Understanding Derivatives: formula sheet

Key formulas

Derivative value

Value of derivative = f(price of underlying asset)

No fixed numeric formula here. The point is that value depends on the underlying.

Hedger

Existing exposure + opposite derivative position → lower risk

Aim is risk reduction, not profit.

Speculator

No existing exposure + derivative position → risk taken for profit

Takes a view on price direction and bears risk.

Arbitrageur

Buy in cheaper market + sell in dearer market at the same time → locked-in gain

Aims at riskless profit and pushes prices towards equality.

Main derivative types

Forwards | Futures | Options | Swaps

Forwards and swaps are OTC. Standard futures and options trade on exchanges.

Cost of carry (continuous compounding)

F = S × e^(r × t)

S = spot price, r = annual risk-free rate, t = time to expiry in years. No income on the asset.

Cost of carry (simple or discrete form)

F = S × (1 + r × t)

Use this when the question gives a simple interest rate and says nothing about continuous compounding.

Cost of carry with known dividend

F = S × (1 + r × t) − D

D = dividend expected before expiry. Income reduces the futures price.

Cost of carry with storage cost

F = S × (1 + r × t) + storage cost

Storage and insurance add to the price for commodities.

Basis

Basis = Spot price − Futures price

Under this convention, contango (futures above spot) gives a negative basis and backwardation (futures below spot) gives a positive basis. Basis moves towards zero at expiry (convergence). Some texts define basis as futures minus spot, which reverses the signs, so follow the question's definition.

Daily MTM

MTM = (Today's settlement price − Previous settlement price) × Quantity

For a long position. A positive figure is a credit; for a short position the sign reverses.

Profit or loss on futures

Long: (Exit price − Entry price) × Quantity; Short: (Entry price − Exit price) × Quantity

Quantity = lot size × number of lots.

Call payoff to buyer at expiry

Max(S − K, 0)

S is spot price at expiry, K is strike. Net profit = payoff − premium.

Put payoff to buyer at expiry

Max(K − S, 0)

Net profit = payoff − premium.

Writer's payoff

Writer's profit = − (buyer's payoff) + premium

Writer's result is the exact opposite of the buyer's. Options are a zero-sum game before costs.

Break-even, call

Strike + Premium

Same for buyer and writer. Above this the call buyer gains.

Break-even, put

Strike − Premium

Below this the put buyer gains.

Intrinsic value

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

Premium = intrinsic value + time value.

Maximum loss and gain

Buyer: loss = premium. Call writer: gain = premium.

Call buyer gain and call writer loss are unlimited in theory. Put gain and loss are limited by the price falling to zero.

Premium

Option premium = Intrinsic value + Time value

Time value = premium − intrinsic value. It is zero at expiry.

Intrinsic value of a call

Max(Spot − Strike, 0)

ITM when spot is above strike.

Intrinsic value of a put

Max(Strike − Spot, 0)

ITM when spot is below strike.

Delta

Delta = Change in premium ÷ Change in spot price

Call delta lies between 0 and +1. Put delta lies between −1 and 0. ATM call delta is roughly 0.5.

Gamma

Gamma = Change in delta ÷ Change in spot price

Highest for ATM options near expiry. Same sign for calls and puts (positive for buyers).

Theta

Theta = Change in premium ÷ Change in time

Negative for option buyers, positive for sellers. Time decay speeds up near expiry for ATM options.

Vega

Vega = Change in premium ÷ Change in volatility

Positive for buyers of both calls and puts. Highest for ATM options with longer time to expiry.

Rho

Rho = Change in premium ÷ Change in interest rate

Positive for calls, negative for puts. Usually the least important greek.

Number of index futures contracts to hedge

Contracts = (Portfolio value × Portfolio beta) ÷ (Index futures price × Lot size)

Round to the nearest whole lot. Sell futures to hedge a long portfolio.

Covered call payoff at expiry

Profit = (Sale price of stock − Purchase price) + Premium received − Max(0, Spot − Strike)

Maximum profit = (Strike − Purchase price) + Premium. Breakeven = Purchase price − Premium.

Protective put

Maximum loss = (Purchase price − Strike) + Premium paid; Breakeven = Purchase price + Premium

Upside is unlimited. Premium is the cost of insurance.

Bull call spread

Max loss = Net premium paid; Max profit = (Higher strike − Lower strike) − Net premium; Breakeven = Lower strike + Net premium

Net premium = premium of bought call − premium of sold call.

Long straddle

Max loss = Call premium + Put premium; Breakevens = Strike ± Total premium

Loss is greatest if spot equals strike at expiry. Profit is unlimited on the upside and large on the downside.

Cash-and-carry arbitrage

Arbitrage exists if Futures price > Spot × (1 + cost of carry) for the period

Buy spot, sell futures, hold to expiry. Reverse the trade if futures are too cheap.

Net swap payment per period

Net = Notional × (Floating rate − Fixed rate) × (days ÷ 365)

Positive means the fixed-rate payer receives. Negative means the fixed-rate payer pays. Many questions use a simple annual or half-yearly fraction, so follow the period given.

Fixed-rate payer view

Fixed payer gains when floating rate > fixed rate

Fixed receiver gains when floating rate < fixed rate.

Gain on hedge for importer using USD long

Gain = (Spot at expiry − Contract rate) × Contract size in USD

Positive when the rupee weakens against the dollar.

Swap versus futures rule

Swap = OTC, customised, counterparty risk; Futures = exchange-traded, standardised, margined

This is the usual comparison tested.

Novation

Buyer ↔ Clearing Corporation ↔ Seller

The clearing corporation becomes the counterparty to both sides, which removes bilateral counterparty risk and guarantees settlement.

Contract value

Contract value = Price × Lot size

Use this to find exposure. Example: a lot of 50 units at ₹100 gives ₹5,000 exposure.

Profit or loss on a long futures position

(Exit price − Entry price) × Lot size × Number of lots

For a short position, reverse the sign: (Entry price − Exit price) × Lot size × Lots.

Daily MTM settlement

MTM = (Today's settlement price − Previous settlement price) × Quantity

A positive MTM on a long position is credited. A negative amount is debited in cash.

Option buyer's maximum loss

Maximum loss = Premium per unit × Lot size × Number of lots (equals the total premium paid)

The premium is quoted per unit of the underlying, so multiply it by the lot size and the number of lots. The option seller's loss can be much larger. The buyer's loss is capped at the total premium paid.

Settlement type

Index derivatives: cash-settled; stock derivatives: physical settlement mandated by SEBI in phases from 2018-19, starting with a select list of stocks

Index derivatives remain cash-settled. The details of stock derivative settlement are revised over time, so use the settlement type stated in the question if it gives one.

Quick revision

• A derivative derives its value from an underlying asset such as a share, index, currency or commodity.
• Hedgers reduce risk, speculators take risk for profit, arbitrageurs exploit price gaps.
• Futures are standardised, exchange-traded and marked to market; forwards are customised and over the counter.
• Long futures gains when the price rises; short futures gains when the price falls.
• Option buyer pays the premium and has limited loss; option seller receives the premium and can face large loss.
• Call gives the right to buy; put gives the right to sell.
• Option premium = intrinsic value + time value.
• Delta measures price change of the option for a change in the underlying; theta measures time decay; vega measures sensitivity to volatility.
• Strategy questions: identify the legs, the market view, and whether profit and loss are capped.
• Swaps are over-the-counter agreements to exchange cash flows.
• Check the question for who is buyer or seller before reading the options.

Common mistakes

• Saying a derivative has its own independent value. Fix: Remember the value is derived from the underlying. If the underlying price is unchanged in the relevant way, there is no separate source of value.
• Calling anyone who uses futures a speculator. Fix: Check for an existing exposure first. A holder of shares who sells index futures to protect the portfolio is a hedger.
• Saying a futures contract has no counterparty risk for the exchange to worry about, or that a forward has a clearing corporation. Fix: Remember: forward = OTC, bilateral, counterparty risk. Futures = exchange, clearing corporation guarantees performance, margins and MTM manage risk.
• Using months directly as t in the cost of carry formula. Fix: Convert to years first: 3 months = 3 ÷ 12 = 0.25 years.
• Treating a put as ITM when spot is above strike. Fix: For a put, ITM means spot is below strike. Ask: would I gain by selling at the strike?
• Forgetting to subtract the premium from the buyer's payoff. Fix: Payoff is not profit. Always compute net profit = payoff − premium when the premium is given.
• Treating the whole premium as intrinsic value. Fix: Always compute intrinsic value from spot and strike first, then subtract it from the premium to get time value.
• Giving a put a positive delta. Fix: Put delta is between −1 and 0 because a put gains when spot falls.
• Buying index futures to hedge a long portfolio. Fix: A long portfolio loses when the market falls, so you need a position that gains then. Sell futures.
• Saying a covered call has unlimited profit. Fix: The short call caps gains at the strike. Maximum profit = (Strike − Purchase price) + Premium.

Exam tips

• Questions are usually definition or scenario based. Decide the participant from intent and existing exposure, not from the instrument used.
• Watch for absolute words such as 'always', 'guaranteed' or 'risk-free'. They are usually wrong, except when describing arbitrage in its theoretical sense.
• Know the four types of contracts and which are OTC and which are exchange-traded.
• Remember the underlying can be many things, including shares, indices, currencies, commodities and interest rates.
• X-A has negative marking of 25% of a question's marks, so on 2-mark caselet questions avoid guessing blindly.
• Learn the forward versus futures comparison as a list: customised or standard, OTC or exchange, counterparty risk or guaranteed, no MTM or daily MTM, illiquid or liquid.
• In pricing questions, check whether the question gives a dividend or storage cost before you pick the formula.
• Wrong answers cost marks, and a 2-mark caselet question costs twice as much as a 1-mark one, so avoid guessing blindly on long calculations. Eliminate options first.