Chapter 11

Hard

Feynman–Kac and the Black-Scholes PDE

00 · Symbol Glossary

$V(t,S)$option value as a function of time and price

The no-arbitrage value of a derivative at time $t$ when the underlying is at level $S$ . Chapters 08–10 studied $S_t$ as a random process; this chapter studies $V$ , a deterministic function of the two variables $(t,S)$ , which is the object a PDE can be written about.

$\dfrac{\partial V}{\partial t}$theta — time decay

The rate $V$ changes as calendar time passes with $S$ held fixed. Distinct from $dV_t$ , which also includes the randomness coming from $S_t$ moving — $\partial V/\partial t$ is only the frozen- $S$ slice, exactly as in Chapter 15 of Calculus's partial-derivative convention.

$\dfrac{\partial V}{\partial S}$delta — hedge ratio

The sensitivity of $V$ to the underlying, holding $t$ fixed. This is the number of shares a hedger holds to offset the option's exposure to $S$ — Section 03 derives exactly why.

$\dfrac{\partial^2V}{\partial S^2}$gamma — convexity

The curvature of $V$ in $S$ . Chapter 06's Itô correction term contributes exactly this second derivative, multiplied by $\tfrac12\sigma^2S^2$ , whenever $S$ follows a diffusion.

$\Pi_t$hedged portfolio value

The value at time $t$ of a portfolio long the option and short $\Delta$ shares of the underlying: $\Pi_t=V(t,S_t)-\Delta S_t$ . Section 03 builds this to remove randomness.

01 · PDE ↔ Expectation

Two ways to price a claim that pays a known function $f(S_T)$ at time $T$ : (a) solve a PDE for $V(t,S)$ backward from the terminal condition $V(T,S)=f(S)$ , or (b) compute today's price as a discounted expectation, $V(t,S)=\mathbb{E}[e^{-r(T-t)}f(S_T)\mid S_t=S]$ under the risk-neutral measure of Chapter 10. These look like different mathematical objects — a PDE is a statement about derivatives of a deterministic function, an expectation is an integral against a probability distribution — yet Feynman–Kac says they are the same number, for the right pairing of PDE and diffusion.

The direction of time is worth pausing on. Chapters 05–09 built processes forward: start at $t=0$ , run the SDE, watch $S_t$ evolve. Pricing runs the other way: you know the payoff at the end, $T$ , and want the value today, $t<T$ . Feynman–Kac is the bridge that lets a forward-in-time SDE answer a backward-in-time valuation question.

Example — the driftless, undiscounted case

Let $u(t,x)$ solve the heat equation $\partial u/\partial t+\tfrac12\partial^2u/\partial x^2=0$ with $u(T,x)=f(x)$ . Feynman–Kac's simplest instance says $u(t,x)=\mathbb{E}[f(x+W_{T-t})]$ — the solution at $(t,x)$ is just the expected payoff after running a Brownian motion for the remaining time $T-t$ , starting at $x$ . Every ingredient of the full theorem below (drift, discounting) is an extra term layered on top of this one identity.

02 · The Feynman–Kac Formula

Definition — Feynman–Kac Theorem

Let $X_t$ solve $dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dW_t$ , and suppose $u(t,x)$ solves the PDE

\frac{\partial u}{\partial t} + \mu(t,x)\frac{\partial u}{\partial x} + \frac12\sigma(t,x)^2\frac{\partial^2 u}{\partial x^2} - r\,u = 0, \qquad u(T,x)=f(x)

Then $u$ has the probabilistic representation

u(t,x) = \mathbb{E}\left[e^{-r(T-t)}f(X_T)\ \middle|\ X_t=x\right]

Plain language: the PDE's first three terms are exactly Itô's lemma's drift-and-convexity correction (Chapter 06) applied to $u(t,X_t)$ , set to zero on average; the $-ru$ term accounts for discounting the payoff back to today. Solving the PDE and computing the expectation are two routes to the same $u$ — one is analysis, one is probability, and Feynman–Kac is the dictionary between them.

❌ Dropping (or duplicating) the discount term

Writing the PDE as $\partial u/\partial t+\mu u_x+\tfrac12\sigma^2u_{xx}=0$ (no $-ru$ ) while still discounting the payoff in the expectation, or the reverse — including $-ru$ in the PDE but computing $u(t,x)=\mathbb{E}[f(X_T)]$ with no discount factor.

Why it breaks: the $-ru$ term and the $e^{-r(T-t)}$ factor are the same discounting, appearing once on each side of the correspondence. Feynman–Kac pairs a specific PDE with a specific expectation; changing one side without the matching change on the other breaks the identity.

Consequence: a PDE solved without $-ru$ gives the expected undiscounted payoff, not a price. Plugging that into a formula that also discounts double-counts nothing, or — worse — silently prices as if $r=0$ while the market clearly has $r\neq0$ .

03 · Replication: The Hedged Portfolio

Feynman–Kac says a PDE solution equals an expectation; it does not yet say that expectation is an enforceable price. That requires an argument that trading can replicate the payoff, so that any other price permits arbitrage. Build a portfolio long one option, short $\Delta$ shares:

\Pi_t = V(t,S_t) - \Delta S_t

If $\Delta$ can be chosen so that $\Pi_t$ 's change over $[t,t+dt]$ has no $dW_t$ term, then $\Pi_t$ is momentarily riskless, and a riskless portfolio in a no-arbitrage market must earn exactly the risk-free rate $r$ . That single requirement — no free lunch — pins down $V$ completely, which is the content of the next section.

Self-financing, one instant at a time

"Hedged" here means locally riskless over the next instant $dt$ , not riskless forever. $\Delta=\partial V/\partial S$ changes as $S_t$ and $t$ move, so the hedge must be rebalanced continuously — a self-financing, dynamically rebalanced replicating strategy, formalized in full in Chapter 13.

Practice — Why the hedge ratio, not some other number

Suppose a hedger used $\Delta=1$ (short one share per option) regardless of $S$ or $t$ , instead of $\partial V/\partial S$ . Explain, in terms of the $dW_t$ coefficient in $d\Pi_t$ , why this portfolio is not riskless, and what would have to be true about $V$ for $\Delta=1$ to accidentally work.

04 · Deriving the Black-Scholes PDE

Step-by-step — From Itô's lemma to the Black-Scholes PDE

Apply Itô's lemma (Chapter 06) to

V(t,S_t)

under GBM

dS_t=\mu S_t\,dt+\sigma S_t\,dW_t

dV = \left(\dfrac{\partial V}{\partial t}+\mu S\dfrac{\partial V}{\partial S}+\dfrac12\sigma^2S^2\dfrac{\partial^2V}{\partial S^2}\right)dt + \sigma S\dfrac{\partial V}{\partial S}\,dW_t

Form the hedged portfolio

\Pi=V-\Delta S

and choose

\Delta=\dfrac{\partial V}{\partial S}

so the

dW_t

terms from

dV

and

-\Delta\,dS_t

exactly cancel:

d\Pi = \left(\dfrac{\partial V}{\partial t}+\dfrac12\sigma^2S^2\dfrac{\partial^2V}{\partial S^2}\right)dt

Invoke no-arbitrage: a riskless portfolio earns the risk-free rate, so

d\Pi = r\Pi\,dt = r\left(V-S\dfrac{\partial V}{\partial S}\right)dt

Equate the two expressions for $d\Pi$ and cancel

dt

\dfrac{\partial V}{\partial t}+\dfrac12\sigma^2S^2\dfrac{\partial^2V}{\partial S^2} = rV - rS\dfrac{\partial V}{\partial S}

Rearrange into the standard form:

\dfrac{\partial V}{\partial t} + rS\dfrac{\partial V}{\partial S} + \dfrac12\sigma^2S^2\dfrac{\partial^2V}{\partial S^2} - rV = 0

— the Black-Scholes PDE.

Notice what disappeared: $\mu$ , the real-world drift, is nowhere in the final equation. It canceled in step 2 along with the $dW_t$ term, because the hedge that removes randomness removes the drift's partner risk premium along with it. The PDE only ever sees $r$ — which is exactly Chapter 10's Girsanov statement, arrived at here by a hedging argument instead of a measure change.

05 · Reading the PDE Term by Term

\underbrace{\frac{\partial V}{\partial t}}_{\text{time decay}} + \underbrace{rS\frac{\partial V}{\partial S}}_{\text{risk-neutral drift}} + \underbrace{\frac12\sigma^2S^2\frac{\partial^2V}{\partial S^2}}_{\text{convexity / Itô term}} - \underbrace{rV}_{\text{discounting}} = 0

$\partial V/\partial t$ is how value erodes or accrues purely from the passage of time, $S$ fixed. $rS\,\partial V/\partial S$ is the drift contribution, but at the risk-free rate $r$ , not the stock's real expected return $\mu$ — the PDE has already priced as if every asset drifted at $r$ , consistent with the risk-neutral measure of Chapter 10. $\tfrac12\sigma^2S^2\,\partial^2V/\partial S^2$ is the Itô correction (Chapter 06): it is present precisely because $S_t$ has nonzero quadratic variation, and it is the term that makes convex payoffs (like options) worth strictly more than their linear approximation suggests. $-rV$ discounts the value itself, since a claim worth $V$ today must earn $r$ like any riskless asset once perfectly hedged.

Same conclusion, two routes

Chapter 10 reached "replace $\mu$ with $r$ " through a change of measure — reweight probabilities so the drift becomes $r$ , leaving $\sigma$ untouched. This chapter reaches the identical replacement through hedging — build a riskless portfolio, and $\mu$ cancels algebraically. Feynman–Kac is the formal statement that these two routes must agree.

06 · Exercises

EXERCISE 11.1

Set $\mu=0$ , $\sigma=0$ , $f(x)=1$ in the Feynman–Kac PDE, and check the constant function $u(t,x)=e^{-r(T-t)}$ satisfies both the PDE and the terminal condition.

With no drift or diffusion, the PDE reduces to $\partial u/\partial t - ru=0$ , $u(T,x)=1$ . The function $u(t,x)=e^{-r(T-t)}$ has $\partial u/\partial t = re^{-r(T-t)}=ru$ , so $\partial u/\partial t - ru=0$ holds, and $u(T,x)=e^0=1$ matches the terminal condition. This is just the price of a riskless zero-coupon bond, recovered as the degenerate case of Feynman–Kac with a deterministic $X_t\equiv x$ .

Use Feynman–Kac to recover the price of a riskless bond paying $1$ at $T$ : what PDE and expectation correspond to $\mu=\sigma=0$ , $f(x)=1$ ?

EXERCISE 11.2

Redo steps 1–2 of Section 04 but leave $\Delta$ general (not yet set to $\partial V/\partial S$ ), and look at the coefficient of $dW_t$ in $d\Pi$ .

$d\Pi = dV - \Delta\,dS_t$ has $dW_t$ -coefficient $\sigma S\dfrac{\partial V}{\partial S} - \Delta\sigma S = \sigma S\left(\dfrac{\partial V}{\partial S}-\Delta\right)$ . This vanishes for every possible path only if $\Delta=\partial V/\partial S$ exactly — any other choice leaves a nonzero multiple of $dW_t$ , an unpredictable term that no amount of averaging removes from a single realized path.

Show directly that $\Delta=\partial V/\partial S$ is the only hedge ratio that eliminates the $dW_t$ term from $d\Pi$ .

EXERCISE 11.3

Compare the Black-Scholes PDE's convexity term for a claim with $\partial^2V/\partial S^2>0$ everywhere (a call) against one with $\partial^2V/\partial S^2=0$ (a forward contract, $V=S-Ke^{-r(T-t)}$ ).

For a forward, $V=S-Ke^{-r(T-t)}$ has $\partial V/\partial S=1$ and $\partial^2V/\partial S^2=0$ , so the convexity term vanishes and the PDE reduces to an ordinary linear discounting relation — no Itô correction needed because the payoff is linear in $S$ . A call's payoff $\max(S_T-K,0)$ is convex, so $\partial^2V/\partial S^2>0$ , and that term contributes strictly positive value beyond what a naive linear (forward-like) valuation would give — the source of an option's time value.

Explain why the convexity term $\tfrac12\sigma^2S^2\partial^2V/\partial S^2$ vanishes for a forward contract but not for a call option, and what that implies about each instrument's "time value."

07 · Chapter Summary

Concept	Meaning
Feynman–Kac	PDE with $-ru$ term $\Leftrightarrow$ discounted expectation $\mathbb{E}[e^{-r(T-t)}f(X_T)]$
Hedged portfolio	$\Pi=V-\Delta S$ ; riskless only when $\Delta=\partial V/\partial S$
No-arbitrage	riskless $\Pi$ must earn exactly $r$
Black-Scholes PDE	$V_t+rSV_S+\tfrac12\sigma^2S^2V_{SS}-rV=0$
Missing ingredient	$\mu$ cancels — hedging reaches the same $r$ -only drift as Girsanov (Chapter 10)
Convexity term	present whenever the payoff is nonlinear in $S$

Next: Chapter 12 — The Black-Scholes Formula, Greeks, and Put-Call Parity, which solves this PDE explicitly for a call and a put.