FRM Part I · FRM Exam Part I

Options Markets: formula sheet

Full chapter guide

Key formulas

Long call payoff: max(S_T − K, 0); S_T is the underlying price at expiration. Never negative.
Long put payoff: max(K − S_T, 0); Never negative.
Short call payoff: −max(S_T − K, 0) = min(K − S_T, 0); Mirror image of the long call.
Short put payoff: −max(K − S_T, 0) = min(S_T − K, 0); Mirror image of the long put.
Profit for the buyer: Profit = Payoff − Premium; Ignores interest on the premium unless stated.
Profit for the seller: Profit = Premium − Buyer's payoff; Maximum gain is the premium.
Breakeven price: Call: K + premium. Put: K − premium; For the long position. Short has the same breakeven.
Maximum gain on a long put: K − premium; Occurs when S_T = 0.
Split adjustment (n-for-m): New strike = Old strike × m ÷ n; New contracts' shares = Old shares × n ÷ m; In an n-for-m split, you get n new shares for m old. A 3-for-2 split multiplies the strike by 2/3 and shares by 3/2.
Stock dividend as split: x% stock dividend = (100 + x)-for-100 split; A 25% stock dividend is 5-for-4, so strike × 4/5 and shares × 5/4.
Cash dividend adjustment: No adjustment to strike or size for exchange-traded options; Exceptions exist only for very large special dividends, so a standard question assumes none.
Naked equity call margin (US-style, as in Hull): Margin = greater of (1) 100% of option proceeds + 20% of underlying value − out-of-the-money amount, and (2) 100% of option proceeds + 10% of underlying value; Per share, then multiply by contract size. Out-of-the-money amount for a call = max(K − S, 0). Use the percentages given in the question.
Naked put margin (US-style, as in Hull): Margin = greater of (1) 100% of proceeds + 20% of underlying value − out-of-the-money amount, and (2) 100% of proceeds + 10% of exercise price; Out-of-the-money amount for a put = max(S − K, 0).
Contract value: Premium paid = quoted premium × shares per contract × number of contracts; Quotes are per share.
Effect summary (increase in the factor): S0 ↑: call ↑, put ↓ | K ↑: call ↓, put ↑ | σ ↑: call ↑, put ↑ | r ↑: call ↑, put ↓ | dividends ↑: call ↓, put ↑; Holds for all other factors fixed. Time to expiry is ↑ for American options; ambiguous for some European ones.
Upper bound, call: c ≤ S0 and C ≤ S0; A call cannot be worth more than the stock.
Upper bound, put: American: P ≤ K | European: p ≤ K e^(−rT); The most a put can pay is K, or K discounted for a European put.
Lower bound, European call (no dividends): c ≥ max(S0 − K e^(−rT), 0); Also the lower bound for an American call on a non-dividend stock.
Lower bound, European put (no dividends): p ≥ max(K e^(−rT) − S0, 0); American put: P ≥ max(K − S0, 0).
Lower bounds with known dividends: c ≥ max(S0 − D − K e^(−rT), 0) | p ≥ max(K e^(−rT) + D − S0, 0); D is the present value of dividends during the option's life.
American versus European: C ≥ c and P ≥ p; The early exercise right can only add value.
Put-call parity (no dividends, European): c + K·e^(−rT) = p + S0; Same strike, same expiry, same underlying. r is the continuously compounded risk-free rate.
Parity with discrete dividends: c + D + K·e^(−rT) = p + S0; D is the present value of dividends paid during the option's life.
Parity with continuous dividend yield: c + K·e^(−rT) = p + S0·e^(−qT); q is the continuous dividend yield.
American options, no dividends: S0 − K ≤ C − P ≤ S0 − K·e^(−rT); An inequality only. C and P are American prices.
American options, with dividends: S0 − D − K ≤ C − P ≤ S0 − K·e^(−rT); D is the present value of dividends. This form is stated for discrete dividends.
Discrete compounding version: c + K ÷ (1 + r)^T = p + S0; Use when the question gives an annual effective rate.
Bull call spread payoff: Payoff = max(S − K1, 0) − max(S − K2, 0), K1 < K2; Long K1 call, short K2 call. Maximum payoff is K2 − K1. Profit = payoff − net premium.
Bull call spread: max profit and max loss: Max profit = (K2 − K1) − net debit; Max loss = net debit; Breakeven = K1 + net debit.
Bull put spread: Max profit = net credit; Max loss = (K2 − K1) − net credit; Long K1 put, short K2 put. Breakeven = K2 − net credit.
Bear put spread payoff: Payoff = max(K2 − S, 0) − max(K1 − S, 0), K1 < K2; Long K2 put, short K1 put. Maximum payoff is K2 − K1. Breakeven = K2 − net debit.
Butterfly spread with calls: Payoff = max(S − K1, 0) − 2 max(S − K2, 0) + max(S − K3, 0); Maximum payoff is K2 − K1 at S = K2. With evenly spaced strikes (K2 − K1 = K3 − K2), the minimum payoff is 0. With unequal spacing, the payoff for S ≥ K3 is 2K2 − K1 − K3, which is negative if K3 − K2 > K2 − K1.
Butterfly breakevens: Lower = K1 + net debit; Upper = K3 − net debit; Max profit = (K2 − K1) − net debit. Max loss = net debit.
Long straddle profit (strike K): Profit = max(S_T − K, 0) + max(K − S_T, 0) − (c + p) = |S_T − K| − (c + p); Break-evens: K + (c + p) and K − (c + p). Maximum loss is c + p at S_T = K. Gain is unlimited on the upside.
Long strangle profit (put strike K1 < call strike K2): Profit = max(S_T − K2, 0) + max(K1 − S_T, 0) − (c + p); Break-evens: K2 + (c + p) and K1 − (c + p). Maximum loss is c + p for any S_T between K1 and K2.
Strip (1 call + 2 puts, strike K): Profit = max(S_T − K, 0) + 2 × max(K − S_T, 0) − (c + 2p); Bearish tilt. Break-evens: K + (c + 2p) above, and K − (c + 2p) ÷ 2 below.
Strap (2 calls + 1 put, strike K): Profit = 2 × max(S_T − K, 0) + max(K − S_T, 0) − (2c + p); Bullish tilt. Break-evens: K + (2c + p) ÷ 2 above, and K − (2c + p) below.
Covered call profit: Profit = (S_T − S_0) − max(S_T − K, 0) + c; Maximum profit = K − S_0 + c, reached when S_T ≥ K. Loss is large if the stock falls.
Protective put profit: Profit = (S_T − S_0) + max(K − S_T, 0) − p; Maximum loss = S_0 − K + p, when S_T ≤ K. Upside is unlimited.
Put-call parity link (European, no dividends): S_0 + p = c + K × e^(−rT); Shows a protective put is equivalent to a long call plus cash. A covered call is equivalent to a short put plus cash.
Vanilla call and put payoff: Call = max(S_T − K, 0); Put = max(K − S_T, 0); The base case to compare every exotic against.
Cash-or-nothing binary call payoff: Q if S_T > K; 0 otherwise; Q is the fixed cash amount. The put pays Q if S_T < K.
Asset-or-nothing binary call payoff: S_T if S_T > K; 0 otherwise; A vanilla call equals an asset-or-nothing call minus K cash-or-nothing calls.
Average-price Asian call payoff: max(S_avg − K, 0); S_avg is the average price over the averaging period. Average-strike call: max(S_T − S_avg, 0).
Floating lookback payoffs: Call = S_T − S_min; Put = S_max − S_T; Always zero or positive. S_min and S_max are observed over the option's life.
Barrier in-out parity: Knock-in + Knock-out = Vanilla (same K, T, barrier); Holds when there is no rebate. Lets you price one from the other.

Quick revision

Long call payoff = max(S_T − K, 0); long put payoff = max(K − S_T, 0).
Profit = payoff − premium paid for a long position; the short position is the mirror image.
A call gains from a higher underlying price; a put gains from a lower one.
Higher volatility raises the value of both calls and puts.
Higher interest rates raise call values and lower put values, other things equal.
Expected dividends lower call values and raise put values.
Put-call parity (European, no dividends): c + K·e^(−rT) = p + S₀.
With a known present value of dividends D: c + D + K·e^(−rT) = p + S₀.
Bull spread: long a lower strike, short a higher strike; bear spread is the reverse.
Straddle: long a call and a put at the same strike; it gains from a large move either way.
Covered call: long the stock and short a call; protective put: long the stock and long a put.
Barrier options are path dependent; Asian options depend on an average price.

Common mistakes

Forgetting the premium when asked for profit. Fix: Read whether the question says payoff or profit. For profit, always subtract the premium if long.
Confusing a short put with a long call. Fix: A long call has a capped loss and unlimited gain. A short put has a capped gain (premium) and a large loss if the price falls.
Adjusting the strike for ordinary cash dividends on exchange-traded options. Fix: Exchange-traded options are protected against splits and stock dividends only. Cash dividends are handled by the market price of the option.
Multiplying the strike by n/m instead of m/n in a split. Fix: Check that the strike falls after a split. A 2-for-1 split halves the strike. Verify strike × shares is unchanged.
Saying higher volatility lowers the value of a put. Fix: The holder's loss is capped at the premium while the gain is not. Higher volatility raises both calls and puts.
Using K instead of K e^(−rT) in the lower bound of a European option. Fix: Discount the strike from expiry to today. Write S0 − K e^(−rT) every time.
Using S0 instead of the discounted strike, or forgetting to discount K Fix: Parity is stated today. The strike is paid at T, so discount it: K·e^(−rT).
Applying the equality to American options Fix: Check the option style. For American options only the bound S0 − K ≤ C − P ≤ S0 − K·e^(−rT) holds.
Calling a bull put spread a bearish trade because you sell a put Fix: Bull means long the lower strike and short the higher strike. The bull put spread has the same payoff shape as the bull call spread but is a credit trade.
Forgetting the net premium when finding profit Fix: Always compute net premium first. Profit = payoff + net credit, or payoff − net debit.

Exam tips

Read whether the question asks for payoff or profit, and whether you are long or short. These two reads decide the answer.
Use the kink at K and the premium level to match a diagram to a position quickly.
Remember the put's maximum gain is K minus the premium, because the price cannot go below zero.
Check moneyness against S and K only. Distractor answers often mix in the premium.
Premium is usually not discounted in these questions. Only adjust for interest if the question states it.
Expect numeric split adjustments. Practice n-for-m and stock dividend conversions until the strike × shares check is automatic.
Know which events adjust the contract: splits and stock dividends yes, ordinary cash dividends no.
Questions on the clearing house test the idea that it is the counterparty to both sides and removes bilateral credit risk. Only OTC trades carry it, unless centrally cleared.