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
Click a status chip to cycle: Not started → In progress → Studied → Practiced → Needs review → Mastered.
- 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- A trading strategy whose value changes only through market moves, with no external cash injected or withdrawn.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
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.
- The Volatility Surface (Jim Gatheral, 2006) foundational - Ch. 1 - Ch. 1: Black–Scholes, the Greeks, and the hedging interpretation of the pricing PDE.
- Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 4–5 - Ch. 4–5: Itô calculus, the Black–Scholes–Merton equation, and risk-neutral pricing.
- Methods of Mathematical Finance (Karatzas & Shreve, 1998) foundational - Ch. 2 - Ch. 2: hedging, replication, and the martingale approach to European claims.