From PDE to Price: Solving Black-Scholes and Delta Hedging

Solve $\mathcal{L}V=0$ , go risk-neutral, then hedge with $\Delta = \partial V/\partial S$ so $d\Pi_t = r\Pi_t dt$ .

5/15/2023

This is the capstone for the series: we turn the PDE into a price, swap to the risk-neutral lens, and sanity-check the hedge.

Attention Conservation Notice

read the warning

Final post in the five-part series--math-heavy, implementation-light, complete derivation.

Recap (where we have been)

We built Brownian motion as a random-walk limit: /brownian-motion-as-a-symmetric-random-walk-limit.
We framed SDEs and argued for geometric Brownian motion (GBM): /stochastic-differential-equations-geometric-brownian-motion.
We used Ito's Lemma to understand transforms of diffusions: /itos_lemma.
We produced the Black-Scholes PDE by delta-hedging a small portfolio: /black-scholes-pde.

Today we close the loop: solve the PDE, hop into the risk-neutral world, write down the closed form, and tie it back to hedging.

The PDE we need to solve

For a European price $V(t,S)$ with maturity $T$ and payoff $\Phi(S_T)$ , the Black-Scholes PDE (no dividends, constant $r,\sigma$ ) is

\frac{\partial V}{\partial t} + \tfrac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - r V = 0,\qquad V(T,S)=\Phi(S).

We'll solve it two ways:

transform to the heat equation (classical PDE route), and
apply a change of measure (risk-neutral route / Feynman-Kac).

Same destination, two viewpoints. Two proofs keep us honest; they're the algebraic version of diversification.

Route A: Heat-equation transform (the classical solve)

Set time-to-maturity $\tau = T-t$ and work with the call $C(t,S;K)$ where $\Phi(S)=(S-K)^+$ . Define the log-moneyness

x = \ln\!\frac{S}{K},

and the re-scaled function

u(x,\tau) = e^{\alpha x + \beta \tau}\,\frac{C(t,S;K)}{K}.

Choose $\alpha,\beta$ to cancel the first-derivative and zero-order terms in the PDE. A standard calculation gives

\alpha = \frac{1}{2} - \frac{r}{\sigma^2},\qquad \beta = -\left(\frac{1}{2}\sigma^2\alpha^2 + r\alpha - r\right).

With that choice, $u$ satisfies the heat equation

\frac{\partial u}{\partial \tau} = \tfrac{1}{2}\sigma^2 \frac{\partial^2 u}{\partial x^2},

with terminal condition

u(x,0) = e^{\alpha x}(e^x - 1)^+.

The heat equation solution is a Gaussian convolution. Evaluate the convolution, undo the change of variables, and out drops the Black-Scholes formula:

C(t,S;K) = S\,N(d_1) - K e^{-r\tau} N(d_2),

d_{1,2} = \frac{\ln(S/K) + (r \pm \tfrac{1}{2}\sigma^2)\tau}{\sigma\sqrt{\tau}},\qquad \tau=T-t,

where $N(\cdot)$ is the standard normal CDF. The put follows by put-call parity,

P(t,S;K) = K e^{-r\tau} N(-d_2) - S N(-d_1).

PDE intuition. The change of variables strips out drift and discounting so only diffusion remains in log-moneyness. It's the same bell curve from the random-walk limit humming in a new key.

Route B: Risk-neutral valuation (the probabilistic solve)

From post #2, under the physical measure $\mathbb P$ ,

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

Define the market price of risk $\theta = (\mu - r)/\sigma$ . Girsanov's theorem says the density process

\Lambda_t = \exp\!\Bigl(-\theta W_t - \tfrac{1}{2}\theta^2 t\Bigr)

tilts $\mathbb P$ to a new measure $\mathbb Q$ so that

W_t^{\mathbb Q} = W_t + \theta t

is a Brownian motion under $\mathbb Q$ . Under $\mathbb Q$ , the stock has drift $r$ :

dS_t = r S_t\, dt + \sigma S_t\, dW_t^{\mathbb Q}.

Feynman-Kac then yields the valuation formula

V(t,S) = e^{-r\tau}\, \mathbb E^{\mathbb Q}\bigl[\Phi(S_T)\mid S_t=S\bigr].

For a European call, $S_T = S \exp\!\bigl((r-\tfrac{1}{2}\sigma^2)\tau + \sigma\sqrt{\tau}\,Z\bigr)$ with $Z\sim\mathcal N(0,1)$ under $\mathbb Q$ . Compute the expectation explicitly to recover the same $C = S N(d_1) - K e^{-r\tau} N(d_2)$ .

Measure-change intuition. Risk premiums move from the drift into the measure. Pricing becomes "discount the payoff under a world where every asset earns $r$ ." In that world, volatility is the only knob the option cares about; $\mu$ vanishes like a stage prop.

Delta hedging, replication, and where the PDE came from

Reprising the portfolio from post #4, hold +1 option and $-\Delta_t$ shares with $\Delta_t = \partial V/\partial S$ . Ito + the product rule give

d\Pi_t = dV_t - \Delta_t\, dS_t = \Bigl(\frac{\partial V}{\partial t} + \tfrac{1}{2}\sigma^2 S_t^2 \frac{\partial^2 V}{\partial S^2}\Bigr)dt.

The $dW_t$ term vanishes by construction. No instantaneous risk $\Rightarrow$ no-arbitrage $\Rightarrow d\Pi_t = r \Pi_t dt$ $\Rightarrow$ the PDE above. Solve the PDE and the delta that makes the hedge work is, mechanically,

\Delta_{\text{call}}(t,S) = \frac{\partial C}{\partial S} = N(d_1),\qquad \Delta_{\text{put}}(t,S) = \Delta_{\text{call}} - 1 = N(d_1) - 1.

In discrete time the hedge is imperfect. Over a small $\Delta t$ ,

\text{P\&L}_{\text{hedged}} \approx \tfrac{1}{2}\,\Gamma\,(\Delta S)^2 - \Theta\,\Delta t - \text{financing},

where $\Gamma=\partial^2 V/\partial S^2$ and $\Theta=-\partial V/\partial t$ . With continuous rebalancing and no costs this error vanishes; with discrete rebalancing and frictions it does not--which is where real trading lives.

Greeks at a glance (for intuition, not worship)

Let $\phi(\cdot)$ be the standard normal pdf.

Delta. $\Delta = N(d_1)$ (call), $N(d_1)-1$ (put). Sensitivity to spot. It is the replicating weight.
Gamma. $\Gamma = \dfrac{\phi(d_1)}{S\sigma\sqrt{\tau}}.$ Curvature in spot; source of hedging error when you hedge discretely.
Vega. $\mathcal V = \dfrac{\partial V}{\partial \sigma} = S \phi(d_1)\sqrt{\tau}.$ Sensitivity to implied vol; symmetric for calls and puts.
Theta. $\Theta_{\text{call}} = -\dfrac{S\phi(d_1)\sigma}{2\sqrt{\tau}} - r K e^{-r\tau} N(d_2).$ Time decay (typically negative).
Rho. $\rho_{\text{call}} = \tau K e^{-r\tau} N(d_2).$ Sensitivity to rates (small unless long-dated).

Connection. $\Theta + \tfrac{1}{2}\sigma^2 S^2 \Gamma = r(V - S\Delta)$ . It's the PDE rearranged--the "theta-gamma-carry" identity traders scribble on whiteboards.

Interpretations worth keeping

Replication. The option is the present value of a self-financing stock+cash strategy that reproduces the payoff. That's "no arbitrage" in plain clothes.
Risk-neutrality. Prices are discounted $\mathbb Q$ -expectations. Pick the bank account as numeraire and everything earns $r$ on average; switch numeraires if a forward or bond makes life easier.
Diffusion worldview. The heat-equation proof says Black-Scholes is a diffusion with Gaussian smoothing in log-space. That's why the log shows up in $d_1,d_2$ , and why the CLT from post #1 keeps humming in the background.

Extensions (the small print)

Continuous dividends (or carry $q$ ). Replace $r$ with $r-q$ in drift terms and $S$ with $S e^{-q\tau}$ in the formula: $C = S e^{-q\tau} N(d_1) - K e^{-r\tau} N(d_2)$ with $d_{1,2}$ using $r-q$ .
Forwards as numeraire. Pricing in the forward measure makes moneyness $\ln(F/K)$ central, with $F = S e^{(r-q)\tau}$ . It cleans up term-structure effects and generalizes well.
Beyond GBM. Smiles and skews reveal that markets want stochastic/local volatility and jumps. Black-Scholes is the zeroth-order approximation around which we perturb and calibrate.

Extra notes

Library of identities. With $Z\sim\mathcal N(0,1)$ , $N(-z)=1-N(z)$ and $\phi(z)=\frac{1}{\sqrt{2\pi}}e^{-z^2/2}$ . Handy when massaging Greeks or parity.
Feynman-Kac links the routes. The PDE solution and the $\mathbb Q$ -expectation are the same object; FK is the bridge between calculus and probability.
On limitations. Continuous rebalancing, zero costs, constant $\sigma$ , lognormal tails, complete markets. Realistic models relax at least one. More realism = more work, and usually better hedges.
Series map. If you landed here first, start at /brownian-motion-as-a-symmetric-random-walk-limit, then /stochastic-differential-equations-geometric-brownian-motion and /itos_lemma; the PDE machinery lives in /black-scholes-pde.