Properties of Options: formula sheet

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)

Maximum possible payoff is K, when S_T = 0.

Short call payoff

−max(S_T − K, 0) = min(K − S_T, 0)

Never positive. Loss is unlimited.

Short put payoff

−max(K − S_T, 0) = min(S_T − K, 0)

Never positive. Worst payoff is −K (when S_T = 0). Worst profit is −(K − premium received).

Profit for long position

Profit = Payoff − Premium paid

Ignores time value of money unless the question says to compound the premium.

Profit for short position

Profit = Premium received − Buyer's payoff

Equals the negative of the long position's profit.

Breakeven price

Call: S_T = K + premium. Put: S_T = K − premium

Same breakeven for long and short in the same option.

Intrinsic value and time value

Call intrinsic = max(S − K, 0). Put intrinsic = max(K − S, 0). Time value = Premium − Intrinsic value

Time value of an option before expiry is normally not negative for the options in this topic.

Call payoff at expiry

c = max(S_T − K, 0)

Rises with S, falls with K.

Put payoff at expiry

p = max(K − S_T, 0)

Falls with S, rises with K.

Direction table (increase in the factor)

S: European call ↑, put ↓ | K: call ↓, put ↑ | σ: call ↑, put ↑ | r: call ↑, put ↓ | Dividends: call ↓, put ↑

Memorise this. For time T: American call and put ↑; European usually ↑ but can fall.

Time to expiry, American options

C(longer T) ≥ C(shorter T); P(longer T) ≥ P(shorter T)

Holds because the longer option can be exercised whenever the shorter one can.

European lower bound, call, no dividends

c ≥ max(S₀ − K·e^(−rT), 0)

Shows why a higher r raises the call's lower bound.

European lower bound, put, no dividends

p ≥ max(K·e^(−rT) − S₀, 0)

Shows why a higher r lowers the put's lower bound.

Dividend-adjusted put-call parity (European)

c + D + K·e^(−rT) = p + S₀

D is the present value of dividends during the option's life.

Upper bound, calls

c ≤ S0 and C ≤ S0

Applies to European (c) and American (C) calls. Holds with or without dividends.

Upper bound, puts

p ≤ K·e^(−rT) and P ≤ K

European put is capped by the present value of the strike. American put is capped by K because it can be exercised now.

Lower bound, European call (no dividends)

c ≥ max(S0 − K·e^(−rT), 0)

Derived by comparing call plus cash with one share.

Lower bound, European put (no dividends)

p ≥ max(K·e^(−rT) − S0, 0)

Derived by comparing put plus share with cash K·e^(−rT).

Lower bounds with 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 life.

American options

C ≥ max(S0 − K, 0); P ≥ max(K − S0, 0)

Intrinsic value is a floor because early exercise is allowed.

American call, no dividends

C = c

Early exercise is never optimal, so the American call equals the European call.

Put-call parity bracket for American options (no dividends)

S0 − K ≤ C − P ≤ S0 − K·e^(−rT)

With dividends the bounds become S0 − D − K ≤ C − P ≤ S0 − K·e^(−rT).

Put-call parity (European, no dividends)

c + K e^(−rT) = p + S0

Same strike K and expiry T. Use continuous compounding unless the question gives another convention.

Parity with discrete dividends

c + K e^(−rT) = p + S0 − D (equivalently c + D + K e^(−rT) = p + S0)

D is the present value of dividends paid during the option's life. The two forms are the same equation. This page uses the first form: subtract D from S0.

Parity with continuous dividend yield q

c + K e^(−rT) = p + S0 e^(−qT)

Use when the question gives a yield, such as an index.

American options, no dividends

S0 − K ≤ C − P ≤ S0 − K e^(−rT)

Bounds, not an equality. 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 during the life.

Synthetic positions

c = p + S0 − K e^(−rT); p = c − S0 + K e^(−rT)

Long call = long put + long stock + borrowing PV of K. Use to build synthetics.

American vs European values

C ≥ c and P ≥ p

The American option is worth at least as much as the European option with the same strike and expiry.

No-dividend call result

C = c

Holds for a non-dividend-paying stock with positive interest rates. Never exercise early.

Lower bound on European call (no dividends)

c ≥ max(S₀ − K·e^(−rT), 0)

Since K·e^(−rT) < K, this exceeds the exercise value S₀ − K when r > 0.

Lower bound on European put (no dividends)

p ≥ max(K·e^(−rT) − S₀, 0)

The American put must be worth at least K − S₀, its exercise value.

Bounds for American options (no dividends)

S₀ − K ≤ C − P ≤ S₀ − K·e^(−rT)

Put-call parity holds only as an inequality for American options.

Early exercise payoff

Call: S − K; Put: K − S

Early exercise is optimal only if this value exceeds the value of holding the option.

Covered call profit at expiry

Profit = (S_T − S_0) − max(S_T − K, 0) + c

Long stock, short call with strike K, premium c received. Maximum profit = K − S_0 + c. Breakeven = S_0 − c.

Protective put profit at expiry

Profit = (S_T − S_0) + max(K − S_T, 0) − p

Long stock plus long put with strike K and premium p paid. Maximum loss = S_0 − K + p. Breakeven = S_0 + p.

Bull call spread (K1 < K2)

Payoff = max(S_T − K1, 0) − max(S_T − K2, 0); Max profit = (K2 − K1) − net premium; Max loss = net premium; Breakeven = K1 + net premium

Buy the K1 call, sell the K2 call. Net premium is positive because the K1 call costs more.

Bear put spread (K1 < K2)

Payoff = max(K2 − S_T, 0) − max(K1 − S_T, 0); Max profit = (K2 − K1) − net premium; Max loss = net premium; Breakeven = K2 − net premium

Buy the K2 put, sell the K1 put.

Butterfly spread with calls (K2 = (K1 + K3) ÷ 2)

Payoff = max(S_T − K1, 0) − 2 × max(S_T − K2, 0) + max(S_T − K3, 0); Max payoff = K2 − K1 at S_T = K2

Net premium is positive. Max profit = (K2 − K1) − net premium. Max loss = net premium. Breakevens = K1 + net premium and K3 − net premium.

Straddle (same strike K)

Payoff = |S_T − K|; Profit = |S_T − K| − (c + p); Breakevens = K ± (c + p)

Long call and long put. Maximum loss = c + p, at S_T = K.

Strangle (put strike K1 < call strike K2)

Payoff = max(K1 − S_T, 0) + max(S_T − K2, 0); Breakevens = K1 − (c + p) and K2 + (c + p)

Max loss = c + p, for any S_T between K1 and K2.

Quick revision

• Long call payoff = max(S_T − K, 0); long put payoff = max(K − S_T, 0).
• Profit equals payoff minus the premium paid for a long position. The seller's profit is the reverse.
• A higher spot price raises call values and lowers put values.
• A higher strike lowers call values and raises put values.
• Higher volatility raises both calls and puts.
• Put-call parity for European options with no dividends: c + K·e^(−rT) = p + S₀.
• With known dividends, subtract their present value from S₀ in the parity formula.
• A European call is worth at least max(S₀ − K·e^(−rT), 0) on a non-dividend stock.
• An American option is worth at least as much as the matching European option.
• An American call on a non-dividend stock should not be exercised early.
• A bull call spread has limited profit and limited loss; a straddle profits from large moves in either direction.
• If a price breaks a bound or parity, form the arbitrage: buy the cheap side, sell the dear side.

Common mistakes

• Treating the premium as part of the payoff. Fix: Payoff is the value at expiration only. Profit equals payoff minus the premium paid (or plus premium received for a writer).
• Using the call rule for a put moneyness test. Fix: A put is ITM when S < K. Ask whether exercising would give you something.
• Saying higher volatility lowers put values because puts are a bearish bet. Fix: Remember the payoff floor at zero. More volatility widens the upside for both calls and puts, so both increase.
• Claiming a higher interest rate raises both options. Fix: A higher rate lowers the present value of the strike. That helps calls and hurts puts.
• Using K instead of K·e^(−rT) in the European lower bound. Fix: European means the strike is paid at T, so discount it. Use plain K only for American intrinsic value.
• Forgetting to floor the bound at zero. Fix: Always write max(…, 0). A negative bound means the minimum price is zero.
• Using K instead of K e^(−rT). Fix: Always discount the strike to time 0 before adding it to the call.
• Applying the equality to American options. Fix: Check the style. For American options use the bounds only.
• Saying an American call on a non-dividend stock should be exercised early because it is deep in the money. Fix: Remember that selling the call captures at least S − K plus time value. Exercising throws the extra value away.
• Saying American and European puts have the same value. Fix: Only the non-dividend call is equal. The American put is worth more than the European put.

Exam tips

• Read the question for the side: long or short, call or put. Most wrong answers come from the wrong sign.
• Check whether the question asks for payoff or profit. The numeric difference is the premium.
• Use the bounds as a filter: a long option's loss is limited to its premium, and a short option's gain is limited to its premium.
• For moneyness questions, apply the test to the option type, and use today's price S, not the premium.
• Unless told otherwise, ignore interest on the premium. If the question asks to compound it, multiply the premium by the growth factor before subtracting.
• Memorise the direction table and be able to recite it for calls and puts, European and American.
• Watch for 'always' and 'must': exceptions exist for time to expiry on European options.
• Treat dividends and rates as a pair: they have opposite effects on the forward price, so opposite effects on options.