DERIVATIVES — CFA LEVEL II CHEAT SHEET

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FORWARDS & FUTURES: NO-ARBITRAGE PRICING

Cost of Carry Model — Forward Price

|---|---| 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

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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. ---

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. ---

OPTIONS GREEKS — RISK SENSITIVITIES

|---|---|---|---| High-Yield Rule: Long call/put = long vega & gamma (benefits from volatility rise & large moves); short options = exposed to vega bleed & large moves. ---

COMMON OPTIONS STRATEGIES

|---|---|---|---|---| Key: Spreads reduce cost but cap profit; directional view needed. ---

OPTION VALUATION NUANCES

Asset Type	Formula
No income/costs	F₀ = S₀ · (1 + r)^T or S₀ · e^(rT)
Dividend yield (equity/index)	F₀ = S₀ · e^((r−q)T)
Storage costs (commodity)	F₀ = (S₀ + PV(storage)) · (1 + r)^T
Currency (CIP)	F₀ = S₀ · [(1 + r_d)/(1 + r_f)]^T
Condition	Action	Profit Lock
Futures too high	Buy spot, sell forward	Cash-and-carry
Futures too low	Sell spot, buy forward	Reverse cash-and-carry
Greek	Meaning	Call	Put
Δ (Delta)	∂Price/∂S	0 to +1	−1 to 0
Γ (Gamma)	∂Δ/∂S (convexity)	Always +	Always +
Θ (Theta)	∂Price/∂T (time decay)	Often −	Often + (OTM)
ν (Vega)	∂Price/∂σ	Always +	Always +
ρ (Rho)	∂Price/∂r	+	−
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
Put Spread (bear)	Buy put (X₁), sell put (X₂	X₁−X₂ − premium	Debit paid	Moderate bearish, lower cost
Straddle	Buy call + put (same X, T)	Unlimited	Both premiums	High volatility bet
Strangle	Buy OTM call + OTM put	Unlimited	Both premiums	High vol, cheaper than straddle
Collar	Long stock + buy put (X₁) + sell call (X₂)	X₂ − initial cost	X₁ − cost	Downside protection, funded by call
Calendar Spread	Buy long-dated, sell short-dated (same X)	Vega bleed + decay	Strike moves away	Implied vol expansion bet
Factor	Effect on Call	Effect on Put	Note

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