE is just that integral written in differential form.

This page gives the full, honest derivation of the BSM PDE (delta-neutral replication) plus the risk-neutral/expectation derivation (Girsanov change of measure), and then shows they coincide via Feynman–Kac. This is the mathematical spine of the whole topic-folder.

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2. Mathematical Ground Truth & Derivations

2.1 Route A - Delta-Neutral Replicating Portfolio (PDE)

Underlying follows GBM under physical measure $\mathbb{P}$:

$$d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t.$$

Let $V(t,S)$ be the derivative price. By Itô–Doeblin (Shreve II Thm 4.4.6; Hull eq. 14.12):

$$dV=\left(\frac{\partial V}{\partial t}+\mu S\frac{\partial V}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}\right)dt+\sigma S\frac{\partial V}{\partial S}dW.$$

Form $\Pi =V-\Delta S$, the hedged portfolio. Substituting and choosing $\Delta =\partial V/\partial S$ kills the $dW$ term:

$$d\Pi =\left(\frac{\partial V}{\partial t}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}\right)dt.$$

A riskless portfolio must earn the risk-free rate (Shreve II §4.5; Björk Prop 7.6 - “only one short rate on an arbitrage-free market”): $d\Pi =r(V-\Delta S)dt$. Equating and rearranging gives the BSM PDE:

$$\boxed{\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}-rV=0}(0\le t<T,S\ge 0).$$

Terminal & boundary conditions (Shreve II 4.5.14):

• Terminal payoff: $V(T,S)=\max (S-K,0)$ (call), $\max (K-S,0)$ (put).
• Boundary: $V(t,0)=0$ for a call (a worthless stock, worthless option).
• Growth: $V(t,S)\sim S-e^{-r(T-t)}K$ as $S\to \infty$.

The drift $\mu$ is absent - volatility (not drift) is the pricing parameter (Shreve II §4.5).

2.2 Route B - Risk-Neutral Expectation (Girsanov / Martingale)

Shreve II Ch 5 and Björk Ch 7. Define the money-market account $D(t)=e^{-{\int }_0^{t}Rds}$ and the market price of risk $\Theta (t)=(\alpha (t)-R(t))/\sigma (t)$. Girsanov’s theorem (Shreve II Thm 5.2.3) builds the equivalent measure $\mathbb{Q}$ via the Radon–Nikodym derivative

$$Z(t)=\exp \{-{\int }_0^t\Theta (u)dW(u)-\frac{1}{2}{\int }_0^t{\Theta }^2(u)du\},$$

under which $\tilde{W}(t)=W(t)+{\int }_0^t\Theta (u)du$ is a Brownian motion. Under $\mathbb{Q}$ the stock becomes

$$dS=RSdt+\sigma Sd\tilde{W},$$

and the discounted stock $D(t)S(t)$ is a $\mathbb{Q}$-martingale. The risk-neutral pricing formula (Shreve II 5.2.30/31) is then

$$V(t)=\tilde{\mathbb{E}}\left[e^{-{\int }_t^{T}R(u)du}V(T)|F(t)\right].$$

For constant $r,\sigma$ and payoff $(S_T-K{)}^{+}$, this expectation has a closed form - the BSM formula (next page). Crucially, the change of measure changes the mean rate ($\alpha \to r$) but not the volatility or the paths.

2.3 The Feynman–Kac Bridge (Björk Props 5.5/5.6; Shreve II §4.5)

Theorem. If $F(t,x)$ solves the parabolic PDE

$$F_t+\mu F_x+\frac{1}{2}{\sigma }^2{F}_{xx}-rF=0,F(T,x)=\Phi (x),$$

for the SDE $dX=\mu dt+\sigma dW$, then

$$F(t,x)=e^{-r(T-t)}{\mathbb{E}}_{t,x}\left[\Phi (X_T)\right].$$

Applying it to the BSM PDE ($\mu =r$, the $\mathbb{Q}$-drift) says exactly: the PDE solution equals the discounted risk-neutral expectation of the payoff. The two routes in §2.1 and §2.2 are the same number by construction.

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3. Computational Implementation - verifying the bridge by Monte Carlo

Direct proof of §2.3: simulate the risk-neutral dynamics (drift $r$, not $\mu$) and compare the discounted average payoff to the closed-form BSM price. Stdlib only.

The MC expectation converges to the PDE solution - the Feynman–Kac bridge, verified numerically. (Remaining gap at $n=10^5$ is $O(1/\sqrt{n})$ sampling error, exactly as Glasserman Ch 1 predicts.)

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4. Failure Modes & First-Principles Breakdowns

1. Girsanov requires an integrability condition. The theorem needs $\tilde{\mathbb{E}}{\int }_0^T{\Theta }^2(u)du<\infty$ (equivalently a Novikov-type condition) for $Z$ to be a genuine martingale - this was a flagged omission in early extractions (Shreve II 5.2.13). Without it the change of measure is formal only.
2. Martingale Representation (MRT) needs the right filtration. Replicating an $F(T)$-claim requires $F$ to be generated by the Brownian motion (Shreve II Thm 5.3.1). Extra randomness (jumps, an untraded source) breaks completeness - the deep root of the failure modes in 05 · Failure Modes.
3. The delta-neutral “riskless portfolio” is riskless only if rebalancing is continuous. The PDE is derived for $dt\to 0$. In discrete time there is hedging error proportional to gamma and squared returns - the gamma–theta residual (04 · Greeks).
4. $\frac{1}{2}{\sigma }^2$ is not optional. Itô’s lemma uses $d{W}^2=dt$; forgetting the half-variance term breaks the martingale property and the pricing formula.

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5. References

• Shreve, Stochastic Calculus for Finance II
• Björk, Arbitrage Theory in Continuous Time
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 01 · From Zero
• Forward: 03 · Pricing Formulas · Index Hub
• Theory: Stochastic Calculus & Itô · Implied Volatility Surfaces