Delta Hedging in Practice

The Greeks gave you a formula, \Delta = \Phi(d_1), and a slogan: "the shares to hold against one call." A trading desk turns that slogan into a job. Every bank that quotes options prices — sells a call here, a put there, to clients who want to buy or sell risk — ends up holding a huge, lopsided book of option exposures it never chose and doesn't want to bet on. Delta hedging is the day-to-day discipline of trading the underlying stock against that book so the desk's fortunes stop depending on which way the stock moves.

This isn't a one-time calculation. \Delta changes every time the stock price moves, so the hedge has to be maintained — checked and adjusted, often several times a day on an active book. That ongoing adjustment is what this lesson is about.

The hedge, in one line

Suppose a market maker sells N call option contracts, each covering m shares (in the US, typically m = 100). Selling a call is a short position in something with positive delta, so the desk's option book alone has delta exposure -N m \Delta — it loses money if the stock rises. To cancel that out, the desk buys shares:

H = N \, m \, \Delta \quad \text{shares held long.}

With H shares long and the short calls, a one-dollar move in the stock changes the shares by +H dollars and the option book by approximately -Nm\Delta dollars — and by construction these cancel. The combined position is, for a small move, delta-neutral: indifferent to the direction the stock takes next.

The catch is the word approximately. \Delta is itself a function of S, so the moment the stock moves, H is already slightly wrong. Keeping the hedge good means recomputing \Delta and re-trading shares — over and over, for as long as the position is open.

Worked example — five days on a market maker's book

A desk has sold N = 100 call contracts on stock ABC, strike K = \$50, m = 100 shares per contract. Every morning the desk recomputes \Delta from the latest spot price and volatility, works out how many shares it now needs (H = 10{,}000\,\Delta, since N m = 10{,}000), and trades the difference from what it already holds.

Day Spot S Delta \Delta Shares needed 10{,}000\Delta Trade to rebalance
0 (inception)$50.000.525,200Buy 5,200
1$51.200.585,800Buy 600 more
2$49.800.505,000Sell 800
3$52.750.636,300Buy 1,300 more
4$54.100.696,900Buy 600 more

Look at the "Trade" column: every time the stock rose, the desk bought more stock; the one day it fell, the desk sold. Rebalancing a short-call hedge means buying as the price rises and selling as it falls — chasing the price, not the other way around. That is not a coincidence of these particular numbers; it is a structural feature of hedging a position with this curvature, and it is expensive. We'll put a name and a number on exactly how expensive in the next lesson.

A natural objection: if the desk sold the call at its fair Black–Scholes value, isn't it already compensated for the risk? Why not just hold the short call naked and let the odds even out?

Because "the odds even out" is a statement about many independent bets, and a single short call is one very large, very direction-dependent bet. A market maker's actual business isn't forecasting whether ABC goes up or down — it's earning the bid-ask spread on option flow, deal after deal, all day. Delta hedging strips out the part of the position the desk has no edge in (which way the stock moves) and leaves only the part it's actually paid for (the spread, and the volatility view embedded in the price). A hedged book turns "did we guess the direction right?" into "did we price the option correctly?" — a much better question for a market maker to be answering forty times a day.

Seeing the hedge move

Below is H(S) = 10{,}000\,\Phi(d_1(S)) — the shares the desk from the example above needs to hold, plotted against the current stock price, for a K=\$50 strike. It's the same rising S-curve as \Delta itself, just rescaled to share counts: near zero far out of the money, near 10,000 (a full share-for-share hedge) deep in the money, steepest right around the strike. Shorten time to expiry and watch the curve steepen into a near-vertical step — exactly where a hedge needs the most frequent attention.

It's tempting to memorize "delta hedging a short option position means buying stock" from the call example above and apply it everywhere. That's backwards for puts.

A put's delta is negative (\Delta_{\text{put}} = \Phi(d_1) - 1 \in (-1,0)) — a put gains value as the stock falls. If the desk sells put contracts, the option book's delta exposure is -Nm\Delta_{\text{put}}, and since \Delta_{\text{put}} is negative, that exposure is positive — the short-put book already behaves like being long stock. To flatten a positive delta exposure you need to hold fewer shares than zero: the correct hedge is to go short stock. Get the sign of \Delta wrong, or forget that "short an option" flips the sign again, and you'll hedge in exactly the wrong direction — doubling your exposure instead of cancelling it. Always work the sign through explicitly: (long or short the option) \times (sign of that option's own delta) tells you which way to trade the stock.