Derivatives Analysis and Valuation: formula sheet

Key formulas

Futures price (discrete, no income)

F = S × (1 + r × T) or F = S × (1 + r)^T

Use the compounding style the question gives. T is in years, e.g. 3 months = 0.25.

Futures price (continuous compounding)

F = S × e^(rT)

Use only when the question says continuous compounding or gives e^ values.

Futures price with known dividend

F = S × (1 + r × T) − D × (1 + r × t) or F = (S − PV of D) × (1 + r × T)

D is the dividend and t is the time from dividend receipt to expiry, so the dividend earns interest until expiry. Use one method consistently. With annual compounding, replace (1 + r × T) with (1 + r)^T.

Futures price with dividend yield

F = S × e^((r − q)T) or F = S × [1 + (r − q) × T]

q is the dividend yield. If the question states simple rates, use the simple form.

Cost of carry

Cost of carry = Futures price − Spot price (net of income)

Includes interest and storage, less income. Basis = Spot − Futures; it is negative when futures exceed spot, and it converges to zero at expiry.

Mark-to-market (long)

Daily gain = (Today's settlement price − Previous settlement price) × Lot size × Contracts

For a short position the sign reverses.

Number of contracts for beta hedge

N = (β of portfolio × Portfolio value) ÷ (Index futures price × Lot size)

Sell N contracts to hedge a long portfolio. To change beta: N = (Target β − Current β) × Portfolio value ÷ (Futures price × Lot size); positive means buy.

Hedge ratio (minimum variance)

h = ρ × (σS ÷ σF)

Contracts = h × Exposure ÷ Value of one futures contract.

Long call payoff and profit

Payoff = max(S − X, 0); Profit = max(S − X, 0) − Premium

Breakeven = X + Premium. Maximum loss = premium. Profit is unlimited.

Long put payoff and profit

Payoff = max(X − S, 0); Profit = max(X − S, 0) − Premium

Breakeven = X − Premium. Maximum loss = premium. Maximum profit = X − Premium (when S = 0).

Short positions

Short payoff = − (long payoff); Short profit = Premium received − long payoff

Writer of a call has unlimited loss. Writer of a put has loss up to X − Premium.

Intrinsic value and time value

Call intrinsic = max(S − X, 0); Put intrinsic = max(X − S, 0); Time value = Premium − Intrinsic value

S is the current spot price for a live option, and the expiry price at expiry.

Long straddle

Net profit = max(S − X, 0) + max(X − S, 0) − (Call premium + Put premium)

Breakevens: X ± total premium. Maximum loss = total premium at S = X.

Long strangle

Buy put at X1 and call at X2 (X1 < X2). Breakevens: X1 − total premium and X2 + total premium

Maximum loss = total premium, for any S between X1 and X2.

Bull call spread

Buy call at X1, sell call at X2 (X1 < X2). Max profit = (X2 − X1) − Net premium; Max loss = Net premium

Breakeven = X1 + Net premium.

Bear put spread

Buy put at X2, sell put at X1 (X1 < X2). Max profit = (X2 − X1) − Net premium; Max loss = Net premium

Breakeven = X2 − Net premium.

Long butterfly (calls)

Buy call X1, sell 2 calls X2, buy call X3, with X2 midway between X1 and X3. Max profit = (X2 − X1) − Net premium at S = X2

Max loss = Net premium. Breakevens: X1 + Net premium and X3 − Net premium.

Protective put, covered call, collar

Protective put profit = S − S0 + max(X − S, 0) − Put premium; Covered call profit = S − S0 − max(S − X, 0) + Call premium; Collar = share + long put (lower X) + short call (higher X)

S0 is the price at which the share was bought. Protective put floor = X − S0 − Premium.

Put-call parity (European, no dividend)

C + X × e^(−rT) = P + S

Same strike, same expiry, same underlying. With discrete compounding use X ÷ (1 + r)^T. Use whichever the question gives.

Put price from parity

P = C − S + X × e^(−rT)

Rearranged for the missing put. Check the sign of each term before substituting.

Call price from parity

C = P + S − X × e^(−rT)

Use when the put price is given.

Parity with known dividend

C + PV(X) = P + S − PV(D)

PV(D) is the present value of dividends paid before expiry. The share price is reduced by it.

European call bounds

max(0, S − PV(X)) ≤ C ≤ S

For a no-dividend share. A price below the lower bound gives arbitrage.

European put bounds

max(0, PV(X) − S) ≤ P ≤ PV(X)

The maximum put value is PV(X) because the share price cannot fall below zero.

American call on non-dividend share

C(American) = C(European)

Early exercise is never optimal when no dividend is paid, so the two have the same value. Exercising early gives only S − X, which is less than the lower bound S − PV(X).

American option vs European

C(Am) ≥ C(Eu); P(Am) ≥ P(Eu)

The right to exercise early cannot reduce value.

American parity inequality (no dividend)

S − X ≤ C − P ≤ S − PV(X)

Parity is an inequality for American options.

Up and down factors

u = Su ÷ S0 ; d = Sd ÷ S0

Use when the question gives up and down prices. If given percentage moves, u = 1 + rise and d = 1 − fall.

Call payoff at expiry

C = max(S − K, 0)

S is the share price at the node and K is the exercise price.

Put payoff at expiry

P = max(K − S, 0)

Compute it separately at every end node.

Hedge ratio (delta)

Δ = (Cu − Cd) ÷ (Su − Sd)

Number of shares in the replicating portfolio for a call. For a put, Δ comes out negative, meaning you short shares.

Replicating portfolio borrowing

B = (Δ × Sd − Cd) ÷ (1 + r)

Amount borrowed at the risk-free rate for one period. Check with Δ × Su − Cu, which gives the same loan repayment.

Option value by replication

C = Δ × S0 − B

Cost of shares less the loan.

Risk-neutral probability

p = ((1 + r) − d) ÷ (u − d)

r is the risk-free rate for one period. With continuous compounding use e^(r × t) in place of (1 + r). Probability of a down move is 1 − p.

Option value by risk-neutral method

C = [p × Cu + (1 − p) × Cd] ÷ (1 + r)

Applied at each node while rolling back. For a two-period tree you can also use p², 2p(1 − p) and (1 − p)² on the end payoffs and discount by (1 + r)².

No-arbitrage condition

d < (1 + r) < u

If this fails, p falls outside 0 to 1 and the data is inconsistent.

d1

d1 = [ln(S ÷ X) + (r + σ² ÷ 2) × t] ÷ (σ × √t)

S = spot, X = strike, r = continuously compounded risk-free rate, σ = annual volatility, t = years to expiry. ln is the natural log.

d2

d2 = d1 − σ × √t

Compute d1 first, then subtract σ√t.

Call value

C = S × N(d1) − X × e^(−rt) × N(d2)

European call, no dividends.

Put value

P = X × e^(−rt) × N(−d2) − S × N(−d1)

N(−d) = 1 − N(d). Check with put-call parity.

Put-call parity

C + X × e^(−rt) = P + S

For European options on a share with no dividends, same strike and expiry. Use it for a quick put value.

Dividend adjustment (discrete)

Use S* = S − PV of dividends expected before expiry in place of S

Discount each dividend at the risk-free rate from its date.

Dividend adjustment (continuous yield q)

Use S* = S × e^(−qt) in place of S

Use this only if the question gives a continuous dividend yield.

Delta

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

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

Gamma

Gamma = N′(d1) ÷ (S × σ × √t), where N′(d1) = e^(−d1²÷2) ÷ √(2π)

Same for a call and a put.

Vega

Vega = S × √t × N′(d1)

Per 1.00 (100%) change in σ. Divide by 100 for a 1% change. Same for call and put.

Rho (call)

Rho = X × t × e^(−rt) × N(d2)

For a put: −X × t × e^(−rt) × N(−d2).

Approximate price change

ΔC ≈ Delta × ΔS + ½ × Gamma × (ΔS)²

Delta alone is a first approximation. Gamma corrects for curvature.

Net settlement in an interest rate swap

Net payment = Notional × (Fixed rate − Floating rate) × (days ÷ 360 or the period fraction given)

The fixed payer pays this if it is positive and receives it if it is negative. Use the day count given in the question.

Total gain from comparative advantage

Total gain = |Fixed rate differential − Floating rate differential|

Differential = rate for the weaker-rated firm minus rate for the stronger-rated firm in that market. Total gain is before any bank fee.

Gain to each party with a bank

Gain per party = (Total gain − Bank fee) ÷ 2

This holds only when the question says the balance is shared equally. Otherwise use the stated ratio.

Who borrows where

Each firm borrows in the market where its comparative advantage is greater (or its disadvantage is smaller)

Then the firm swaps into the stream it actually wants.

Currency swap principals

Principal in currency B = Principal in currency A × Spot rate (B per A)

Principals are exchanged at the start and re-exchanged at maturity at the same rate, unless the question says otherwise.

Swap valuation

Value to a party = PV of cash flows it receives − PV of cash flows it pays

Discount each flow at the current rate for its date. Right after a floating reset, the floating leg is worth the notional.

Fixed swap rate from discount factors

Par swap rate = (1 − Z_n) ÷ Σ Z_t

Z_t is the discount factor for period t, and the sum runs over all payment periods to maturity. Fixed payments are per period, so annualise if needed.

FRA settlement amount

Settlement = (SR − FR) × N × (d ÷ 360) ÷ [1 + SR × (d ÷ 360)]

SR = reference rate at settlement, FR = FRA rate, N = notional, d = days in the FRA period. Positive means the buyer receives; negative means the buyer pays. Use the day count given in the question (360 or 365).

Effective rate with FRA (borrower)

Effective rate = FRA rate (the loan costs SR, the FRA settlement offsets SR − FR)

This is exact in this framework, ignoring credit and basis effects. Check: the settlement received at the start, invested for the FRA period at SR, grows to (SR − FR) × N × d/360. That offsets the extra interest over the FRA rate.

Interest rate futures price

Price = 100 − implied annual rate (in %)

Rate rise means price fall. Borrowers sell, lenders buy.

Futures gain or loss

Gain = Change in price (in %) ÷ 100 × Contract size × (Contract period in months ÷ 12) × No. of contracts

A one basis point move (0.01) on a 3-month contract is 0.0001 × size × 3/12.

Number of futures contracts

Contracts = (Exposure ÷ Contract size) × (Exposure period ÷ Contract period)

Round to a whole number. The second ratio adjusts for tenor mismatch.

Cap payoff per period

Cap payoff = Max(0, Reference rate − Cap strike) × N × (d ÷ 360)

Usually paid at the end of the period. Use the discounted version only if the question says it is settled in advance.

Floor payoff per period

Floor payoff = Max(0, Floor strike − Reference rate) × N × (d ÷ 360)

Received by the floor buyer, paid by the floor seller.

Collar (borrower)

Buy cap at higher strike + Sell floor at lower strike; Net premium = Cap premium − Floor premium

Effective rate stays between the floor strike and the cap strike (plus the net premium cost).

Quick revision

• Forward: customised OTC contract, settled at maturity. Future: standardised, exchange-traded, marked to market daily with margins.
• Cost-of-carry futures price (no income, continuous time): F = S × e^(rT). With annual compounding: F = S × (1 + r)^T.
• Long call payoff = max(S - X, 0). Long put payoff = max(X - S, 0). Profit = payoff minus premium paid.
• Call breakeven = X + premium. Put breakeven = X - premium.
• Writer's payoff is the mirror image of the buyer's. The writer's maximum profit is limited to the premium received, while the loss is unlimited for a short call and up to X minus the premium for a short put.
• Put-call parity (European, no dividends): C + X × e^(-rT) = P + S.
• Binomial: u and d are the up and down factors. Risk-neutral p = (e^(rT) - d) ÷ (u - d) when using continuous compounding. With discrete rates, p = [(1 + r) - d] ÷ (u - d) per period.
• Binomial value today = discounted expected end-node value using risk-neutral probabilities.
• 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.
• Plain vanilla swap: one party pays fixed, the other pays floating, usually on a notional that is not exchanged in an interest rate swap.
• FRA settles the difference between the agreed rate and the reference rate on the notional, discounted to the settlement date. A cap pays when the rate is above the strike, a floor when below.

Common mistakes

• Using the full-year interest rate for a 3 or 6 month contract. Fix: Always multiply by T in years first, such as 3/12 = 0.25, before computing.
• Adding dividends to the futures price instead of subtracting them. Fix: Dividends are income from holding the asset, so they reduce the cost of carry and the fair price.
• Ignoring the premium when asked for profit. Fix: Payoff is before premium. Always subtract premium paid or add premium received before stating profit or breakeven.
• Treating time value as premium minus strike, or giving negative intrinsic value. Fix: Intrinsic value is never below zero. Time value = premium − intrinsic value. For an out-of-the-money option, the whole premium is time value.
• Using the undiscounted strike price X in the parity equation. Fix: Always discount X to today. Compute PV(X) before writing the equation.
• Applying parity to American options as an equality. Fix: Check the option type first. Parity is exact only for European options. For American options use the inequality.
• Using the real-world probability of an up move in place of the risk-neutral p. Fix: Unless the question tells you to use p from the formula, compute p = ((1 + r) − d) ÷ (u − d). The real-world probability is not needed for pricing.
• Forgetting to discount, or discounting by the wrong power in a two-period tree. Fix: Discount by (1 + r) at every step. If you use the direct formula, discount by (1 + r)² for two periods.
• Using a percentage like 10 instead of 0.10 for r and σ Fix: Convert r and σ to decimals before starting. Write them as 0.10 and 0.20 in your working.
• Forgetting to compute the discount factor and using X instead of X × e^(−rt) Fix: Write PV(X) = X × e^(−rt) as a separate line before applying the formula.

Exam tips

• Write the fair price formula first with numbers. Marks are given for the method even if the arithmetic slips.
• If the question does not state compounding, use the style used in its other data, and say which one you chose.
• In hedging answers, always state the position (long or short futures), the number of contracts, and the net outcome.
• For margin questions, draw a small table of day, settlement price, gain or loss, and balance. Show the margin call amount clearly.
• Add one line on basis risk in the conclusion. Examiners often give a mark for it in theory-linked parts.
• Always give the final answer in profit terms, with breakevens, maximum profit and maximum loss stated. Marks are given for each.
• Draw a neat sketch even when not asked, and label strikes and breakevens. In case-scenario MCQs, check each leg's direction before computing.
• Read the view stated in the question (bullish, bearish, volatile, range-bound) and match it to the strategy. Many theory parts ask when to use a strategy.