Stochastic Calculus Notes 05 | Black–Scholes: Replication, the PDE and the Greeks

21 Oct 2026 • Ziyuan Zhang

Overview

The price of an option is the capital needed to replicate it. This post sets up the Black–Scholes market, writes down the wealth of a self-financing trader, and matches it term by term (\(dW\) terms and \(dt\) terms) against the option’s price process. Matching the \(dW\) terms gives the delta hedge; matching the \(dt\) terms gives the Black–Scholes PDE. Solving the PDE gives the Black–Scholes formula, and differentiating it gives the Greeks.

A surprise worth noticing early: the stock’s expected return \(\mu\) disappears from the price.

1. The market

Fix \((\Omega,\mathcal F,\mathbb F,\mathbb P)\) and a Brownian motion \(W\). Two assets trade:

Stock (risky), \(S\sim GBM(\mu,\sigma)\): \(\dfrac{dS_t}{S_t}=\mu\,dt+\sigma\,dW_t\), \(S_0>0\). \(S_t(\omega)\) is the price of one share at time \(t\) in scenario \(\omega\).
Money market (riskless), \(\dfrac{dB_t}{B_t}=r\,dt\), \(B_0=1\): one dollar at time \(0\) grows to \(e^{rt}\) dollars at time \(t\), with constant \(r>0\).

2. Self-financing strategies and wealth

A trading strategy \(\Delta\) is an adapted process; \(\Delta_t(\omega)\) is the number of shares held at \((t,\omega)\). Start with capital \(x\). Self-financing: all money not in the stock sits in the money market, and no money enters or leaves after time \(0\).

The wealth \(X=X^{x,\Delta}\) splits as

\[X_t=\underbrace{\Delta_tS_t}_{\text{stock}}+\underbrace{X_t-\Delta_tS_t}_{\text{money market}},\]

and the money-market part buys \((X_t-\Delta_tS_t)/B_t\) “shares” of \(B\).

Wealth dynamics. Fix positions at the start of \([t,t+dt]\):

\[X_{t+dt}-X_t=\Delta_t(S_{t+dt}-S_t)+\frac{X_t-\Delta_tS_t}{B_t}(B_{t+dt}-B_t)\ \xrightarrow{dt\to0}\ dX_t=\Delta_t\,dS_t+(X_t-\Delta_tS_t)\frac{dB_t}{B_t}.\]

Substituting the dynamics of \(S\) and \(B\):

\[dX_t=\big(rX_t+(\mu-r)\Delta_tS_t\big)dt+\sigma\Delta_tS_t\,dW_t,\]

so \(X\) is an Itô process. Read it as: earn \(r\) on everything, plus the excess return \(\mu-r\) on the part in stock, plus noise.

3. What “price” means

We want to price a European call: the right to buy \(S\) at strike \(K\) at maturity \(T\), with payoff \((S_T-K)^+=\max(S_T-K,0)\).

Arbitrage-free price = replication cost. If starting from \(x\) and trading self-financingly with some \(\Delta\) we end with \(X^{x,\Delta}_T=(S_T-K)^+\) with probability one, then the call’s price must be \(x\). Any other price would let you sell the expensive one, buy the cheap one, and pocket the difference risk-free.

Same start + same dynamics = same process. If \(dY_t=\Theta^Y_tdt+\Delta^Y_tdW_t\) and \(dZ_t=\Theta^Z_tdt+\Delta^Z_tdW_t\) with \(Y_0=Z_0\), \(\Theta^Y\equiv\Theta^Z\), \(\Delta^Y\equiv\Delta^Z\), then \(Y_t=Z_t\) for all \(t\le T\) with probability one. This is the replication argument in one line: match the starting point and match both the \(dt\) and \(dW\) terms.

4. Matching wealth against the option price

Let \(c(t,s)\) be the call price at time \(t\) when \(S_t=s\). We know \(c(T,s)=(s-K)^+\) and want \(c\) for \(t<T\). Work with discounted processes (it simplifies the algebra): \(e^{-rt}X_t\) and \(e^{-rt}c(t,S_t)\).

Discounted wealth. By the discounting rule from Part 04,

\[d(e^{-rt}X_t)=e^{-rt}(\mu-r)\Delta_tS_t\,dt+e^{-rt}\sigma\Delta_tS_t\,dW_t .\]

Discounted price. Assume \(c\) is smooth (strictly speaking the payoff is not differentiable at \(s=K\), but \(c\) is smooth for \(t<T\)). With \(d[S,S]_t=\sigma^2S_t^2dt\) and writing \((\cdot)=(t,S_t)\):

\[d\big(e^{-rt}c(t,S_t)\big)=e^{-rt}\Big(-rc+c_t+c_s\,\mu S_t+\tfrac12c_{ss}\,\sigma^2S_t^2\Big)(\cdot)\,dt+e^{-rt}c_s(\cdot)\,\sigma S_t\,dW_t .\]

Set \(x=c(0,S_0)\) and match:

\(dW\) terms: \(e^{-rt}\Delta_t\sigma S_t=e^{-rt}c_s(t,S_t)\sigma S_t\), so
\[\boxed{\ \Delta_t=c_s(t,S_t)\ }\quad\text{— the delta hedge.}\]
\(dt\) terms: \(\Delta_t(\mu-r)S_t=-rc+c_t+c_s\mu S_t+\frac12c_{ss}\sigma^2S_t^2\). Plug in \(\Delta_t=c_s\); the \(c_s\mu S_t\) terms cancel, leaving
\[0=-rc(t,S_t)+c_t(t,S_t)+rS_t\,c_s(t,S_t)+\tfrac12\sigma^2S_t^2\,c_{ss}(t,S_t).\]

It suffices to require this for every \(t<T\) and \(s>0\), which gives the Black–Scholes PDE:

\[\begin{cases}0=-rc(t,s)+c_t(t,s)+rs\,c_s(t,s)+\tfrac12\sigma^2s^2\,c_{ss}(t,s),&t<T,\ s>0,\\[4pt] c(T,s)=(s-K)^+,&s>0.\end{cases}\]

\(\mu\) is gone. The hedge cancels the stock’s risk exactly, so how fast the stock is expected to grow is irrelevant to the price.

5. Solving the PDE

Two steps: (1) assume a solution \(c\) exists and find a formula for it; (2) check that the formula solves the PDE.

Step 1. Let \(\widetilde S\sim GBM(r,\sigma)\) under some measure \(\widetilde{\mathbb P}\) with Brownian motion \(\widetilde W\) — note the new measure and the drift \(r\) instead of \(\mu\). By Itô,

\[d\big(e^{-rt}c(t,\widetilde S_t)\big)=e^{-rt}\big(\underbrace{-rc+c_t+r\widetilde S_tc_s+\tfrac12\sigma^2\widetilde S_t^2c_{ss}}_{=0\ \text{by the PDE}}\big)dt+e^{-rt}c_s\,\sigma\widetilde S_t\,d\widetilde W_t,\]

so the discounted price is a martingale under \(\widetilde{\mathbb P}\). Together with the terminal condition,

\[e^{-rt}c(t,\widetilde S_t)=E^{\widetilde{\mathbb P}}\big[e^{-rT}c(T,\widetilde S_T)\mid\widetilde{\mathcal F}_t\big]\quad\Longrightarrow\quad c(t,\widetilde S_t)=E^{\widetilde{\mathbb P}}\big[e^{-r(T-t)}(\widetilde S_T-K)^+\mid\widetilde{\mathcal F}_t\big].\]

Write \(\widetilde S_T\) in terms of \(\widetilde S_t\):

\[\widetilde S_T=\widetilde S_t\exp\Big\{\big(r-\tfrac12\sigma^2\big)(T-t)+\sigma(\widetilde W_T-\widetilde W_t)\Big\}.\]

\(\widetilde S_t\) is known at time \(t\) and \(\widetilde W_T-\widetilde W_t\perp\!\!\!\perp\widetilde{\mathcal F}_t\), so we may freeze \(\widetilde S_t=s\). With \(\widetilde W_T-\widetilde W_t\sim N(0,T-t)\):

\[c(t,s)=\frac{e^{-r(T-t)}}{\sqrt{2\pi(T-t)}}\int_{-\infty}^{\infty}\Big(s\,e^{(r-\frac12\sigma^2)(T-t)+\sigma z}-K\Big)^+e^{-\frac{z^2}{2(T-t)}}\,dz .\]

Computing the integral gives the Black–Scholes formula:

\[\boxed{\ c(t,s)=s\,N\big(d_+(T-t,s)\big)-Ke^{-r(T-t)}N\big(d_-(T-t,s)\big)\ }\] \[d_\pm(\tau,s)=\frac1{\sigma\sqrt\tau}\Big(\log\frac sK+\big(r\pm\tfrac12\sigma^2\big)\tau\Big),\qquad N(y)=\frac1{\sqrt{2\pi}}\int_{-\infty}^ye^{-z^2/2}\,dz .\]

Only the time to maturity \(\tau=T-t\) enters, not calendar time.

Step 2. Verify directly that this \(c\) satisfies the PDE and \(c(T,s)=(s-K)^+\) (a homework exercise; the identity checked in Section 7 below is the key step).

6. The Greeks

Think of \(c=c(t,s;T,K,r,\sigma)\). Sensitivities to these inputs are the Greeks. Write \(\tau=T-t\) and \(d_\pm=d_\pm(\tau,s)\).

Greek	Definition	Formula	Sign
Delta	\(c_s\)	\(N(d_+)\)	\(>0\)
Gamma	\(c_{ss}\)	\(\dfrac{N'(d_+)}{\sigma s\sqrt\tau}\)	\(>0\)
Theta	\(c_t\)	\(-rKe^{-r\tau}N(d_-)-\dfrac{\sigma s}{2\sqrt\tau}N'(d_+)\)	\(<0\)
Vega	\(c_\sigma\)	\(s\sqrt\tau\,N'(d_+)\)	\(>0\)

Delta. A call is worth more when the stock is higher. The replicating strategy is \(\Delta_t=c_s(t,S_t)\): to first order in \(S\) (pretend \(dt=0\)), \(dX_t=c_s\,dS_t=dc(t,S_t)\), so the hedge matches the option’s moves — we are delta neutral.

Gamma. \(c_{ss}>0\): the call is convex in the stock price. To second order in \(S\),

\[dX_t=c_s\,dS_t,\qquad dc(t,S_t)=c_s\,dS_t+\tfrac12c_{ss}\,d[S,S]_t .\]

Intuition for “long convexity”. Own the call and short \(c_s\) shares. A delta hedge is a tangent line to the convex price curve; whichever way the stock jumps, the curve ends up above the tangent, so the hedged position gains \(\frac12c_{ss}(dS)^2\ge0\) from moves of either sign. Gamma is largest near the strike and becomes more concentrated as maturity approaches.

Theta. \(c_t<0\): with everything else fixed, the call loses value as time passes. That is the price paid for long gamma: with all second-order changes included,

\[dX_t=c_s\,dS_t+(c-c_sS_t)\,r\,dt,\qquad dc(t,S_t)=c_t\,dt+c_s\,dS_t+\tfrac12c_{ss}\sigma^2S_t^2\,dt,\]

and the PDE says these are equal — a perfect hedge. Theta exactly pays for the gamma gains plus financing.

Vega. \(c_\sigma>0\): higher volatility means bigger moves; up-moves help the call, down-moves hurt it only down to zero. Positive vega = long volatility.

7. Check: the hedge is perfect

With \(c=SN(d_+)-Ke^{-r\tau}N(d_-)\), \(c_s=N(d_+)\), \(c_{ss}=\frac{N'(d_+)}{\sigma S\sqrt\tau}\) and \(c_t\) from the table, we need \((c-c_sS)\,r=c_t+\frac12c_{ss}\sigma^2S^2\):

\[\text{LHS}=\big[SN(d_+)-Ke^{-r\tau}N(d_-)-SN(d_+)\big]r=-rKe^{-r\tau}N(d_-),\] \[\text{RHS}=-rKe^{-r\tau}N(d_-)-\frac{\sigma S}{2\sqrt\tau}N'(d_+)+\frac12\cdot\frac{\sigma S}{\sqrt\tau}N'(d_+)=-rKe^{-r\tau}N(d_-).\ \checkmark\]

The theta term and the gamma term cancel exactly.

8. Implied volatility

Under Black–Scholes, conditional mean and variance of returns do not change over time:

\[E\Big[\frac{dS_t}{S_t}\,\Big\vert\,\mathcal F_t\Big]=\mu\,dt,\qquad\mathrm{Var}\Big[\frac{dS_t}{S_t}\,\Big\vert\,\mathcal F_t\Big]=\sigma^2\,d[W,W]_t=\sigma^2\,dt .\]

In reality volatility is not constant. Two ways to estimate it:

historical volatility from past returns (backward-looking);
implied volatility from option prices (forward-looking).

For a European call with maturity \(T\) and strike \(K\), let \(C_{BS}(\sigma,K,T)\) be the Black–Scholes price and \(C_m(K,T)\) the observed market price. The implied volatility \(\sigma_{imp}=\sigma_{imp}(K,T)\) solves

\[C_{BS}(\sigma_{imp},K,T)=C_m(K,T).\]

Because vega \(>0\), \(C_{BS}\) is increasing in \(\sigma\), so the solution is unique. Options are typically quoted in implied volatility, not price. If the world were exactly Black–Scholes, \(\sigma_{imp}(K,T)=\sigma\) for every \(K\) and \(T\). Real data instead show \(\sigma_{imp}\) varying with strike and maturity — the volatility smile / skew — which is the market telling us the model is only an approximation.

Series contents

01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales
02 · Random Walks, Quadratic Variation and Brownian Motion
03 · Stochastic Integration
04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
05 · Black–Scholes: Replication, the PDE and the Greeks (this post)
06 · Risk-Neutral Pricing, Change of Measure and Girsanov
07 · SDEs, Feynman–Kac and Stopping Times

← Previous: Itô’s Formula, Itô Processes and Geometric Brownian Motion　·　Next: Risk-Neutral Pricing, Change of Measure and Girsanov →

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.