Introduction to Derivatives: formula sheet

Key formulas

Variation margin (long futures)

Variation margin = (F_t − F_t−1) × contract size × number of contracts

A positive amount is credited to a long position; a short position receives the opposite sign. Settled daily.

Exposure to a counterparty without netting

Exposure = Σ max(V_i, 0)

Sum only the positive-value trades. Negative-value trades do not offset in the sum.

Exposure with legally enforceable netting

Net exposure = max(Σ V_i, 0)

Netting never increases exposure. Netted exposure is less than or equal to the un-netted exposure.

Collateralized exposure

Exposure after collateral = max(Net exposure − collateral held, 0)

Ignores thresholds and the margin period of risk unless the question gives them.

Number of bilateral links among n dealers

n × (n − 1) ÷ 2

With a CCP, links fall to n. Useful for explaining why central clearing simplifies the network.

Daily futures gain or loss

Gain/Loss = (F_t − F_t−1) × contract size × number of contracts (long position)

Reverse the sign for a short position. This is the variation margin flow for the day.

Margin account balance

New balance = previous balance + daily gain − daily loss − withdrawals + deposits

Update this every day before testing against maintenance margin.

Margin call trigger

Margin call if balance < maintenance margin; deposit = initial margin − balance

The top-up restores the account to initial margin, not to maintenance margin.

Price move that triggers a call

Adverse move per contract = (balance − maintenance margin) ÷ contract size

A call occurs once the adverse move exceeds this amount (strictly, it takes the balance below maintenance).

Default waterfall order

Defaulter's initial margin → defaulter's default fund share → CCP capital → surviving members' default fund

This is the typical order. Exact details vary by CCP.

Long forward payoff at maturity

Payoff (long) = S_T − K

Per unit of the asset. Multiply by the contract size for the total.

Short forward payoff at maturity

Payoff (short) = K − S_T

Exactly the negative of the long payoff.

Zero-sum check

Payoff (long) + Payoff (short) = 0

Ignores credit default and transaction costs.

Total payoff

Total = (S_T − K) × contract size for the long

Use the opposite sign for the short.

Daily gain or loss

Daily P&L (long) = (F_t − F_(t−1)) × contract size × number of contracts

For a short position, reverse the sign. F is the daily settlement price.

Total gain over holding period

Total P&L (long) = (F_exit − F_entry) × size × contracts

The sum of daily settlements equals this, ignoring interest on margin.

Basis

Basis = Spot price − Futures price

Check which sign convention the question uses. Basis converges to zero at maturity.

Margin call rule

If balance < maintenance margin, variation margin = initial margin − balance

The top-up restores the initial margin level, not the maintenance level.

Convergence

F_T = S_T at delivery

Holds at maturity for the deliverable asset, ignoring delivery options and frictions.

Long call payoff

max(S − K, 0)

S is the asset price at expiry, K the strike.

Long put payoff

max(K − S, 0)

Rises as the asset price falls below K.

Short call payoff

−max(S − K, 0) = min(K − S, 0)

Never positive. Loss is unlimited as S rises.

Short put payoff

−max(K − S, 0) = min(S − K, 0)

Maximum loss is K before the premium.

Profit, long option

Payoff − premium paid

For a short option: premium received − payoff owed.

Break-even price

Call: K + premium; Put: K − premium

Same for the long and short side of the same option.

Moneyness

Call ITM if S > K; put ITM if S < K

At the money when S = K. Otherwise out of the money.

Intrinsic value

Call: max(S − K, 0); Put: max(K − S, 0)

Option price = intrinsic value + time value. American options have non-negative time value because they can be exercised early, so they never trade below intrinsic value. European options cannot be exercised early, so they can trade below intrinsic value. Examples are a deep in-the-money put, or a deep in-the-money call on an asset that pays large dividends.

Fixed leg payment

Fixed payment = Notional × Fixed rate × Accrual fraction

For semi-annual payments the accrual fraction is usually 0.5 (adjust for the day count given in the question).

Floating leg payment

Floating payment = Notional × Floating rate set at start of period × Accrual fraction

The rate is normally set at the start of the period and paid at the end.

Net swap payment

Net to fixed payer = Floating payment − Fixed payment

A positive result means the fixed-rate payer receives. A negative result means it pays.

Swap as bond positions

Value to fixed payer = B(floating) − B(fixed)

The fixed payer is long a floating-rate bond and short a fixed-rate bond. Reverse this for the fixed receiver.

Swap as forward positions

Swap = portfolio of forward rate agreements (interest rate swap) or forward FX contracts (currency swap)

Useful for valuation and for understanding why initial value is about zero.

Long hedge (futures)

Hedged cost = Spot at purchase − Gain on futures = Initial futures price + Final basis

Basis = Spot − Futures. A long hedge protects a planned purchase. Gain on a long futures = (F_end − F_start) × quantity.

Short hedge (futures)

Hedged proceeds = Spot at sale + Gain on short futures = Initial futures price + Final basis

A short hedge protects an asset you own or will sell. Gain on a short futures = (F_start − F_end) × quantity.

Basis

Basis = Spot price − Futures price

If the basis at close is zero, the hedge is perfect. Otherwise basis risk remains.

Return on margin (leverage)

Return on margin = Profit or loss ÷ Initial margin

Leverage multiplies percentage returns. The same ratio applies to losses.

Cost-of-carry forward price (no income)

F₀ = S₀ × e^(rT)

Continuous compounding. If the market forward differs from this, an arbitrage exists.

Option payoffs at expiry

Long call = max(S_T − K, 0); Long put = max(K − S_T, 0)

Buyers risk only the premium. Option sellers keep the premium but face large possible losses.

Quick revision

• A derivative's value depends on an underlying asset, rate or index.
• Exchange-traded contracts are standardised and cleared; OTC contracts are customised and traditionally carry bilateral counterparty risk.
• A central counterparty stands between buyer and seller, reducing bilateral credit exposure.
• Initial margin is posted when a position opens; variation margin moves daily with price changes.
• A margin call arises when the account falls below the maintenance level and must be topped up as the exchange requires.
• Long forward payoff at expiry = S(T) − K; short forward payoff = K − S(T).
• Futures are marked to market daily; forwards are usually settled at maturity.
• Long call payoff = max(S(T) − K, 0); long put payoff = max(K − S(T), 0).
• The option buyer's loss is limited to the premium; the seller of an uncovered call faces potentially unlimited loss.
• A plain vanilla interest rate swap exchanges fixed payments for floating payments on a notional, with the notional usually not exchanged.
• Hedgers reduce an existing exposure; speculators take risk for gain; arbitrageurs seek riskless profit from price differences.
• Option profit = payoff minus premium paid (for the buyer); the seller's profit is the mirror image.

Common mistakes

• Saying exchange-traded derivatives have no credit risk. Fix: Say the risk is much lower and concentrated in the CCP. Margin and default funds reduce it but do not remove it.
• Adding negative-value trades into gross exposure. Fix: Without netting, exposure is the sum of positive values only. Only with netting do negatives offset positives.
• Topping up only to maintenance margin after a margin call. Fix: Futures margin calls restore the account to initial margin. Deposit = initial margin − balance.
• Confusing initial and variation margin. Fix: Initial margin is a upfront buffer for potential future loss. Variation margin is the daily transfer of realized mark-to-market gains and losses.
• Using K − S_T for the long position. Fix: Remember that the long profits when prices rise, so the long payoff is S_T − K.
• Treating the forward payoff like an option payoff with max(S_T − K, 0). Fix: A forward is an obligation. The payoff can be negative with no floor at zero.
• Topping up to the maintenance margin after a margin call. Fix: The call restores the balance to the initial margin. Calculate initial margin minus current balance.
• Getting the sign wrong for short positions. Fix: Write long or short first. A short gains when the futures price falls.
• Reporting payoff when profit is asked, or the reverse. Fix: Underline the word payoff or profit first. Profit always includes the premium.
• Using S − K for a put. Fix: Put payoff is K − S. The put holder sells at K and profits when S is below K.

Exam tips

• Expect conceptual statements: pick the one that correctly pairs a feature (customization, margin, transparency) with the market.
• In exposure problems, check whether the question says netting is enforceable. That one phrase decides whether you sum positives or net everything.
• Remember that central clearing shifts risk to the CCP and creates liquidity strain from margin calls. Questions on reforms often test this.
• For margin questions, write the position and the sign before multiplying. Then use the cumulative price change as a quick check.
• Always write the direction (long or short) before any arithmetic. Sign errors are the most common loss of marks.
• Remember that the call restores the account to initial margin. Distractors often use the maintenance level.
• For conceptual items, match words to margin type: buffer or default coverage means initial margin, daily settlement means variation margin.
• Expect trade-off questions: CCPs cut bilateral exposure through netting but concentrate risk and raise liquidity demands.