The Hedge

A hedge gives one person a claim whose payoff offsets a risk elsewhere in that person's affairs. The loss and the compensating payment are joined by a contract, so that the event which makes the first position worse makes the second more valuable. A farmer can arrange a price for a crop before harvest; an importer can arrange an exchange rate before an invoice falls due; the holder of a share can buy the right to sell it at a specified price. The word covers several arrangements, and an option adds a particular freedom: its buyer acquires a right that need be exercised only when it is useful. A call gives the right to buy, a put the right to sell, under the contract's terms. Their seller takes the corresponding obligation in return for a payment now. The question is what that payment prices, and how a claim against the same uncertain future can make the future easier for its buyer to bear.¹

A protective put makes the division of the outcome visible. Suppose a person owns one share and buys a put permitting its sale for one hundred at a single future date. If the share is then worth eighty, the right supplies twenty of value; if it is worth one hundred and twenty, the right can expire unused and the owner keeps the more valuable share. At that date the combined share and put are worth at least one hundred, provided the promise is honoured. This floor concerns the specified asset, exercise date and contractual units. The initial premium has already been paid, and its financing or forgone interest belongs in the total result; purchasing-power changes and other exposures remain. Protection also has a shape. The put preserves the owner's participation in a rise while placing a floor under the asset's terminal value. A sale at a fixed price would give up that participation. The option's value reflects the particular division of good and bad outcomes that the buyer has purchased, rather than a general exemption from uncertainty.

The price becomes especially sharp when the option's payoff can be reproduced by trading assets whose prices are already known. Consider a deliberately small world: the share costs one hundred today and will be worth either one hundred and twenty or eighty tomorrow; interest is zero, there are no dividends or trading costs, and borrowing and short selling are allowed. A put with a strike of one hundred pays zero in the upper state and twenty in the lower. The same payoff can be made by holding sixty in cash and selling half a share short. In the upper state, the half-share liability costs sixty to close and uses up the cash; in the lower, it costs forty and leaves twenty. Setting up this replica costs ten: sixty of cash less fifty received from the short sale. An option with exactly those enforceable payoffs must also cost ten if both positions can be bought and sold without friction. A price above ten permits the expensive claim to be sold and the cheaper replica bought; a price below ten reverses the trade. Either discrepancy leaves an initial gain and matching future obligations. The argument fixes a price through reproducibility.²

Risk-neutral pricing expresses this reproducibility in the language of probabilities, without requiring the people who trade to be indifferent to risk. In the small world, equal weights on the two future share prices give today's price when interest is zero: one half of one hundred and twenty plus one half of eighty is one hundred. Applying those same weights to the put gives ten. These are pricing weights implied by the traded share and cash, not an estimate that the upper and lower states are equally likely in life. If the actual probability of the upper state were sixty per cent, the put's expected cash payoff would be eight; the replication price would still be ten under the stated conditions. The difference reflects the greater price attached to a payment in the lower state. In the continuous model of Black and Scholes, with Merton's contribution to the hedging argument, an option can likewise be replicated by adjusting a holding of the underlying share and a borrowing or lending position as its price changes. The formula uses the current share price, strike, remaining time, interest rate and volatility. The share's expected return disappears from the option-price formula because the dynamically matched position removes the exposure that would have required it.³

The option premium is the price of the whole contingent claim, whereas the risk premium examined in the earlier plates is compensation in expected return or certainty-equivalent value for bearing an exposure. The two uses of the word belong together but cannot be substituted for one another. A put can command a positive purchase price and offer a low expected return because it pays in states where its owner especially wants resources. The person selling it receives the price and undertakes an obligation; the receipt alone is no measure of profit or of a reward for courage. Where replication is possible, the seller can use the receipt to construct the matching position, and the model fixes what that construction costs. Where it is incomplete or costly, the remaining exposure, capital, financing and competition among counterparties matter. The existence of an option price therefore neither proves that the seller has personally carried the full risk to maturity nor tells us how much of the price is payment for doing so. We have to follow the positions created behind the promise.

The conditions that make a hedge exact also mark the places where an actual transfer can fail. A price jump can cross the interval over which a local hedge was matched; a change in volatility can disturb a calculation that held it fixed; transaction costs and interruptions prevent continuous adjustment at no charge. A contract on a related asset may leave a mismatch with the loss it was meant to cover, and a payment at maturity may arrive too late for a collateral demand today. There is also the counterparty: a right to receive a large sum is exposed to the payer's ability to deliver precisely when that sum becomes due. Brunnermeier's analysis of the crisis shows how offsetting contractual positions can coexist with additional funding needs when participants doubt one another's solvency. Collateral and netting can change these exposures, but each requires an account of the assets, timing and obligations it actually covers. A hedge may substantially reduce a chosen risk while moving the residual into funding, execution or another balance sheet. The achievement is measured by the combined position and the conditions of its performance.⁴

The service of the hedge lies in the arrangement of burdens across persons and states. Before costs and default, a derivative payment received by one party is owed by another, but the transfer can still make both parties better situated: their other exposures, resources and needs can differ. A claim on wheat matters differently to the grower and the bakery; protection against a market fall matters differently to someone whose livelihood also falls with it and to someone with resources available in that state. The contract can redistribute purchasing power without rebuilding a destroyed harvest or restoring the output lost in a common disaster. It works through the positions and capacities of those who stand behind it. This is why the premium cannot be read as a quantity of fear bottled and passed across a counter. It is a price for a specified pattern of rights and obligations, carried by particular people through time. The question that follows is whether the person bearing the remaining risk will still be there to take the next step. The Ruin asks what becomes of a favourable average when a loss removes the very capacity to continue.⁵