CFA Level III · Level III Core

Options Strategies: formula sheet

Full chapter guide

Key formulas

Long call payoff and profit: Payoff = max(0, S_T − X); Profit = payoff − c₀; c₀ is the call premium. Maximum loss is c₀. Gain is unlimited.
Short call payoff and profit: Payoff = −max(0, S_T − X); Profit = c₀ − max(0, S_T − X); Maximum gain is c₀. Loss is unlimited as S_T rises.
Long put payoff and profit: Payoff = max(0, X − S_T); Profit = payoff − p₀; Maximum loss is p₀. Maximum gain is X − p₀, reached when S_T = 0.
Short put payoff and profit: Payoff = −max(0, X − S_T); Profit = p₀ − max(0, X − S_T); Maximum gain is p₀. Maximum loss is X − p₀.
Breakeven, call (long or short): S_T* = X + c₀; Long call profits above this price; short call profits below it.
Breakeven, put (long or short): S_T* = X − p₀; Long put profits below this price; short put profits above it.
Put-call parity (European options, same X and expiry): c₀ + X ÷ (1 + r)^T = p₀ + S₀; Left side is the fiduciary call; right side is the protective put. With a dividend-paying underlying, subtract the present value of dividends from S₀. With continuous compounding use X·e^(−rT).
Synthetic positions from parity: Long call = long put + long underlying + short bond (borrow PV of X); Long underlying = long call − long put + long bond; In the first identity the bond is short: you borrow the present value of X. In the second identity the bond is long: you lend the present value of X.
Covered call value at expiration: Value = S_T − max(0, S_T − X); S_T is the stock price at expiry and X is the strike. Subtract S_0 and add the premium received to get total profit.
Covered call profit: Profit = (S_T − S_0) − max(0, S_T − X) + C_0; S_0 is the initial stock price and C_0 is the call premium received.
Covered call maximum profit: Max profit = X − S_0 + C_0; Reached when S_T ≥ X. The profit is capped.
Covered call maximum loss: Max loss = S_0 − C_0; Occurs if the stock falls to zero. The loss is large but finite.
Covered call breakeven: Breakeven = S_0 − C_0; Below this stock price at expiry, the position loses money.
Protective put value at expiration: Value = S_T + max(0, X − S_T); The put gives a floor value of X if S_T is below X.
Protective put profit: Profit = (S_T − S_0) + max(0, X − S_T) − P_0; P_0 is the put premium paid.
Protective put maximum loss: Max loss = S_0 + P_0 − X, if positive; If S_0 + P_0 − X is positive, it is the maximum loss. If it is negative, the position has a guaranteed minimum profit of X − S_0 − P_0. The worst result occurs when S_T ≤ X.
Protective put breakeven: Breakeven = S_0 + P_0; The stock must rise above this level for the position to profit.
Protective put maximum profit: Unlimited; Profit = S_T − S_0 − P_0 when S_T ≥ X, and it grows without limit as S_T rises.
Bull call spread: net cost: Net debit = Premium of long call (low strike) − Premium of short call (high strike); The lower-strike call is always worth at least as much as the higher-strike call, so the net is a debit.
Bull call spread: max gain, max loss, breakeven: Max gain = (High strike − Low strike) − Net debit; Max loss = Net debit; Breakeven = Low strike + Net debit; Max gain occurs at or above the high strike. Max loss occurs at or below the low strike.
Bear put spread: net cost: Net debit = Premium of long put (high strike) − Premium of short put (low strike); Same logic as the bull call spread. The higher-strike put costs more.
Bear put spread: max gain, max loss, breakeven: Max gain = (High strike − Low strike) − Net debit; Max loss = Net debit; Breakeven = High strike − Net debit; Max gain occurs at or below the low strike. Max loss occurs at or above the high strike.
Collar on a held asset: value at expiry: Floor = Put strike; Cap = Call strike; Net cost = Put premium − Call premium; Net cost is positive for a net debit and negative for a net credit. Per unit, the value of the stock plus options at expiry is bounded between the put strike and call strike, and the net cost is then added to or subtracted from the result.
Collar: profit and loss bounds: Max loss = (S0 − Put strike) + (Put premium − Call premium); Max gain = (Call strike − S0) − (Put premium − Call premium); S0 is the price at which the position was entered. Put premium − Call premium is positive for a net debit and negative for a net credit, so a net credit reduces the max loss and raises the max gain.
Zero-cost collar condition: Put premium = Call premium; Choose the call strike that makes this hold for a given put strike.
Long straddle payoff at expiry: Profit = max(S − X, 0) + max(X − S, 0) − (c + p); Same strike X for call and put. c and p are the premiums paid. Equivalent to |S − X| − (c + p).
Straddle breakevens: Upper = X + (c + p); Lower = X − (c + p); Same breakevens for long and short. Long loses between them; short gains between them.
Long straddle maximum loss: Maximum loss = c + p (at S = X); Maximum gain is unlimited on the upside. On the downside it is X − (c + p).
Short straddle maximum gain and loss: Maximum gain = c + p; Maximum loss is unlimited; Maximum gain occurs at S = X. Loss is unlimited on the upside and, on the downside, is limited to X − (c + p) as the price falls to zero.
Strangle breakevens: Upper = XC + (c + p); Lower = XP − (c + p); XC is the call strike and XP is the put strike, with XP < XC. c and p are the premiums.
Long strangle maximum loss: Maximum loss = c + p, when XP ≤ S ≤ XC; Loss is the same total premium but occurs across a range, not at a single point.
Greek profile: Long straddle or strangle: vega > 0, gamma > 0, theta < 0; Short positions reverse every sign. Delta is close to zero at the start for an at-the-money straddle.
Delta: Delta = ΔOption price ÷ ΔUnderlying price; Call: 0 to +1. Put: -1 to 0. Valid for small moves.
Estimated price change using delta: ΔC ≈ Delta × ΔS; A linear estimate. It is inaccurate for large moves in S.
Delta plus gamma approximation: ΔC ≈ Delta × ΔS + ½ × Gamma × (ΔS)²; The gamma term is always positive for a long option, so delta alone understates the gain from a large move.
Gamma: Gamma = ΔDelta ÷ ΔUnderlying price; Positive for long calls and long puts. Negative for short options.
Vega effect: ΔOption price ≈ Vega × Δ Implied volatility (in percentage points); Vega is positive for long options of either type.
Number of options for delta neutrality: Contracts = -(Shares held) ÷ (Option delta × Contract size); Share delta is +1. Use the option's own signed delta (call +, put -). With a call delta, a negative result means write calls. Writing puts adds positive delta, so it does not hedge long shares. To hedge long shares with puts, buy puts: with a negative put delta the result is positive. With a contract size of 1, the result is the number of options. For 50,000 shares, a call delta of 0.40 and a contract size of 100, the result is -1,250, so write 1,250 contracts.
Number of underlying units to hedge options: Underlying units = -(N options × Option delta × Contract size); Negative means sell the underlying, positive means buy it.
Portfolio delta: Portfolio delta = Σ (position size × delta); Short positions carry the opposite sign. Target is zero for a delta-neutral position.
Delta and gamma neutral hedge: Choose option quantity to set net gamma = 0, then use the underlying to set net delta = 0; The underlying has zero gamma. Use a second option to fix gamma first.
Protective put payoff at expiry: Value = S_T + max(X − S_T, 0) − put premium (paid); Maximum loss = S_0 + premium − X. Breakeven = S_0 + premium. Upside is unlimited.
Covered call payoff at expiry: Value = S_T − max(S_T − X, 0) + call premium (received); Maximum gain = X − S_0 + premium. Breakeven = S_0 − premium. Downside is only cushioned by the premium.
Collar: Long stock + long put (strike X_P) + short call (strike X_C); Worst outcome = X_P − S_0 − net premium paid. Best outcome = X_C − S_0 − net premium paid. Zero-cost if the premiums are equal.
Interest rate cap payoff (per period): Payoff = max(Reference rate − Cap strike, 0) × Notional × (days ÷ 360); Usually paid in arrears, at the end of the period. Use the day-count convention given.
Interest rate floor payoff (per period): Payoff = max(Floor strike − Reference rate, 0) × Notional × (days ÷ 360); Used to protect against falling rates. A floating-rate borrower who sells a floor reduces cap cost.
Effective borrowing cost with a cap: Effective rate = min(Reference rate, Cap strike) + premium cost (annualised); Borrower pays the floating rate, receives cap payoff, so the net rate is limited to the strike plus premium.
Implied volatility rule of thumb: Implied vol > expected realised vol → sell options; implied vol < expected realised vol → buy options; This is a decision guide, not a guarantee. It also depends on the client's objective.

Quick revision

Long call profit = max(0, S − X) − premium paid; breakeven = X + premium.
Long put profit = max(0, X − S) − premium paid; breakeven = X − premium.
Short option payoffs mirror long ones: the seller's maximum gain is the premium received, and the potential loss is large (unlimited for a short call, strike minus premium for a short put).
Covered call = long underlying + short call: income and a capped upside, with downside only partly cushioned by the premium.
Protective put = long underlying + long put: a floor on losses, paid for by the premium.
Bull spread profits when the underlying rises; bear spread profits when it falls; both have capped gain and capped loss.
Collar = long underlying + long put + short call, so the call premium offsets some or all of the put cost.
Long straddle (call + put, same strike) profits from a large move either way and loses if the price stays near the strike.
Strangle uses different strikes, so it costs less than a straddle but needs a bigger move to profit.
Calendar spread exploits time decay differences, with the near-term option decaying faster than the longer-term one.
Delta is the change in option value per unit change in the underlying; gamma is the change in delta; vega measures sensitivity to volatility; theta measures time decay.
Always tie the strategy to the client's objectives and constraints before recommending it.

Common mistakes

Reporting payoff when the question asks for profit, or the reverse. Fix: Read the command word. Payoff ignores the premium. Profit subtracts it for a buyer and adds it for a writer.
Using X + premium as the breakeven of a put. Fix: A put gains as the price falls, so the breakeven is X − premium. Check by asking which way the price must move.
Adding the call premium to the breakeven of a covered call. Fix: You receive the premium in a covered call, so it lowers your cost. Breakeven = S_0 − C_0.
Saying a covered call has unlimited upside or a fully protected downside. Fix: A covered call caps gains at the strike and only cushions losses by the premium. A protective put keeps the upside and sets a floor.
Forgetting to subtract the net premium from the strike width when finding max gain. Fix: Always compute max gain = width − net debit. Check that max gain + max loss = width.
Mixing up which leg is long in a bear put spread. Fix: Buy the put you want to profit from, the higher strike, and sell the lower strike put to cut cost.
Using only one premium when finding breakevens. Fix: Both options are bought or sold. Always add the call and put premiums, then apply that total to the strikes.
Applying the total premium to the wrong side for a strangle. Fix: Add the total premium to the call strike for the upper breakeven and subtract it from the put strike for the lower one.
Treating delta as fixed after hedging. Fix: State that the hedge holds only for small moves and must be rebalanced. Link frequency to gamma.
Wrong direction for a put hedge. Fix: Think in share-equivalents. A long put acts like a short stock position, so you hold stock to offset it.

Exam tips

Read for the side: long or short. A wrong sign is the most common lost-point error in item sets.
In essay sets, show each leg's payoff, then the premium adjustment, then the total. A correct number alone earns credit, but working protects you if you slip.
When asked to justify a position for a client, tie the payoff shape to the client's objective, such as limited downside or income, in one sentence.
For parity items, check the question for dividends and the compounding method before calculating.
Round only at the end. Parity answers can be sensitive to rounding of the discount factor.
Read the command words. If asked to 'calculate', show the formula and the numbers, then type the final figure. If asked to 'justify', link the strategy to the client's objective and constraint in one or two sentences.
Write the stock leg and the option leg on separate lines. This earns partial credit if you make an arithmetic slip.
In multiple-choice items, check signs first. Eliminate any option that gives unlimited profit for a covered call or a breakeven above S_0 for a covered call.