How Level II tests Black-Scholes-Merton, Black model applications, option Greeks, delta hedging, and implied volatility interpretation.
Once the binomial logic is clear, Level II moves to closed-form models and trading interpretation. The exam is rarely asking you to become an options trader. It is asking whether you can identify the right model family, interpret the Greeks, and explain what a hedge or volatility quote means economically.
Candidates often know the names but not the choice logic:
Level II rewards model selection and interpretation more than formula recitation.
flowchart TD
A["European option valuation problem"] --> B["Spot equity or currency"]
A --> C["Futures or rate underlying"]
A --> D["Early exercise or flexible state model"]
B --> E["Use BSM logic"]
C --> F["Use Black model logic"]
D --> G["Use a tree or another flexible method"]
That decision path is often the real test hiding inside a vignette.
A standard call-option form is:
$$ C_0=S_0N(d_1)-Xe^{-rT}N(d_2) $$
where
$$ d_1=\frac{\ln(S_0/X)+(r-q+\tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} \quad\text{and}\quad d_2=d_1-\sigma\sqrt{T} $$
| Input | What it is doing |
|---|---|
| (S_0) | Current underlying price |
| (X) | Exercise price |
| (r) | Discounting rate |
| (q) | Yield or carry adjustment such as dividends |
| (\sigma) | Volatility input |
| (T) | Time to expiration |
Level II often tests interpretation more than pure calculation. If volatility rises, the option is usually worth more because the payoff asymmetry becomes more valuable.
For a European call on a futures contract, a common Black-style expression is:
$$ c_0=e^{-rT}\bigl(F_0N(d_1)-XN(d_2)\bigr) $$
The key point is not just the formula. It is recognizing when the underlying is better framed as a forward or futures price than as a spot asset.
| Instrument | Typical model frame |
|---|---|
| European equity option | Black-Scholes-Merton |
| European currency option | Black-Scholes-Merton style with carry adjustment |
| Option on futures | Black model |
| Caplet, floorlet, swaption | Black-model logic in the curriculum treatment |
| Greek | What it measures | Why Level II cares |
|---|---|---|
| Delta | Sensitivity to the underlying price | First-order hedge ratio |
| Gamma | Change in delta as the underlying moves | Shows why delta hedges drift |
| Vega | Sensitivity to volatility | Connects price to implied-volatility changes |
| Theta | Sensitivity to time passing | Explains time decay |
| Rho | Sensitivity to interest rates | Matters more for some maturities and contract types than others |
The exam usually wants you to say which risk changed, not just to recite the label.
If an option position has delta (\Delta), the first-order hedge is to take the offsetting amount in the underlying.
That sounds simple, but gamma explains why the hedge does not stay correct after the underlying moves.
| If gamma is high | The delta changes quickly, so the hedge needs more frequent rebalancing |
|---|---|
| If implied volatility changes | Vega-driven value changes may still matter even when delta is hedged |
That is why a “delta-neutral” position is not “risk-free.”
Implied volatility is the volatility input that makes the model match the observed market price.
| If implied volatility rises | Option values generally rise |
|---|---|
| If two options have different implied volatilities | The market is pricing their uncertainty or payoff asymmetry differently |
Level II often tests whether you can back out the economic meaning of an implied-volatility change without pretending it is a pure forecast of realized volatility.
A portfolio manager says an option book is safe because it is delta neutral.
A weak answer agrees and stops there.
A stronger answer asks about gamma and vega. If the underlying moves sharply or implied volatility shifts, the book can still change value materially even though the first-order price sensitivity was hedged at one moment in time.
Why does a high-gamma option position usually require more frequent rebalancing of a delta hedge?
Best answer: Because delta changes quickly as the underlying price moves, so the original hedge ratio becomes stale faster.
Why: Gamma measures the instability of delta, not just the size of the option position.