The Black–Scholes PDE and Formula
Delta-hedging a continuously-traded option leads to a PDE whose solution is the closed-form price.
Leads to: Phase 15 relaxes constant volatility (local/stochastic vol, Gatheral).
Learning Objectives
Click a status chip to cycle: Not started → In progress → Studied → Practiced → Needs review → Mastered.
- Derive the Black–Scholes PDE by constructing a risk-free delta-hedged portfolio.
- State the Black–Scholes call and put formulas and identify each term.
- Compute the Greeks (delta, gamma, vega, theta, rho) and interpret their signs.
- Explain the assumptions of the model and how each can fail in real markets.
- Implement the formula and its Greeks and verify against known benchmark values.
Key Vocabulary
- Black–Scholes PDE
- The PDE \(V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0\) governing any European derivative under GBM.
- Delta-hedged portfolio
- Long the option, short \(\Delta=V_S\) shares; instantaneously risk-free, so it must earn \(r\).
- d1, d2
- The standardized moneyness terms \(d_{1,2}=\tfrac{\ln(S/K)+(r\pm\tfrac12\sigma^2)T}{\sigma\sqrt T}\).
- Vega
- Sensitivity of price to volatility, \(\partial V/\partial\sigma\); positive for vanilla calls and puts.
- Implied volatility
- The \(\sigma\) that makes the Black–Scholes price match a quoted market price; the market’s language for option prices.
- Model assumptions
- Constant \(\sigma\) and \(r\), GBM dynamics, no jumps, continuous frictionless trading, no dividends - idealizations, each violable.
Intuition & Motivation
Deriving the PDE by delta-hedging
Assume \(dS=\mu S\,dt+\sigma S\,dW\) and a derivative price \(V(t,S)\). By Ito’s lemma,
Form the portfolio \(\Pi=V-\Delta S\) with \(\Delta=V_S\). The \(dW\) terms cancel:
Now \(d\Pi\) is deterministic over \(dt\), so by no-arbitrage it must equal the risk-free growth of the money invested, \(d\Pi=r\Pi\,dt=r(V-SV_S)\,dt\). Equating the two expressions and cancelling \(dt\) gives the Black–Scholes PDE:
The Black–Scholes formula
Solving (14.9) with the call payoff (or evaluating the risk-neutral integral over the lognormal \(S_T\)) gives
with \(N\) the standard normal CDF. Interpret \(N(d_2)=\mathbb{Q}(S_T\gt K)\) (risk-neutral exercise probability) and \(S_0N(d_1)\) as the discounted expected stock received conditional on exercise. Puts follow from put–call parity \(C-P=S_0-Ke^{-rT}\).
The Greeks
Differentiating (14.10) gives the sensitivities used for hedging. With \(\varphi\) the normal density:
| Greek | Symbol | Call formula | Sign (call) |
|---|---|---|---|
| Delta | \(\partial_S V\) | \(N(d_1)\) | + |
| Gamma | \(\partial_{SS}V\) | \(\varphi(d_1)/(S_0\sigma\sqrt T)\) | + |
| Vega | \(\partial_\sigma V\) | \(S_0\varphi(d_1)\sqrt T\) | + |
| Theta | \(\partial_t V\) | \(-\tfrac{S_0\varphi(d_1)\sigma}{2\sqrt T}-rKe^{-rT}N(d_2)\) | − (usually) |
| Rho | \(\partial_r V\) | \(KTe^{-rT}N(d_2)\) | + |
Interactive: the Black–Scholes surface and Monte Carlo check
- Putting the real-world drift \(\mu\) in the PDE or formula; hedging removes it, leaving only \(r\) and \(\sigma\).
- Reading \(N(d_2)\) as the real probability of finishing in the money; it is the risk-neutral probability \(\mathbb{Q}(S_T\gt K)\).
- Forgetting the discount factor \(e^{-rT}\) on the strike term, or mis-signing \(d_2=d_1-\sigma\sqrt T\).
- Trusting a single Black–Scholes \(\sigma\) across strikes - real option prices imply a volatility smile, evidence the constant-vol assumption fails.
- Memorize \(d_1,d_2\) and the parity \(C-P=S_0-Ke^{-rT}\); together they reconstruct most vanilla results fast.
- Vega and gamma are largest near the money and (for gamma) near expiry - where hedging is hardest and most valuable.
- Quote and think in implied volatility, not price; it is the market’s common currency and exposes the smile the model ignores.
- Treat Black–Scholes as a coordinate system, not truth: its assumptions (constant \(\sigma\), no jumps, frictionless continuous trading) are approximations to be stress-tested, and it makes no promise of trading profit.
Knowledge Check
Practical Exercise
For \(S_0=100,K=100,r=0.03,\sigma=0.2,T=1\) you are given the call value \(9.4134\). (a) Use put–call parity to find the put. (b) State the call’s delta and explain what hedge it prescribes. (c) Vega is \(38.67\); estimate the price change if implied volatility rises from 20% to 21%.
(a) Parity: \(P=C-S_0+Ke^{-rT}=9.4134-100+100e^{-0.03}=9.4134-100+97.0446=6.458\).
(b) Delta \(=N(d_1)=0.5987\): to hedge one long call, short \(0.5987\) shares; the position is then locally insensitive to small moves in \(S\).
(c) \(\Delta C\approx\text{vega}\times\Delta\sigma=38.67\times0.01=0.387\), so the call rises by about \(\$0.39\) to roughly \(9.80\) (a first-order estimate; gamma-of-vega and higher terms adjust it slightly).
Lesson Summary
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: \(V_t+\tfrac12\sigma^2S^2V_{SS}+rSV_S-rV=0\); \(C=S_0N(d_1)-Ke^{-rT}N(d_2)\) with \(d_1=\tfrac{\ln(S_0/K)+(r+\tfrac12\sigma^2)T}{\sigma\sqrt T}\), \(d_2=d_1-\sigma\sqrt T\).
A: The delta hedge cancels the \(dW\) term and eliminates \(\mu\); only \(r\) and \(\sigma\) remain. \(N(d_2)=\mathbb{Q}(S_T\gt K)\), the risk-neutral exercise probability.
Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 4-6 - Ch. 4–6: the Black–Scholes–Merton PDE, delta-hedging derivation, and closed-form solution.
- The Volatility Surface (Jim Gatheral, 2006) foundational - Ch. 1 - Ch. 1: Black–Scholes as a coordinate system, implied volatility, and the smile that reveals its limits.