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
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- 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