flowchart LR D[Derivatives] --> F[Forwards<br/>OTC, customised] D --> FU[Futures<br/>Exchange-traded, standardised] D --> O[Options<br/>Right not obligation] D --> S[Swaps<br/>Periodic exchange] O --> C[Call: right to buy] O --> P[Put: right to sell] style D fill:#FCE4EC,stroke:#AD1457 style F fill:#FFF3E0,stroke:#EF6C00 style FU fill:#E3F2FD,stroke:#1565C0 style O fill:#E8F5E9,stroke:#2E7D32 style S fill:#FFF8E1,stroke:#F9A825
50 Derivatives: Options, Forwards and Futures
50.1 What is a Derivative?
A derivative is a financial contract whose value is derived from the price of an underlying asset, rate, or index. Underlyings can be commodities, equities, currencies, interest rates, bonds, or even other derivatives. The standard reference is John C. Hull’s Options, Futures and Other Derivatives — used by every CFA programme and most graduate finance courses (hull2021?).
Hull’s compact definition: a derivative is “an instrument whose value depends on, or is derived from, the value of another asset” (hull2021?). The Indian Securities Contract (Regulation) Act, 1956 defines a derivative as “a security derived from a debt instrument, share, loan, or any other underlying asset, security, index or contract”.
TipThree Working Definitions
| Source | Definition | What it foregrounds |
|---|---|---|
| John C. Hull | “An instrument whose value depends on, or is derived from, the value of another asset.” | Derived value |
| SCRA, 1956 | “A security derived from a debt instrument, share, loan, security or contract.” | Indian statutory |
| Brealey-Myers | “A claim contingent on the value of one or more underlying assets.” | Contingent claim |
50.1.1 Why derivatives exist — three uses
TipThree Uses of Derivatives
| Use | What the user does | Example |
|---|---|---|
| Hedging | Reduce existing risk | An exporter buys forwards to lock in INR/USD rate |
| Speculation | Take a directional bet on price | A trader buys a call option expecting rise |
| Arbitrage | Lock in risk-less profit from price discrepancies | Cash-and-carry arbitrage between spot and futures |
50.2 Forward Contracts
A forward contract is an OTC, customised agreement to buy or sell an asset at a specified price on a specified future date. Both parties are obligated; no money changes hands at inception.
TipFive Features of Forwards
| Feature | What it captures |
|---|---|
| Customised | Quantity, quality, delivery date are negotiated |
| OTC | Bilateral — direct between buyer and seller |
| No exchange | Counterparty risk is the central concern |
| Settlement at maturity | Usually physical delivery or cash settlement |
| No daily MTM | All gain or loss realised at maturity |
The fair forward price (no arbitrage):
\[F_0 = S_0 \times (1 + r)^T\]
where \(S_0\) = spot price, \(r\) = risk-free rate, \(T\) = time to maturity.
50.3 Futures Contracts
A futures contract is the exchange-traded, standardised cousin of a forward.
TipForwards vs Futures
| Feature | Forward | Futures |
|---|---|---|
| Trading venue | OTC | Exchange |
| Standardisation | Customised | Standardised contract specifications |
| Counterparty | Each other | Clearing house — guarantees performance |
| Margins | None typically | Initial + maintenance margin |
| Settlement | At maturity | Daily mark-to-market |
| Liquidity | Low | High |
| Counterparty risk | High | Low (clearing house) |
The clearing house is what makes futures different from forwards — it acts as the buyer to every seller and the seller to every buyer, eliminating counterparty risk through daily mark-to-market and margin requirements.
50.4 Options
An option is a contract giving the holder the right — but not the obligation — to buy or sell an underlying asset at a specified price (strike) on or before a specified date (maturity).
TipTwo Types of Options
| Type | Right of holder | When valuable |
|---|---|---|
| Call option | Right to buy at the strike | Spot price > strike |
| Put option | Right to sell at the strike | Spot price < strike |
TipAmerican vs European Options
| Type | When the option can be exercised |
|---|---|
| European | Only at maturity |
| American | Any time up to and including maturity |
50.4.1 Payoff Diagrams
For a call option with strike K, the payoff to the holder at expiry is \(\max(S_T - K, 0)\). For a put, it is \(\max(K - S_T, 0)\). The writer (seller) of the option has the opposite payoff.
TipFour Basic Option Positions
| Position | Payoff at expiry | Maximum gain | Maximum loss |
|---|---|---|---|
| Long Call | max(S_T − K, 0) − premium | Unlimited | Premium paid |
| Short Call | premium − max(S_T − K, 0) | Premium received | Unlimited |
| Long Put | max(K − S_T, 0) − premium | K − premium | Premium paid |
| Short Put | premium − max(K − S_T, 0) | Premium received | K − premium |
50.4.2 Moneyness
TipMoneyness of an Option
| Term | Call option | Put option |
|---|---|---|
| In the money (ITM) | Spot > Strike | Spot < Strike |
| At the money (ATM) | Spot ≈ Strike | Spot ≈ Strike |
| Out of the money (OTM) | Spot < Strike | Spot > Strike |
50.4.3 Option price = Intrinsic value + Time value
TipTwo Components of Option Value
| Component | What it captures |
|---|---|
| Intrinsic value | Max(0, S − K) for call; Max(0, K − S) for put — value if exercised now |
| Time value | The remaining premium — reflects volatility, time to expiry, interest rate |
50.5 Black-Scholes Option Pricing Model
Fischer Black, Myron Scholes (Nobel 1997) and Robert Merton (Nobel 1997) derived the pricing formula for European options in 1973 (blackscholes1973?; merton1973?). For a non-dividend-paying stock:
\[C = S_0 N(d_1) - K e^{-rT} N(d_2)\]
\[d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2) T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}\]
where \(C\) = call price, \(N(\cdot)\) = cumulative standard normal, \(\sigma\) = volatility of underlying, \(T\) = time to maturity, \(r\) = risk-free rate.
The five Black-Scholes inputs:
TipThe Five Inputs of Black-Scholes
| Input | Effect on Call price | Effect on Put price |
|---|---|---|
| Spot S | + | − |
| Strike K | − | + |
| Volatility σ | + | + |
| Time to maturity T | + | + (American) |
| Interest rate r | + | − |
50.6 The Greeks
Options are routinely managed using sensitivities called the Greeks:
TipThe Five Option Greeks
| Greek | Captures sensitivity to | Value for long call |
|---|---|---|
| Delta (Δ) | Underlying price | 0 to +1 |
| Gamma (Γ) | Change in delta — convexity | Always positive for long options |
| Theta (Θ) | Time decay | Negative for long options |
| Vega (ν) | Volatility | Positive for long options |
| Rho (ρ) | Interest rate | Positive for call; negative for put |
50.7 Put-Call Parity
For European options on a non-dividend-paying stock:
\[C - P = S_0 - K e^{-rT}\]
Put-call parity is a no-arbitrage relationship — if violated, a riskless profit can be locked in. It is the gateway to a host of synthetic-position constructions.
50.8 Swaps
A swap is an OTC contract to exchange cash flows over time. The two main families:
TipTwo Common Swap Families
| Type | What is exchanged | Use |
|---|---|---|
| Interest-rate swap | Fixed rate vs floating rate (e.g., MIBOR) | Hedge interest-rate risk |
| Currency swap | Cash flows in two different currencies | Hedge or convert currency exposure |
50.9 Indian Derivatives Market
TipIndian Derivatives Market — Milestones
| Year | Milestone |
|---|---|
| 2000 | Index futures launched on BSE and NSE (Sensex / Nifty) |
| 2001 | Index options and stock options launched |
| 2001 | Single-stock futures launched |
| 2008 | Currency futures launched on NSE |
| 2010 | Interest-rate futures re-launched |
| 2018-onward | Major commodity-derivatives expansion |
| 2024-25 | SEBI tightens disclosure and reduces retail F&O leverage in response to losses |
The two main exchanges for derivatives in India are NSE (National Stock Exchange) — the dominant venue — and BSE. Commodity derivatives are traded on MCX and NCDEX, with SEBI as the unified regulator since 2015. Currency derivatives are also offered on the NSE/BSE.
50.10 Practice Questions
Q 01
Forward vs Future
Medium
A key difference between a forward and a futures contract is that:
View solution
Correct Option: B
Futures = exchange-traded, standardised, daily MTM, margined. Forwards = OTC, customised, settled at maturity, no daily MTM.
Q 02
Call
Easy
A call option gives the holder the right to:
View solution
Correct Option: B
A call = right to buy at the strike. A put = right to sell.
Q 03
Long Call
Medium
The maximum loss a holder of a long call option can suffer is:
View solution
Correct Option: C
A long-call holder loses at most the premium paid. Maximum gain is unlimited (price can rise without bound).
Q 04
European
Easy
A European-style option:
View solution
Correct Option: B
European options can be exercised only at maturity. American options can be exercised any time up to maturity.
Q 05
Black-Scholes
Medium
An increase in the volatility (σ) of the underlying:
View solution
Correct Option: A
Higher volatility raises the probability of large moves in either direction; both calls and puts become more valuable. Vega is positive for long options.
Q 06
Put-Call Parity
Medium
Put-call parity for European options on a non-dividend-paying stock states:
View solution
Correct Option: B
Put-call parity: C − P = S₀ − K e−rT. A no-arbitrage relationship between calls, puts, the underlying and a zero-coupon bond.
Q 07
Greeks
Medium
"Theta" of an option measures the sensitivity of the option price to a change in:
View solution
Correct Option: C
Theta = time decay. Delta = spot; Vega = volatility; Rho = interest rate; Gamma = convexity (Δ change).
Q 08
India Derivatives
Medium
Index futures were first launched in India in:
View solution
Correct Option: B
Index futures (Sensex, Nifty) launched in India in 2000. Index options and stock options followed in 2001; currency futures in 2008.
ImportantQuick recall
- Derivative = contract whose value derives from an underlying asset / rate / index. Standard text: John C. Hull.
- Three uses: hedging · speculation · arbitrage.
- Forward = OTC, customised, no MTM, counterparty risk. Futures = exchange-traded, standardised, daily MTM, margined, clearing-house cleared.
- Fair forward price: F₀ = S₀ × (1 + r)T.
- Options: call = right to buy; put = right to sell. European = at maturity only; American = any time up to maturity.
- Long-call max loss = premium; max gain = unlimited. Short-call max gain = premium; max loss = unlimited.
- Moneyness: ITM, ATM, OTM. Option price = intrinsic + time value.
- Black-Scholes (1973): five inputs — S, K, σ, T, r. Vega and Theta are positive and negative respectively for long options.
- Greeks: Delta · Gamma · Theta · Vega · Rho.
- Put-call parity: C − P = S₀ − K e−rT.
- Swaps: interest-rate, currency. India: index futures launched 2000; currency futures 2008. Regulators: SEBI; exchanges: NSE, BSE, MCX, NCDEX.