Black–Scholes Assumptions, the Greeks, and Hedging
The pricing PDE as a hedging identity, the sensitivities that run a derivatives book, and where the model’s assumptions break.
Leads to: 15.2 shows the single σ assumption fails across strikes, launching the smile.
Learning Objectives
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- State the Black–Scholes assumptions and identify which one each later model relaxes.
- Derive the Black–Scholes PDE from a delta-hedged portfolio and interpret it as a hedging identity.
- Compute the five principal Greeks for a European option and explain the sign of each.
- Implement the Black–Scholes price and Greeks in code and verify put–call parity.
- Explain the gamma–theta trade-off that a delta-hedged option position actually earns.
Key Vocabulary
- Delta Δ
- Sensitivity of the option value to the underlying, \(\partial V/\partial S\); the hedge ratio.
- Gamma Γ
- Curvature \(\partial^2 V/\partial S^2\); how fast delta moves, so how often you must re-hedge.
- Vega ν
- Sensitivity to volatility \(\partial V/\partial\sigma\); not a Greek letter, but the risk that dominates a smile.
- Theta Θ
- Time decay \(\partial V/\partial t\); the price you pay (or earn) for holding the position as time passes.
- Rho ρ
- Sensitivity to the interest rate \(\partial V/\partial r\); usually the smallest first-order risk.
- Delta-hedging
- Continuously holding \(-\Delta\) units of the underlying so the portfolio is instantaneously insensitive to \(S\).
- Self-financing
- A trading strategy whose value changes only through market moves, with no external cash injected or withdrawn.
Intuition & Motivation
The assumptions - and a map of what fails
The classical model of Black, Scholes and Merton assumes the underlying solves, under the physical measure,
with a constant risk-free rate \(r\), no dividends, no transaction costs, continuous frictionless trading, and no arbitrage. Every model later in this phase relaxes exactly one of these:
| Assumption | Reality | Model that relaxes it |
|---|---|---|
| Constant σ | Options at different strikes imply different σ | Local vol (15.3), Stochastic vol (15.4) |
| Lognormal returns | Fat tails, skew, jumps | Local/stochastic vol, jump models |
| Continuous costless hedging | Discrete hedges, bid–ask, impact | Transaction-cost models (Phase 16) |
| Single flat surface | A whole \(\sigma(K,T)\) surface trades | Volatility surface (15.5) |
The pricing PDE as a hedging identity
Form a portfolio long one option \(V(S,t)\) and short \(\Delta\) shares: \(\Pi=V-\Delta S\). By Itô’s lemma,
Choosing \(\Delta=\partial V/\partial S\) kills the random \(dS\) term. A riskless portfolio must earn \(r\,\Pi\,dt\), giving the Black–Scholes PDE:
For a European call the solution is the Black–Scholes formula
where \(N\) is the standard normal CDF. Put–call parity \(C-P=S-Ke^{-rT}\) then fixes the put.
The Greeks
Differentiating (15.4) gives the sensitivities that a trading desk manages in real time (\(n(\cdot)\) is the normal PDF):
| Greek | Call formula | Sign / meaning |
|---|---|---|
| Δ | \(N(d_1)\) | in \([0,1]\); hedge ratio |
| Γ | \(\dfrac{n(d_1)}{S\sigma\sqrt{T}}\) | >0; peaks near ATM |
| Vega | \(S\,n(d_1)\sqrt{T}\) | >0; largest ATM, long-dated |
| Θ | \(-\dfrac{S n(d_1)\sigma}{2\sqrt{T}}-rKe^{-rT}N(d_2)\) | usually <0 for long options |
| ρ | \(KTe^{-rT}N(d_2)\) | >0 for a call |
Interactive: the Greeks explorer
Drag spot, strike, vol, rate and maturity and watch price and every Greek respond. Confirm the signs in the table above.
Interactive: price by Monte Carlo, check against the formula
The risk-neutral price is \(e^{-rT}\,\E^{\Q}[(S_T-K)^+]\). Simulate lognormal terminal prices and watch the Monte Carlo estimate converge to the closed-form value.
Interactive: implement the price and Greeks
- Confusing the physical drift \(\mu\) with the risk-neutral drift \(r\): pricing and hedging use \(r\), not \(\mu\).
- Reporting vega ‘per 1%’ but coding it per unit vol (a factor of 100). Be explicit about units.
- Believing delta-hedging removes all risk: it removes first-order \(S\) risk only; gamma, vega and jump risk remain.
- Treating theta as pure loss - for a delta-hedged book theta is the premium you collect against gamma.
- Manage a book by net Greeks, not position-by-position: aggregate delta/gamma/vega across all strikes and maturities.
- Near expiry ATM options have exploding gamma and theta - hedging costs and pin risk spike; size accordingly.
- Vega and gamma are largest ATM but at different maturities: gamma is a short-dated risk, vega a long-dated one.
- Always sanity-check a pricer with put–call parity and limiting cases (deep ITM delta→1, \(\sigma\to0\) → intrinsic).
Knowledge Check
Practical Exercise
A dealer sells a one-year ATM call on a non-dividend stock (\(S=K=100\), \(r=5\%\)) at an implied vol of \(20\%\) and delta-hedges daily. (a) Explain in one paragraph what P&L this position earns and on what it depends. (b) If the stock actually realizes \(25\%\) volatility, does the dealer make or lose money, and why? (c) Name two real-world frictions that make the hedge imperfect.
(a) Selling and delta-hedging a call makes the dealer short gamma / long theta. Each day the dealer collects the option’s theta but must pay the gamma cost of re-hedging when the stock moves. The net hedged P&L is approximately \(\tfrac12\Gamma S^2(\sigma_{\text{imp}}^2-\sigma_{\text{real}}^2)\,dt\) summed over the life of the trade (sign flipped because the dealer is short the option). It depends on realized vol versus the \(20\%\) implied sold.
(b) With realized \(25\% \gt 20\%\) implied, the term \(\sigma_{\text{imp}}^2-\sigma_{\text{real}}^2\) is negative, so the short-option dealer loses money: re-hedging costs (buying high, selling low as the stock whips around) exceed the theta collected. The dealer sold vol too cheaply.
(c) Frictions: (i) discrete rather than continuous hedging leaves residual gamma risk between rebalances; (ii) transaction costs / bid–ask spread on each re-hedge; also gaps/jumps and the fact that a single constant \(\sigma\) cannot describe the true dynamics.
Lesson Summary
Formula Sheet Additions
- Did I use \(r\) (not \(\mu\)) in the pricing and hedging?
- Are my vega/theta units stated (per unit vol? per day?)
- Did I verify put–call parity and the ATM limiting cases?
- Did I account for gamma/vega risk that delta-hedging leaves behind?
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: Because the option is replicated by a self-financing stock+cash portfolio; no-arbitrage prices the replica, and the hedge cancels the \(dS\) term, leaving only \(r\) and \(\sigma\).
A: Approximately \(\tfrac12\Gamma S^2(\sigma_{real}^2-\sigma_{imp}^2)dt\): positive when realized vol exceeds the implied vol paid, negative otherwise.
A: Both peak near at-the-money; gamma is a short-dated risk (blows up near expiry) while vega is a long-dated risk (grows with \(\sqrt{T}\)).
Flashcards
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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