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DERIVATIVES — CFA LEVEL II CHEAT SHEET
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FORWARDS & FUTURES: NO-ARBITRAGE PRICING
Cost of Carry Model — Forward Price
| Asset Type | Formula |
|---|
|---|---|
| No income/costs | F₀ = S₀ · (1 + r)^T or S₀ · e^(rT) |
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| Dividend yield (equity/index) | F₀ = S₀ · e^((r−q)T) |
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| Storage costs (commodity) | F₀ = (S₀ + PV(storage)) · (1 + r)^T |
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| Currency (CIP) | F₀ = S₀ · [(1 + r_d)/(1 + r_f)]^T |
|---|
Key: Forward price ≠ expected future spot price. Determined by no-arbitrage only.
Basis & Convergence
- Basis = Spot − Futures Price
- At expiration: basis → 0 (or delivery cost)
- Prior to expiration: basis fluctuates → basis risk
Arbitrage Signals
| Condition | Action | Profit Lock |
|---|
|---|---|---|
| Futures too high | Buy spot, sell forward | Cash-and-carry |
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| Futures too low | Sell spot, buy forward | Reverse cash-and-carry |
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FUTURES FOR PORTFOLIO MANAGEMENT
Equity Portfolio — Beta Adjustment
$$N = \frac{\text{(Target Beta − Current Beta)}}{\text{Beta}_\text{futures}} \times \frac{\text{Portfolio Value}}{\text{Futures Contract Value}}$$- Negative N = sell futures (reduce beta)
- Positive N = buy futures (increase beta)
- Avoids transaction costs of liquidating/buying stocks
Bond Portfolio — Duration Adjustment
$$N = \frac{\text{(Target MD − Current MD)}}{\text{MD}_\text{futures}} \times \frac{\text{Portfolio Value}}{\text{Futures Price}}$$Note: Use modified duration (price sensitivity), not Macaulay duration.
Cross-Hedge (Basis Risk)
$$N^* = -\rho \times \frac{\sigma_S}{\sigma_F} \times \frac{S}{F} \times \frac{\text{Portfolio Size}}{\text{Contract Size}}$$where ρ = correlation between spot & futures changes.
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OPTIONS: BLACK-SCHOLES-MERTON PRICING
Call & Put Formulas
$$\text{Call} = S_0 N(d_1) − X e^{-rT} N(d_2)$$
$$\text{Put} = X e^{-rT} N(−d_2) − S_0 N(−d_1)$$d₁ & d₂
$$d_1 = \frac{\ln(S_0/X) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$$
$$d_2 = d_1 − \sigma\sqrt{T}$$BSM Inputs: S₀, X, r, T, σ (volatility)
NOT INPUT: Expected return (μ) — eliminates in risk-free hedge.
Put-Call Parity (European)
$$C − P = S_0 − X e^{-rT}$$Rearrange: C = P + S₀ − Xe^{-rT}
Arbitrage if violated: long cheap side, short dear side.
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OPTIONS GREEKS — RISK SENSITIVITIES
| Greek | Meaning | Call | Put |
|---|
|---|---|---|---|
| Δ (Delta) | ∂Price/∂S | 0 to +1 | −1 to 0 |
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| Γ (Gamma) | ∂Δ/∂S (convexity) | Always + | Always + |
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| Θ (Theta) | ∂Price/∂T (time decay) | Often − | Often + (OTM) |
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| ν (Vega) | ∂Price/∂σ | Always + | Always + |
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| ρ (Rho) | ∂Price/∂r | + | − |
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High-Yield Rule: Long call/put = long vega & gamma (benefits from volatility rise & large moves); short options = exposed to vega bleed & large moves.
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COMMON OPTIONS STRATEGIES
| Strategy | Construction | Max Profit | Max Loss | Use Case |
|---|
|---|---|---|---|---|
| Call Spread (bull) | Buy call (X₁), sell call (X₂>X₁) | X₂−X₁ − premium | Debit paid | Moderate bullish, lower cost |
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| Put Spread (bear) | Buy put (X₁), sell put (X₂| X₁−X₂ − premium | Debit paid | Moderate bearish, lower cost | |
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| Straddle | Buy call + put (same X, T) | Unlimited | Both premiums | High volatility bet |
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| Strangle | Buy OTM call + OTM put | Unlimited | Both premiums | High vol, cheaper than straddle |
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| Collar | Long stock + buy put (X₁) + sell call (X₂) | X₂ − initial cost | X₁ − cost | Downside protection, funded by call |
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| Calendar Spread | Buy long-dated, sell short-dated (same X) | Vega bleed + decay | Strike moves away | Implied vol expansion bet |
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Key: Spreads reduce cost but cap profit; directional view needed.
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OPTION VALUATION NUANCES
| Factor | Effect on Call | Effect on Put | Note |
|---|
|---|---|---|