CFA Level I · CFA Level I Exam

Pricing and Valuation of Options: formula sheet

Full chapter guide

Key formulas

Call payoff at expiration (buyer): max(0, S − X); S is the underlying price at expiration, X the exercise price.
Put payoff at expiration (buyer): max(0, X − S); Never negative for the buyer.
Profit, option buyer: Profit = Payoff − Premium; Maximum loss for the buyer is the premium.
Profit, option seller: Profit = Premium − Payoff; Seller's payoff is −max(0, S − X) for a call and −max(0, X − S) for a put.
Breakeven price: Call: S = X + premium; Put: S = X − premium; Same breakeven for buyer and seller.
Maximum profit and loss: Long call: loss = premium, profit unlimited. Long put: loss = premium, profit = X − premium (S falls to zero). Short call: profit = premium, loss unlimited. Short put: profit = premium, loss = X − premium.; Assumes the underlying price cannot fall below zero.
Intrinsic value (before expiration): Call: max(0, S − X); Put: max(0, X − S); Uses the current price S.
Time value: Time value = Option premium − Intrinsic value; Zero at expiration.
Put-call parity: c + X ÷ (1 + r)^T = p + S0; European options, same strike X and expiry T, no cash flows on the underlying. r is the annual risk-free rate, T in years.
With continuous compounding: c + X × e^(−rT) = p + S0; Use when the question gives a continuously compounded rate.
Fiduciary call and protective put: Fiduciary call = c + PV(X); Protective put = p + S0; Both pay max(ST, X) at expiry.
Synthetic call: c = p + S0 − X ÷ (1 + r)^T; Long put, long underlying, borrow PV of X.
Synthetic put: p = c − S0 + X ÷ (1 + r)^T; Long call, short underlying, lend PV of X.
Synthetic underlying: S0 = c − p + X ÷ (1 + r)^T; Long call, short put, long bond.
Synthetic bond: X ÷ (1 + r)^T = p + S0 − c; Long put, long underlying, short call.
Parity with known dividends: c + PV(X) = p + S0 − PV(dividends); Subtract the present value of dividends paid before expiry from S0. Equivalent form uses the forward price: c + PV(X) = p + PV(F0(T)).
Exercise value of a call: max(0, S − X); S is the underlying price and X the exercise price.
Exercise value of a put: max(0, X − S); Never negative.
Time value: Option price − Exercise value; Time value is non-negative for American options. A European option can trade below its current exercise value, which gives it negative time value.
European call bounds: max(0, S₀ − X ÷ (1 + r)^T) ≤ c ≤ S₀; Lower bound for an underlying with no income. With income, subtract the present value of the benefits from S₀.
European put bounds: max(0, X ÷ (1 + r)^T − S₀) ≤ p ≤ X ÷ (1 + r)^T; With income, replace S₀ by S₀ − PV of benefits, so the lower bound is max(0, X ÷ (1 + r)^T − S₀ + PV of benefits).
American call bounds: max(0, S₀ − X) ≤ C ≤ S₀ and C ≥ c; Lower bound is at least the exercise value, and C is at least the European call value.
American put bounds: max(0, X − S₀) ≤ P ≤ X and P ≥ p; The upper bound is X because early exercise is possible.
Direction of effects (call / put): S↑: call ↑, put ↓ | X↑: call ↓, put ↑ | Volatility↑: both ↑ | Rate↑: call ↑, put ↓ | Income↑: call ↓, put ↑; Time to expiration: American both ↑ or unchanged; European usually ↑ but not always.
Up and down factors: u = S_up ÷ S0 ; d = S_down ÷ S0 ; d = 1 ÷ u (when given as a recombining tree); The d = 1 ÷ u link holds only when the question builds the tree that way. Use the numbers given.
Risk-neutral probability of an up move: π = (1 + r − d) ÷ (u − d); r is the risk-free rate per period. The down probability is 1 − π. For a stock with a dividend yield, the numerator changes, so use the formula the question gives.
One-period option value: c0 = [π × c_up + (1 − π) × c_down] ÷ (1 + r); Works for calls and puts. Use the payoffs of the option you are pricing (for a put, replace c with p).
Hedge ratio: h = (V_up − V_down) ÷ (S_up − S_down), where V is the option value (c for a call, p for a put); Number of units of the underlying held against one short option to remove risk. For a call, h is between 0 and 1. For a put, h is negative.
Replicating portfolio value: c0 = h × S0 + B, where B = (c_down − h × S_down) ÷ (1 + r) and h = (c_up − c_down) ÷ (S_up − S_down); B is the bond position. It is negative when you borrow. It gives the same price as the risk-neutral method. For a put, use p in place of c.
Terminal payoffs: Call = max(0, S − X) ; Put = max(0, X − S); X is the exercise price. Compute at the final nodes first.
American option node value: Value = max(continuation value, exercise value); Apply at every node before expiry. For European options, use continuation value only.
BSM call price: c = S0 × N(d1) − X × e^(−rT) × N(d2); European call, no income on the underlying. r is the continuously compounded risk-free rate.
BSM put price: p = X × e^(−rT) × [1 − N(d2)] − S0 × [1 − N(d1)]; Equivalent to using put-call parity: p = c − S0 + X × e^(−rT).
d1: d1 = [ln(S0 ÷ X) + (r + σ²÷2) × T] ÷ (σ × √T); σ is annualized volatility and T is in years.
d2: d2 = d1 − σ × √T; d2 is always lower than d1, so N(d2) is lower than N(d1).
Delta: Call delta = N(d1); put delta = N(d1) − 1; Without dividends. Call delta is between 0 and 1, put delta between −1 and 0.
Continuous dividend yield (δ): Replace S0 with S0 × e^(−δT); d1 = [ln(S0 ÷ X) + (r − δ + σ²÷2) × T] ÷ (σ × √T); Call delta becomes e^(−δT) × N(d1). Discrete dividends: use S0 minus the PV of the dividends.
Currency options: Replace S0 with S0 × e^(−rf × T); d1 = [ln(S0 ÷ X) + (r − rf + σ²÷2) × T] ÷ (σ × √T); S0 is the spot rate in domestic currency per unit of foreign currency. rf is the foreign risk-free rate and acts like a dividend yield.
Black model, call on a futures: c = e^(−rT) × [F0 × N(d1) − X × N(d2)]; d1 = [ln(F0 ÷ X) + (σ²÷2) × T] ÷ (σ × √T); d2 = d1 − σ × √T; Put: p = e^(−rT) × [X × (1 − N(d2)) − F0 × (1 − N(d1))]. Discounting applies to the whole bracket.
Delta: Delta = ΔOption price ÷ ΔUnderlying price; Call delta is between 0 and 1. Put delta is between -1 and 0.
Put delta from call delta: Put delta = Call delta − 1; Holds for European options on an underlying with no dividends. The call and put must share strike and expiry.
Gamma: Gamma = ΔDelta ÷ ΔUnderlying price; Highest at the money and near expiry. Same for a call and put with the same terms.
Price change estimate: ΔOption ≈ Delta × ΔS + ½ × Gamma × (ΔS)²; Delta gives the first-order estimate. The gamma term corrects for curvature.
Portfolio delta: Portfolio delta = Σ (number of units × delta of each unit); Shares have delta of 1 each. Short positions carry a negative sign.
Delta hedge size: Options needed = Shares held ÷ Option delta; Assumes each option covers one share. Sell calls against long shares; buy shares against short calls.
Vega, theta and rho: Vega = ΔOption ÷ ΔVolatility; Theta = ΔOption ÷ ΔTime; Rho = ΔOption ÷ ΔRate; Long options: vega positive, theta usually negative. Rho is positive for calls and negative for puts.

Quick revision

Long call payoff at expiry is max(0, S − X); long put payoff is max(0, X − S).
A call is in the money when S > X; a put is in the money when S < X.
Put-call parity: c + X ÷ (1 + r)^T = p + S, for European options on an asset with no cash flows.
Higher underlying price raises call value and lowers put value.
Higher volatility raises both call and put values.
Longer time to expiry generally raises option value; this is not guaranteed for deep in-the-money European puts.
Higher risk-free rate raises call value and lowers put value.
Binomial risk-neutral probability of an up move: π = (1 + r − d) ÷ (u − d).
Option value today is the discounted risk-neutral expected payoff, found by working backward through the tree.
Delta of a call is between 0 and 1; delta of a put is between −1 and 0.
Gamma measures the change in delta; it is highest for at-the-money options near expiry.
Implied volatility is the volatility that makes the model price equal the market price.

Common mistakes

Reporting a negative payoff for a long option that expires out of the money. Fix: Always write max(0, ...). The buyer's payoff is 0, and the profit is minus the premium.
Forgetting the premium when asked for profit. Fix: Underline the word 'profit' or 'net'. Then subtract the premium for the buyer.
Using X instead of its present value Fix: Always discount X at the risk-free rate for time T. Only the bond term is discounted.
Ignoring dividends on the underlying Fix: If the stem gives dividends before expiry, subtract their present value from S0 before using parity.
Saying higher volatility lowers put value because puts profit from price falls. Fix: Volatility widens the range of outcomes. The holder keeps the upside and the loss is limited to the premium, so both calls and puts gain.
Using X instead of the present value of X in European lower bounds. Fix: For European options, discount X at the risk-free rate over the life: X ÷ (1 + r)^T.
Using a real-world probability of an up move instead of the risk-neutral π. Fix: Always compute π from u, d and r. Ignore real-world probabilities when pricing.
Using an annual risk-free rate for each period of a multi-period tree. Fix: Confirm r is per period. If the stem gives an annual rate and a half-year step, convert it as the question instructs.
Treating N(d1) and N(d2) as the same thing, or swapping them in the formula. Fix: N(d1) multiplies the stock (S0) and is the delta. N(d2) multiplies the discounted strike and is the risk-neutral probability of finishing in the money.
Forgetting to discount the strike, or discounting it with the wrong rate or time. Fix: Always write X × e^(−rT) as a separate number first. T is in years, so 6 months is 0.5.

Exam tips

Questions often ask for profit but offer the payoff as a trap option. Read the last line first.
Expect a question on who has limited versus unlimited loss. Only the short call has unlimited loss.
Moneyness questions on puts are common. Test with 'would I gain by exercising now?'.
No penalty for wrong answers: if unsure, eliminate any option where the buyer loses more than the premium, then guess.
Memorise the identity as two named portfolios. Questions often ask which position is equivalent to a fiduciary call or protective put.
For synthetic-position questions, rearrange the equation and read off the signs. Negative terms mean short or borrow.
In arbitrage questions, find which side is cheaper first. Then eliminate options with the wrong direction. This usually leaves one answer.
Watch for dividends and for American-style wording. Either changes the answer, and these are classic trap details.