Principles of option pricing: formula sheet

Key formulas

Long call payoff at expiry

max(S_T − K, 0)

Zero when S_T ≤ K. Rises one-for-one with S_T above K.

Long put payoff at expiry

max(K − S_T, 0)

Zero when S_T ≥ K. Maximum value is K, when S_T = 0.

Short call payoff

−max(S_T − K, 0) = min(K − S_T, 0)

The negative of the long call payoff.

Short put payoff

−max(K − S_T, 0) = min(S_T − K, 0)

The negative of the long put payoff.

Profit of a long option (ignoring interest)

Profit = Payoff − Premium

For the writer, Profit = Premium − Payoff owed. If asked, include interest: premium × (1 + i) or premium × e^(δT).

Break-even share price

Call: S_T = K + c. Put: S_T = K − p

c and p are the premiums paid. The break-even price is where profit equals zero.

Maximum profit and loss, long call

Max loss = c. Max profit = unlimited

Short call is the reverse: max profit = c, max loss unlimited.

Maximum profit and loss, long put

Max loss = p. Max profit = K − p

Short put: max profit = p, max loss = K − p.

Call upper bound

c ≤ S₀

A call gives the right to buy one share, so it cannot be worth more than the share. Also holds for American calls (C ≤ S₀).

Put upper bound (European)

p ≤ K·e^(−rT)

The best payoff of a put at T is K, so its value cannot exceed the present value of K.

Put upper bound (American)

P ≤ K

An American put can be exercised at once for at most K.

European call lower bound (no dividends)

c ≥ max(S₀ − K·e^(−rT), 0)

Needs a non-dividend-paying share. With a known dividend present value D, use S₀ − D − K·e^(−rT).

European put lower bound (no dividends)

p ≥ max(K·e^(−rT) − S₀, 0)

Follows from put-call parity and c ≥ 0, or from a direct portfolio argument.

Put-call parity (European, no dividends)

c + K·e^(−rT) = p + S₀

Holds exactly under no-arbitrage. Use it to move between call and put bounds.

American call, no dividends

C = c

It is never optimal to exercise early, so the American call equals the European call.

American option bounds

S₀ − K ≤ C − P ≤ S₀ − K·e^(−rT)

Holds for non-dividend shares. Note the inequality, since parity is not exact for American options.

Put-call parity, no dividends

C + K e^(−rT) = P + S₀

European options, same K and T, same underlying. r is the continuously compounded risk-free rate.

Rearranged form

C − P = S₀ − K e^(−rT)

Use this to find a missing call or put price.

Discrete dividends

C + K e^(−rT) = P + S₀ − D

D is the present value at time 0 of dividends paid before T, discounted at the risk-free rate.

Continuous dividend yield q

C + K e^(−rT) = P + S₀ e^(−qT)

q is the continuously compounded dividend yield.

Annual effective rate form

C − P = S₀ − K (1 + i)^(−T)

Use when the rate is given as an effective annual rate i. Same as e^(−rT) with r = ln(1 + i).

Call and put payoffs at expiry

Call = max(S_T − K, 0); Put = max(K − S_T, 0)

The base for every directional argument.

Direction summary (European, other factors fixed)

S↑: call ↑, put ↓. K↑: call ↓, put ↑. σ↑: call ↑, put ↑. r↑: call ↑, put ↓. Dividends↑: call ↓, put ↑.

Time is ambiguous for European options. It is positive for American options.

Time to expiry (American)

For T₂ > T₁: C_A(T₂) ≥ C_A(T₁) and P_A(T₂) ≥ P_A(T₁)

A longer American option can be exercised on every date the shorter one can.

Put-call parity (European, no dividends)

C + K e^(−rT) = P + S₀

Use it to check the direction of r and K effects.

Put-call parity with known dividends

C + D + K e^(−rT) = P + S₀, where D is the present value of dividends during the life

Higher D means a lower C or a higher P.

Lower bounds, European, no dividends

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

Violation gives an arbitrage.

Ordering of values

C_A ≥ C_E and P_A ≥ P_E

The American option has all the rights of the European one, so it cannot be worth less.

American call, no dividends

C_A = C_E

Holds for a share with no dividends before expiry and r ≥ 0. Early exercise is not optimal (strictly so when r > 0).

Lower bound for a European call

C_E ≥ max(S − K·e^(−r(T−t)), 0)

Used to show C > S − K, which proves early exercise of the call loses value.

Lower bound for an American put

P_A ≥ max(K − S, 0)

Intrinsic value. If P_A would fall below K − S, exercise is better.

Put-call relation for American options (no dividends)

S − K ≤ C_A − P_A ≤ S − K·e^(−rT)

This is an inequality, not an equality. Exact put-call parity holds only for European options.

Binomial node value (American)

V = max(exercise value, e^(−r·Δt)[q·V_u + (1 − q)·V_d])

q is the risk-neutral probability. Apply at every node before expiry.

Share price nodes

S(up) = S·u, S(down) = S·d; after j ups and n − j downs: S·u^j·d^(n−j)

For a recombining tree, an up then down equals a down then up.

Risk-neutral probability

q = (1 + r − d) ÷ (u − d)

Here r is the rate per period, and the formula is for discrete compounding. If the question gives a continuous rate δ, use e^δ in place of 1 + r.

One-period option price

V0 = [q·Vu + (1 − q)·Vd] ÷ (1 + r)

Vu and Vd are the option values at the up and down nodes. Use the same step at every node.

Delta (shares held)

Δ = (Vu − Vd) ÷ (S·u − S·d)

Positive for a call, negative for a put (a short share position).

Cash in replicating portfolio

B = [Vu − Δ·S·u] ÷ (1 + r)

Negative B means borrowing. Check with the down state: B = [Vd − Δ·S·d] ÷ (1 + r).

Option price by replication

V0 = Δ·S + B

This must equal the risk-neutral price. Use it as a check.

Multi-period European price

V0 = (1 + r)^(−n) · Σ C(n, j) · q^j · (1 − q)^(n−j) · payoff(S·u^j·d^(n−j))

Valid for a recombining tree with constant u, d and r.

American option node value

V = max[exercise value, continuation value]

Continuation value is the discounted q-expectation of the next two node values.

d1

d1 = [ln(S/K) + (r + σ²/2)T] ÷ (σ√T)

S is the current share price, K the strike, r the continuously compounded risk-free rate, σ the volatility, T the time to expiry in years.

d2

d2 = d1 − σ√T

Compute d1 first, then subtract σ√T.

European call price

c = S Φ(d1) − K e^(−rT) Φ(d2)

Φ is the standard normal distribution function. Valid for a share paying no dividends.

European put price

p = K e^(−rT) Φ(−d2) − S Φ(−d1)

Use Φ(−x) = 1 − Φ(x). You can also get p from put-call parity.

Put-call parity

c − p = S − K e^(−rT)

For European options on a non-dividend-paying share with the same K and T.

Delta

Call delta = Φ(d1); Put delta = Φ(d1) − 1

Call delta lies between 0 and 1. Put delta lies between −1 and 0.

Gamma

Γ = φ(d1) ÷ (S σ √T)

Same for a call and a put. φ(x) = e^(−x²/2) ÷ √(2π) is the standard normal density.

Vega

ν = S φ(d1) √T

Same for a call and a put. Always positive.

Share price model

dS = μS dt + σS dZ

Geometric Brownian motion. Under risk-neutral valuation μ is replaced by r.

Quick revision

• Call payoff at expiry is max(S − K, 0); put payoff is max(K − S, 0).
• No arbitrage means identical payoffs must have identical prices.
• European put-call parity for a non-dividend share: C − P = S₀ − K e^(−rT).
• A call price cannot be negative and cannot exceed the share price.
• A higher volatility raises the value of both calls and puts.
• An American option is worth at least as much as the matching European option.
• For a non-dividend share, early exercise of an American call is not optimal.
• Risk-neutral probability in a binomial tree: q = (e^(rΔt) − d) ÷ (u − d).
• Option price at a node = e^(−rΔt) × [q × up value + (1 − q) × down value].
• For American options, compare exercise value with continuation value at every node.
• Black-Scholes assumes constant volatility and interest rate, lognormal share prices, no transaction costs and continuous trading.
• Risk-neutral valuation is a pricing tool; it does not mean investors are risk neutral.

Common mistakes

• Treating profit as the same as payoff. Fix: Always ask: is the question about payoff or profit? Profit = payoff − premium paid. Write it down as a final step.
• Giving a long put an unlimited maximum profit. Fix: A share price cannot go below zero. The maximum put payoff is K, so the maximum profit is K minus the premium.
• Using K instead of K·e^(−rT) in the European lower bound. Fix: The strike is paid at T, so it must be discounted. Only American exercise-now arguments use plain K.
• Forgetting the max with zero in the lower bound. Fix: An option price cannot be negative. Always write max(…, 0).
• Using parity for American options as an equality. Fix: Check the exercise style first. Parity is an equality only for European options. American options give inequalities.
• Forgetting to discount the strike. Fix: Always use K e^(−rT) or K(1 + i)^(−T). The strike is paid at T, not today.
• Saying higher volatility lowers put values because a put is a bearish bet. Fix: The put payoff is capped below at zero. More spread raises the expected payoff, so puts rise with volatility too.
• Saying a higher interest rate raises put values. Fix: A higher r reduces the present value of the strike a put holder receives, so puts fall. Calls rise. Check with parity.
• Saying an American call on a non-dividend share is worth more than the European call because it is more flexible. Fix: The extra right has no value here, since early exercise is never optimal. State C_A = C_E, with the condition of no dividends.
• Saying American puts are never exercised early, by copying the call result. Fix: For a put, exercising gives K now, which earns interest. Deep in-the-money puts can be exercised early, so P_A can exceed P_E.

Exam tips

• In MCQs, check the sign and whether the question asks for payoff or profit before you compute. These are the usual traps.
• In written answers, state your assumptions: European exercise, no dividends, and whether interest on the premium is ignored.
• When asked to sketch, label axes, the strike, the premium, break-even and the maximum gain and loss. Examiners award marks for these labels.
• Practise short and long versions of both options until you can draw all four in under a minute. Combinations in later topics build on them.
• In Paper B style work, you may be asked to compute payoffs in Excel or R. Use a max function on a vector of share prices and subtract the premium for profit.
• Write the assumptions first: no arbitrage, no dividends, no costs, constant r. Examiners award marks for stating them.
• In proofs, tabulate payoffs for S_T > K and S_T ≤ K. A clear table-like list earns the dominance marks.
• In numerical questions, give the trade, the time-0 cash flow and the payoff at T. Do not stop at saying an arbitrage exists.