CMA Final · Strategic Financial Management

Options: formula sheet

Full chapter guide

Key formulas

Intrinsic value of a call: Max(S − X, 0); S = spot price of the underlying, X = strike price. Zero if the call is ATM or OTM.
Intrinsic value of a put: Max(X − S, 0); Zero if the put is ATM or OTM.
Premium split: Premium = Intrinsic value + Time value; So time value = Premium − Intrinsic value. For an option whose premium is below intrinsic value, check the data; it signals an arbitrage or a question error.
Moneyness of a call: ITM if S > X; ATM if S = X; OTM if S < X; Compare spot with strike.
Moneyness of a put: ITM if S < X; ATM if S = X; OTM if S > X; Opposite of a call.
Net profit to buyer at exercise (per unit): Call: Max(S − X, 0) − Premium; Put: Max(X − S, 0) − Premium; Ignores interest on premium and transaction costs unless the question includes them. The writer's profit is the opposite sign.
Long call payoff: max(S − X, 0); S is the spot price at expiry. The premium is ignored.
Long call profit: max(S − X, 0) − Premium; Maximum loss = premium. Profit is unlimited as S rises.
Short call profit: Premium − max(S − X, 0); Maximum gain = premium. Loss is unlimited as S rises.
Long put payoff: max(X − S, 0); Maximum payoff = X, when S falls to zero.
Long put profit: max(X − S, 0) − Premium; Maximum loss = premium. Maximum profit = X − premium.
Short put profit: Premium − max(X − S, 0); Maximum gain = premium. Maximum loss = X − premium.
Break-even price: Call: X + Premium; Put: X − Premium; Same for buyer and writer of the same option.
Zero-sum rule: Buyer's profit + Writer's profit = 0; Holds for the same option, ignoring brokerage and taxes.
Long call profit: max(S − X, 0) − premium; S is the price at expiry, X is the strike. Reverse the sign for a short call.
Long put profit: max(X − S, 0) − premium; Reverse the sign for a short put.
Protective put: Profit = (S − S0) + max(X − S, 0) − put premium; for X ≤ S0, maximum loss = (S0 − X) + put premium; S0 is the purchase price of the share. Break-even = S0 + premium. If X > S0, the position has a profit floor of (X − S0) − premium, so the loss is smaller than the premium and can even turn into a guaranteed profit if X − S0 exceeds the premium.
Covered call: Profit = (S − S0) − max(S − X, 0) + call premium; maximum profit = (X − S0) + premium; Break-even = S0 − premium. Loss is large if the price falls.
Long straddle: Break-evens = X ± (call premium + put premium); maximum loss = total premium; Profit is unlimited on the upside. On the downside it is limited only by the price falling to zero (maximum profit = X − total premium).
Long strangle: Break-evens = X_put − total premium and X_call + total premium; maximum loss = total premium; Put strike is below call strike. Loss is greatest when S lies between the two strikes.
Bull call spread: Maximum profit = (X2 − X1) − net premium; maximum loss = net premium; break-even = X1 + net premium; X1 is the lower strike (bought), X2 the higher strike (sold). Net premium = premium paid − premium received.
Bear put spread: Maximum profit = (X2 − X1) − net premium; maximum loss = net premium; break-even = X2 − net premium; X2 is the higher strike (bought), X1 the lower strike (sold).
Long call butterfly: Maximum profit = (X2 − X1) − net premium; maximum loss = net premium; break-evens = X1 + net premium and X3 − net premium; Strikes X1 < X2 < X3 equally spaced. Maximum profit occurs at S = X2.
Put-call parity (no dividend): C + X ÷ (1 + r)^T = P + S; European options, same strike X and expiry T. Use X × e^(−rT) if the question gives a continuous rate.
Put-call parity with known dividends: C + PV(X) + PV(D) = P + S; PV(D) is the present value of dividends expected before expiry. Alternatively, use S − PV(D) in place of S.
Call from put: C = P + S − PV(X); Use it to find the fair price of a call when the put is known.
Put from call: P = C − S + PV(X); Use it to find the fair price of a put when the call is known.
Arbitrage gap: Profit today = |(C + PV(X)) − (P + S)|; Per unit of the asset. Multiply by the lot size.
Up and down factors: u = Su ÷ S0 ; d = Sd ÷ S0; Su and Sd are the prices after an up and a down move. In a two-period tree: Suu = S0 × u², Sud = S0 × u × d, Sdd = S0 × d².
Call and put payoff at expiry: Call = max(S − K, 0) ; Put = max(K − S, 0); Apply at each end node of the tree. K is the strike price.
Hedge ratio (delta): Δ = (Cu − Cd) ÷ (Su − Sd); Number of shares in the replicating portfolio. Cu and Cd are the option values at the up and down nodes. For a put use Pu and Pd; Δ comes out negative, meaning you short the shares.
Borrowing in the replicating portfolio: Borrowing today = (Δ × Sd − Cd) ÷ (1 + r); r is the risk-free rate for one period. This holds for a call, where the portfolio is Δ shares financed partly by borrowing.
Option value by replication: C0 = Δ × S0 − Borrowing today; Gives the same value as the risk-neutral method.
Risk-neutral probability: p = ((1 + r) − d) ÷ (u − d) ; probability of down move = 1 − p; Use the rate for one step. This p lies between 0 and 1 only if d < 1 + r < u.
Risk-neutral option value: C0 = [p × Cu + (1 − p) × Cd] ÷ (1 + r); Same form for a put. Repeat at every node when working backward in a multi-period tree.
Early exercise check (American option): Value at node = max(continuation value, exercise value); Continuation value comes from the risk-neutral formula. Exercise value is the immediate payoff.
d1: d1 = [ln(S ÷ X) + (r + σ² ÷ 2) × T] ÷ (σ × √T); S = spot price, X = exercise price, r = continuously compounded risk-free rate, σ = annual volatility (standard deviation), T = time in years. ln is the natural log.
d2: d2 = d1 − σ × √T; Always compute d1 first. Use the same σ√T you used in the d1 denominator.
Call value: C = S × N(d1) − X × e^(−rT) × N(d2); e^(−rT) discounts the exercise price at the risk-free rate.
Put value: P = X × e^(−rT) × N(−d2) − S × N(−d1); Use this directly, or find the put from put-call parity.
Negative d values: N(−d) = 1 − N(d); Tables usually show only positive d. This symmetry gives you the rest.
Put-call parity (no dividends, European): C + X × e^(−rT) = P + S; Use it to find the second option value, or to check your answer.
Delta: Δ = change in option price ÷ change in underlying price; Call: 0 to 1. Put: -1 to 0. In Black-Scholes, call delta = N(d1) and put delta = N(d1) - 1.
Put-call delta link: Δ(put) = Δ(call) - 1; Applies to European options on a non-dividend-paying asset with the same strike and expiry.
Gamma: Γ = change in delta ÷ change in underlying price; Same for call and put with the same strike and expiry. Always positive for a long option.
Price change estimate: ΔC ≈ Δ × ΔS + ½ × Γ × (ΔS)²; Delta alone is a first-order estimate. Gamma corrects it for larger moves.
Delta-neutral hedge: Units of underlying to hold = - (option delta × number of options); Short call: buy the underlying. Long put: buy the underlying, as put delta is negative.
Theta, vega, rho: Θ = change in value per day; ν = change per 1% change in volatility; ρ = change per 1% change in rate; Check the unit given in the question: per day or per year, per 1% or per 1 unit.
Direction of effects on a call / put: Call: higher S ↑, higher σ ↑, longer time usually ↑, higher r ↑, higher strike ↓; Put: higher S ↓, higher σ ↑, longer time usually ↑, higher r ↓, higher strike ↑. Longer time to expiry usually raises the value of both calls and puts. For European options it can lower the value of deep in-the-money puts, or of calls when dividends are large.
Payoff of a long call at expiry: Max(S − K, 0); S is the price at expiry and K is the strike. Subtract the premium to get the net result.
Payoff of a long put at expiry: Max(K − S, 0); Used to protect a long position in shares, a portfolio or foreign currency to be received.
Protective put: value at expiry: Portfolio value + Put payoff − Total premium; Below the strike, the portfolio loss is offset by the put gain. Only the premium and any gap between the portfolio and the index are left.
Number of index puts for a portfolio: (Beta × Portfolio value) ÷ (Index level × Lot size); Round to a whole number of lots. State your rounding. Beta scales the index hedge to your portfolio.
Net cost of a currency hedge with a call option: Effective rate = Lower of (spot at expiry, strike) + Premium per unit; For a payable. For a receivable with a put, effective rate = higher of (spot at expiry, strike) − premium. Add interest on the premium if the question asks.
Interest rate cap payoff per period: Notional × Max(0, Reference rate − Cap strike) × (Days ÷ Day-count basis); Paid at the end of the period. Use the day-count basis given in the question (360 or 365).
Interest rate floor payoff per period: Notional × Max(0, Floor strike − Reference rate) × (Days ÷ Day-count basis); Protects a lender or investor in floating-rate assets against falling rates.
Collar net cost: Premium paid on cap − Premium received on floor; Zero-cost collar when the two premiums are equal.

Quick revision

A call gives the right to buy and a put gives the right to sell; the writer has the obligation.
Call payoff at expiry for the holder = max(S − X, 0); put payoff = max(X − S, 0).
Profit = payoff − premium paid for a buyer; for a writer, premium received − payoff paid.
Intrinsic value of a call = max(S − X, 0); time value = premium − intrinsic value.
A long straddle is a call and a put at the same strike; it gains from large moves either way.
A bull call spread buys a lower-strike call and sells a higher-strike call; both profit and loss are limited.
Put-call parity (European, no dividends): C + X × e^(−rT) = P + S.
Binomial risk-neutral probability: p = (e^(rT) − d) ÷ (u − d), or with simple interest, (1 + r − d) ÷ (u − d).
Binomial option value = [p × payoff up + (1 − p) × payoff down] ÷ (1 + r), for one period.
Black-Scholes call: C = S × N(d1) − X × e^(−rT) × N(d2).
d1 = [ln(S ÷ X) + (r + σ²÷2) × T] ÷ (σ × √T); d2 = d1 − σ × √T.
Delta of a call lies between 0 and 1; gamma and vega are positive for long options; theta is usually negative.

Common mistakes

Using S − X for a put's intrinsic value. Fix: For a put, write X − S. Say it as 'strike minus spot' every time you see a put.
Showing negative intrinsic value for an OTM option. Fix: Intrinsic value is Max(gap, 0). An option holder is never forced to exercise at a loss, so the floor is zero.
Treating payoff and profit as the same thing. Fix: Payoff excludes the premium. Profit includes it. Read the question wording and compute both if unsure.
Using X − premium as the break-even of a call. Fix: A call needs S to rise above X by enough to recover the premium, so X + premium. A put needs S to fall, so X − premium.
Ignoring the premium when finding break-even or profit. Fix: Always finish with net profit = total payoff − net premium paid (or + net premium received).
Treating a short option's premium as a cost. Fix: The seller receives the premium. The buyer pays it. Mark each leg first.
Using X instead of PV(X). Fix: Always discount the strike for the time to expiry before comparing with today's prices.
Ignoring dividends paid before expiry. Fix: Read the question for dividends. Subtract PV(D) from S, or add PV(D) to the call side.
Using the real-world probability of an up move in the pricing formula. Fix: Option value uses only the risk-neutral p computed from u, d and r. Real probabilities do not affect the value.
Discounting the expected payoff at the wrong rate, or not discounting at every step in a two-period tree. Fix: Discount at the risk-free rate for one period at each backward step. Two periods mean two discounting steps.

Exam tips

In MCQs, first decide call or put, then apply the direction. Most wrong answers come from reversing the put formula.
Always check that time value is not negative. A negative result usually means you used the wrong intrinsic formula.
Read whether the question wants the buyer's or the writer's position. Writer's profit is the mirror image of the buyer's.
Write the definition in one line before the working in descriptive answers. Examiners reward clear terms such as strike, premium and expiry.
Check whether the question is per share or per lot. Multiply by lot size only if the lot size is given.
Write the position (long or short, call or put) at the top of your answer. It decides every sign.
In MCQs, test the options with one spot price above X and one below. This avoids formula confusion.
Show break-even, maximum profit and maximum loss in a short table-like list even when only one is asked. It earns method marks.