9.4 Introduction to Derivatives

Introduction to Derivatives

FMP–4 opens the derivatives half of Financial Markets and Products. Later readings price and hedge in detail; this section builds vocabulary, payoff intuition, market structure, and participant motives so that futures, options, and swaps chapters have a spine.

Linear Versus Nonlinear Derivatives

A linear derivative’s P&L moves roughly in proportion to the underlying price (constant slope)—classic forwards, futures, and plain interest-rate swaps (viewed as packages of forwards). A nonlinear derivative’s P&L curvature changes with the underlier—options, many exotics, and bonds with embedded options.

Linearity is about payoff geometry, not about whether the product is exchange-traded. A forward is linear and OTC; a listed option is nonlinear and exchange-traded.

Exchange-Traded Versus OTC

Exchange markets excel at fungible risk transfer with multilateral clearing. OTC markets excel at bespoke hedges (exact date, notional, underlier). Post-GFC reforms pushed standardized OTC into central clearing where possible, blurring the old binary—but the customization-versus-standardization trade-off remains exam-relevant.

Options, Forwards, and Futures

Forward contract: agreement to buy/sell an asset at a future date for a price K agreed today. Usually OTC, zero value at initiation if K = forward price F0; at maturity T, long payoff ≈ ST − K (per unit).

Futures contract: similar economic obligation, but exchange-traded, marked to market daily, with margin accounts. Futures price converges to spot at expiry; daily settlement means credit exposure is managed through the clearing system rather than a single terminal payment (in the idealized story).

Call option: right to buy at strike K. Long call payoff at expiry = max(ST − K, 0). Cost = premium paid upfront.

Put option: right to sell at strike K. Long put payoff = max(K − ST, 0).

Forwards/futures bind both parties. Options bind the short (writer) if the long exercises; the long walks away if out-of-the-money (forgoing the premium).

Payoff table (unit notional, ignore marking nuances)

Net P&L for options subtracts (long) or adds (short) the future value of premium.

Worked payoff examples

Assume K = 100.

1. Long forward if ST = 110: payoff = +10. If ST = 90: payoff = −10. Symmetric linear exposure.
2. Long call, premium = 4, ST = 110: gross payoff 10; net ≈ 10 − 4 = +6 (ignoring interest on premium). If ST = 90: gross 0; net ≈ −4.
3. Long put, premium = 3, ST = 90: gross 10; net ≈ +7. If ST = 110: gross 0; net ≈ −3.

Nonlinear products cap downside for the long at the premium while leaving upside open (calls) or downside protection (puts).

Hedgers, Speculators, and Arbitrageurs

Hedgers transfer risk; they do not make it disappear—someone (often a speculator or another hedger with opposite needs) takes the other side. Speculators provide liquidity and risk-bearing capacity. Arbitrageurs enforce law of one price between spots, forwards, and options—subject to frictions, funding, and short-sale constraints.

Hedge example

A wheat farmer will harvest in three months and fears falling prices. Short wheat futures. If spot/futures fall, futures P&L gains offset lower cash crop revenue. If prices rise, futures losses offset higher crop revenue—the hedge stabilizes outcomes, sacrificing upside.

Speculate example

A trader with no wheat inventory buys wheat futures (or calls) purely on a bullish view. Gains if prices rise; losses if prices fall. Leverage via margins amplifies return on equity—and loss.

Arbitrage example (cash-and-carry sketch)

Suppose the traded forward price F0 is too high versus the fair forward from spot S0, storage/funding costs, and benefits. Arbitrage: borrow to buy spot, store, and short the expensive forward. At T, deliver into the forward, repay financing, and pocket the mispricing (if costs were correctly measured). If F0 is too low, reverse: short/sell spot (if possible) and long the cheap forward. Real-world “arb” can fail when hard-to-borrow securities, margin, or uncertain storage break the hedge—then it is relative-value speculation dressed as arb.

Derivative Risks

Derivatives redistribute risk but create new ones:

1. Market risk — delta, duration, vega exposures on open positions.
2. Counterparty credit risk — OTC mark-to-market gains may not be paid (mitigated by collateral, netting, CCPs).
3. Liquidity risk — inability to unwind or roll without large slippage; margin spirals.
4. Basis risk — hedge underlier ≠ exposure (jet fuel vs heating oil; CTD cheapest-to-deliver dynamics).
5. Model risk — wrong vol surface, wrong correlation, wrong CSA discounting.
6. Operational / legal risk — confirmations, settlement fails, disputed terms.
7. Wrong-way risk — exposure rises when counterparty creditworthiness falls.
8. Systemic / procyclical margin risk — cascading variation-margin calls in stress.

A hedger who “eliminates” price risk with an OTC forward still holds counterparty and liquidity risk. A short option position can show stable P&L until gamma and gap risk explode—nonlinear products hide tail exposure.

Synthesis for Later FMP Readings

Carry forward three habits: (1) draw the payoff before debating price; (2) ask exchange or OTC and who is the counterparty; (3) label the trader as hedge / speculate / arb and list residual risks after the trade. Pricing formulas in later chapters assume these distinctions are already clear.