Derivatives — futures, options, swaps

Derivatives are financial contracts whose value depends on an underlying asset (a stock, bond, currency, commodity, interest rate, or index). They're the most powerful risk-transfer instruments in finance — and the most misunderstood. Used well, they protect businesses and enable risk-bearing where it's most efficiently borne. Used badly, they amplify losses catastrophically.

Futures and forwards

• Forward: bilateral contract between two parties to exchange an asset at a future date for a price agreed today. OTC, customised. Used in FX hedging primarily.
• Futures: exchange-traded standardised forward contracts. CME, ICE, etc. Daily mark-to-market through margin accounts. Used heavily by speculators and corporate hedgers.
• Both create symmetric exposure — the buyer profits if the underlying rises, loses if it falls. The seller has the opposite payoff.

Options

An option gives the holder the RIGHT but not the OBLIGATION to buy (call) or sell (put) the underlying at a defined strike price by a defined expiration date. The holder pays a premium upfront for this right. If the option ends out of the money, the holder simply doesn't exercise; max loss is the premium. If it ends in the money, the holder exercises and profits. The seller (writer) of the option receives the premium but has unlimited (or large) potential losses.

Example: Call option on a stock currently at KES 100
  Strike: 105
  Expiry: 3 months
  Premium: KES 4

Payoff at expiry (per share):
   If stock <= 105: option expires worthless. Loss = 4 (premium paid).
   If stock = 110:  exercise. Profit = 110 - 105 - 4 = 1.
   If stock = 120:  exercise. Profit = 120 - 105 - 4 = 11.
   If stock = 90:   don't exercise. Loss = 4.

Leverage: a 10% move in the stock (100 → 110) produces 1/4 = 25% return
on the option premium. But a 10% fall produces 100% loss of premium.
Options are leverage; leverage is double-edged.

Swaps

A swap is an agreement between two parties to exchange cash flows. Most common type: interest-rate swap. Party A pays B a fixed rate; B pays A a floating rate. Used by corporates to convert floating-rate debt to fixed (or vice versa). FX swaps and equity swaps follow similar logic with different underlyings. Swaps are typically OTC with major banks as intermediaries.

Black-Scholes — the formula and its five inputs

Fischer Black, Myron Scholes, and Robert Merton's 1973 option-pricing model is one of finance's foundational results, and Merton-Scholes won the 1997 Nobel Prize in Economics for it (Black had died and was therefore ineligible). The formula prices a European call on a non-dividend-paying stock as a closed-form function of five inputs — and although working traders rarely compute the Black-Scholes price by hand any more, the variables of the model are the language every option market is quoted in.

  C  =  S · N(d₁)  −  K · e^(−rT) · N(d₂)

  P  =  K · e^(−rT) · N(−d₂)  −  S · N(−d₁)

  where:

         ln( S / K )  +  ( r  +  ½ σ² ) · T
  d₁  =  ─────────────────────────────────────
                       σ · √T

  d₂  =  d₁  −  σ · √T

  N(x) =  the cumulative standard-normal distribution function evaluated at x

Variable glossary — every symbol explained

• C — the fair value of the European call option, in currency units per share. The output. P is the analogous put price.
• S — the current spot price of the underlying stock. Observable in the market. Higher S makes calls more valuable and puts less valuable, both linearly through delta.
• K — the strike price, fixed at contract inception. Lower K makes calls more valuable (you can buy more cheaply); higher K makes puts more valuable (you can sell more dearly).
• T — time to expiry, in years. A 30-day option has T ≈ 0.0822. More time generally makes both calls and puts more valuable because the underlying has more opportunity to move favourably; the exception is deep in-the-money European puts in some interest-rate regimes.
• r — the continuously-compounded risk-free interest rate, in decimal form. Higher r makes calls more valuable (the present value of paying K at expiry falls, so the option to buy gets cheaper to execute) and puts less valuable (you would receive K later in present value).
• σ — sigma, the annualised volatility of the underlying's continuously-compounded returns, in decimal form (0.25 means 25% annual vol). This is the only input not directly observable: it must be estimated from history or inferred from market option prices. Higher σ makes both calls and puts more valuable because the option's pay-off is convex in the terminal price — wider distributions of S_T translate into higher expected pay-off.
• d₁ and d₂ — intermediate quantities standing for the distance, in volatility-scaled units, between current spot and the present value of the strike. They are the points at which the normal distribution is evaluated to give the probabilities the option will be exercised; N(d₂) is the risk-neutral probability the call expires in-the-money.
• ln(S/K) — the log-moneyness of the option. Zero when at-the-money; positive when in-the-money for a call; negative when out-of-the-money.
• ½σ² — the Itô drift correction in the lognormal-returns assumption. Comes from the geometric Brownian motion that underlies the model.
• σ · √T — the standard deviation of the log return over the life of the option. Square root because volatility scales with the square root of time under the model's i.i.d. return assumption.

Volatility σ is the only input that is not directly observable in the market. Analysts estimate it from historical price moves ('historical' or 'realised' volatility) or back it out from observed option prices ('implied volatility', the σ that makes Black-Scholes match the market price). The implied-vol surface — how σ varies across strikes K and tenors T for the same underlying — captures the market's view of skewed and time-dependent risk, and is the daily working object of every options trading desk.

Hedging vs speculation

Derivatives are powerful precisely because they can be used for both hedging (transferring risk away) and speculation (taking on risk for return). A wheat farmer selling December wheat futures is hedging — locking in a price for crops not yet harvested. A speculator buying December wheat futures because they think weather will damage the crop is speculating. Same contract, opposite economic intentions. The market needs both to function — speculators provide the liquidity that lets hedgers execute.