Derivatives: formula sheet

Key formulas

Derivative value

Value of derivative = f(price of underlying)

A derivative has no standalone value. It depends on the underlying.

Hedger

Hedger: has an existing exposure → takes an opposite position to reduce risk

Aim is risk reduction, not profit.

Speculator

Speculator: no existing exposure → takes a view on price direction

Accepts risk for possible profit. Provides liquidity.

Arbitrageur

Arbitrageur: buys cheap in one market and sells dear in another at the same time

Aims at risk-free profit and keeps prices aligned across markets.

Spot and derivative link

Futures price ≈ Spot price adjusted for cost of carry

Gaps beyond this attract arbitrage.

Cost of carry (simple, no income)

F = S × (1 + r × t)

S = spot price, r = annual interest rate, t = time in years (days ÷ 365). Used when the question says simple interest.

Cost of carry (annual compounding)

F = S × (1 + r)^t

Use when the question gives annual compounding.

Cost of carry (continuous)

F = S × e^((r − q) × t)

q = dividend yield or income yield. Income on the asset lowers the futures price.

Cost of carry with known dividend

F = S × (1 + r × t) − PV or value of dividends

Subtract expected dividends received before expiry (compounded to expiry if asked).

Basis

Basis = Spot price − Futures price

With this convention, basis is normally negative when futures trade above spot (contango). It moves towards zero as expiry nears and is zero at expiry (convergence). The opposite convention is Basis = Futures price − Spot price, which is normally positive in contango. Read the question to see which one it uses.

Daily MTM

MTM = (Today's settlement price − Previous settlement price) × lot size × number of lots

For a long position. A short position has the opposite sign. On day one, use the entry price as the reference.

Contract value

Contract value = Futures price × lot size

Margin is a percentage of this value.

Intrinsic value of a call

Max(Spot − Strike, 0)

Never negative. Zero for ATM and OTM calls.

Intrinsic value of a put

Max(Strike − Spot, 0)

Never negative. Zero for ATM and OTM puts.

Premium split

Premium = Intrinsic value + Time value

So time value = premium − intrinsic value.

Call buyer profit at expiry

Max(Spot − Strike, 0) − Premium

Maximum loss = premium. Breakeven = Strike + Premium.

Put buyer profit at expiry

Max(Strike − Spot, 0) − Premium

Maximum loss = premium. Breakeven = Strike − Premium.

Writer profit

Writer's profit = − Buyer's profit

Call writer: Premium − Max(Spot − Strike, 0). Put writer: Premium − Max(Strike − Spot, 0).

Intrinsic value of a call

Max(S − K, 0)

S is spot price, K is strike price. Never negative.

Intrinsic value of a put

Max(K − S, 0)

Out-of-the-money options have zero intrinsic value.

Premium

Premium = Intrinsic value + Time value

Time value = Premium − Intrinsic value. It is highest at the money and falls to zero at expiry.

Put-call parity (European, no dividends)

C + K × e^(−rT) = P + S

C and P are call and put premiums. With discrete compounding, use K ÷ (1 + r)^T. Applies to same strike and expiry.

Delta range

Call: 0 to +1; Put: −1 to 0

At-the-money options have delta near 0.5 (call) or −0.5 (put). Deep ITM approaches ±1.

Greek signs for a long option

Gamma +, Vega +, Theta −; Rho: call +, put −

Option buyers gain from volatility and lose from time decay. Sellers are the reverse.

Approximate premium change

ΔPremium ≈ Delta × ΔS

Gamma refines this for larger moves.

Hedge ratio using beta

Number of futures contracts = β × Portfolio value ÷ (Futures price × Lot size)

Sell (go short) this many contracts to hedge. For a full hedge, use the portfolio beta against the index.

Covered call: maximum profit

(Strike − Purchase price of share) + Premium received

Per share. Applies when the share ends at or above the strike. It assumes the strike is at or above the purchase price. If the strike is below the purchase price, the same expression still gives the capped result, but it can be small or negative.

Covered call: breakeven

Purchase price of share − Premium received

Below this price the position makes a loss.

Protective put: maximum loss

(Purchase price − Strike) + Premium paid

Per share. Valid for a strike at or below the purchase price (an OTM or ATM put). If the strike equals the purchase price, the maximum loss is only the premium. If the strike is above the purchase price, the expression is smaller and can turn negative, which means a locked-in floor above cost. Profit is unlimited as the share rises.

Protective put: breakeven

Purchase price + Premium paid

The share must rise above this for a profit.

Long straddle breakevens

Strike + Total premium and Strike − Total premium

Total premium = call premium + put premium. Maximum loss is the total premium.

Long strangle breakevens

Call strike + Total premium and Put strike − Total premium

Maximum loss is the total premium paid.

Bull call spread

Maximum profit = (Higher strike − Lower strike) − Net premium paid; Maximum loss = Net premium paid

Breakeven = Lower strike + Net premium paid.

Notional value of a futures position

Notional value = Price × Lot size × Number of lots

Margin is a percentage of this value, not of the premium.

Option premium outlay (buyer)

Premium paid = Premium per unit × Lot size × Number of lots

An option buyer's maximum loss is this premium. A seller's loss is not capped.

Daily mark-to-market (long futures)

MTM gain or loss = (Today's settlement price − Previous settlement price) × Lot size × Lots

For a short position, reverse the sign.

Role of clearing corporation

Buyer ↔ Clearing corporation ↔ Seller (novation)

Each party's contract is with the clearing corporation, not with the counterparty.

Regulatory rule for PMS use

Derivatives use must be within the PMS agreement, disclosure document and SEBI rules

Use for hedging and rebalancing is the safe answer in the exam.

Quick revision

• A derivative derives its value from an underlying asset, index, rate or currency.
• Hedgers reduce risk, speculators take risk for profit, arbitrageurs exploit price gaps.
• Forwards are customised, traded over the counter, and carry counterparty risk.
• Futures are standardised, exchange traded, and settled daily through a clearing corporation.
• A call gives the buyer the right to buy; a put gives the buyer the right to sell.
• The option buyer pays the premium and has limited loss; the seller receives the premium and takes the larger risk.
• Buyer's loss on a long option is limited to the premium paid.
• Intrinsic value is the in-the-money amount; premium minus intrinsic value is time value.
• Delta measures premium change for a small change in the underlying price.
• Theta measures time decay; vega measures sensitivity to volatility.
• Strategies combine legs: always note the view, the maximum gain and the maximum loss.
• Check who regulates and who clears: SEBI regulates, exchanges list contracts, clearing corporations guarantee settlement.

Common mistakes

• Thinking a derivative is an asset with its own independent value. Fix: Remember the price is derived from the underlying. If the underlying is removed, the contract has no meaning.
• Classifying a participant by the instrument used. Fix: Classify by purpose and existing exposure. Any instrument can serve any role.
• Saying forwards have daily mark-to-market settlement. Fix: Link daily MTM and margins only to futures. Forwards are normally settled once, at maturity.
• Believing the buyer alone pays margin. Fix: Both long and short positions can lose, so both pay margin to the clearing corporation.
• Treating a put as ITM when spot is above strike. Fix: Remember the put rule is the mirror image: a put is ITM when spot is below strike.
• Showing negative intrinsic value for an OTM option. Fix: Intrinsic value is Max(…, 0). OTM and ATM options have zero intrinsic value.
• Saying higher volatility lowers one of the premiums, such as the put. Fix: Volatility raises both call and put premiums because the buyer's loss is capped at the premium while gains can grow.
• Treating theta as positive for an option buyer. Fix: Time decay erodes the buyer's premium, so theta is negative for long positions and positive for writers.
• Saying a covered call has unlimited profit. Fix: Remember the call you sold caps your gain at the strike plus the premium.
• Thinking a protective put costs nothing. Fix: The premium paid is a cost. It raises the breakeven above the purchase price.

Exam tips

• Questions often give a short scenario. Decide the role from purpose and existing exposure, not the instrument.
• Watch for absolute words like always, guaranteed, eliminates and no risk. They usually mark a wrong option.
• Know the OTC versus exchange-traded split: forwards and swaps are OTC, futures and options are exchange-traded.
• PMS negative marking is 10% of the marks for a question, so guess only when you can remove at least one option.
• Expect difference questions: OTC vs exchange-traded, customised vs standardised, counterparty risk vs clearing guarantee, and settlement timing.
• In calculations, check whether the question says simple or compounded interest before choosing the formula.
• The PMS Distributors paper has negative marking of 10% of the marks for a question, so eliminate wrong options first, then guess if two remain.
• Remember that cumulative MTM equals final price minus entry price, times quantity. Use it to cross-check daily figures.