Most people buy a call because they think the stock is going up. A volatility trader buys the same call and then sells stock against it, so that the position no longer cares which way the stock goes. That one step, called delta hedging, is the dividing line between betting on direction and betting on movement. Everything in this module builds on it.
Delta as a number of shares
Delta tells you how much an option's price moves for a $1 move in the stock. Since one contract covers 100 shares, delta also tells you how many shares the contract behaves like. A call with a delta of 0.54 on one contract moves like 54 shares. Sell 54 shares short against it and the two pieces cancel: for a small move in either direction the call's gain matches the shares' loss, or the other way round. The position is delta neutral.
Take a stock at $120 and a 30-day $120 call with a delta of 0.54. You buy one contract and sell 54 shares. The stock rises $1. The call gains about 0.54 times 100, which is $54. The 54 short shares lose $54. Net, nothing. The stock falls $1 and the same thing happens with the signs flipped. Direction has been removed.
The hedge is right at one price only
Delta does not stay at 0.54. As the stock climbs, the call moves further into the money and its delta grows, say to 0.57 after the $1 rise. You are still short 54 shares, but the call now behaves like 57, so the position is net long 3 shares' worth, and it got that way after the stock went up. Fall $1 instead and delta shrinks to about 0.51; now you are net short 3 shares' worth, after the stock went down. The hedge was the delta at the price you set it, and only there.
Net delta of one $100 call hedged with 50 shares sold short, the hedge you would set when the call's delta is one half. The line crosses zero there, just below the strike. Hover at $95 and $105 to read how many shares out the hedge is, then compare the 7-day line.
Read the chart as your hedge's report card. Where the line crosses zero, the hedge is exact. Everywhere else the position has picked up delta, positive above and negative below, and the slope of the line is the call's gamma. With a week left the line is far steeper, so the same $2 move knocks the hedge much further out. Short-dated hedges need trading much more often.
What is left is the curve
Whichever way the stock moves, the hedged position ends up leaning the right way: long after a rally, short after a fall. That bending of delta is gamma, and it is the reason a hedged long option makes money from movement in either direction.
Why does the hedge not cancel the bend too? A share always has a delta of exactly one, whatever the price, so stock has no curve at all. Selling shares can only remove the straight-line part of the option's value. The curved part, the gamma, is left behind untouched. That is the whole idea of dynamic hedging in one sentence: strip out the slope, keep the bend.
One $100 call bought with 30 days left and 50 shares sold against it. The entry figure counts the cash from the short sale, so it shows as a credit. The live curve bends up on both sides of the strike: that is gamma. Drag the days slider down and the smile sinks, fastest at the bottom: that is theta.
Plot the hedged position's profit and loss against the stock price and the result is a smile. It sits lowest near the price where you hedged and rises on both sides, by about half of gamma times the move squared. A $2 move earns four times what a $1 move earns, because the move is squared. Big moves are worth far more than small ones.
The cost of that shape is time decay, theta. Every day the option loses a little value whether or not the stock moves, and on the chart the whole smile sinks, fastest at its bottom. So the hedged position is a trade-off: gamma pays when the stock moves, theta charges when it does not. It is not long the stock and not short the stock. It is long volatility.
Sizing the hedge from the chain
The chain gives you delta for every strike, so the hedge is one multiplication: delta times 100 shares per contract. Ten contracts of a 0.54 delta call need 540 shares sold. A 0.25 delta call needs 25 shares per contract. A long put has a negative delta, so you hedge it by buying shares: one contract of a minus 0.40 delta put needs 40 shares bought. If you own both a call and a put, add the deltas first, then hedge the total.
Round to whole shares; a hedge off by a share or two is fine. The next lesson is about what to do when the price moves.
The maths: the hedged P&L over one day
Write the hedged position as Π = C - Δ·S: one call, short Δ shares. Over a short time dt the stock moves by dS. The call changes by about Δ·dS + ½·Γ·dS² + θ·dt: the slope times the move, the bend times the move squared, and the day's decay. The shares change by -Δ·dS. Add them and the delta terms cancel, leaving dΠ ≈ ½·Γ·dS² + θ·dt. No dS is left on its own, only dS squared, which is positive whichever way the stock went. That is direction dropping out, in one line.
Worked example with the $120 call, taking a gamma of 0.03 (delta 0.54 to 0.57 on a $1 rise) and a theta of minus $0.09 a day per share. A $3 move either way earns ½ × 0.03 × 9 = $0.135 per share, less $0.09 of theta: $0.045 a share, $4.50 on the contract. A $1 move earns ½ × 0.03 × 1 = $0.015 against the same $0.09: a loss of $7.50 on the contract. Set the two terms equal and the break-even move is √(2 × 0.09 ÷ 0.03) = √6, about $2.45 a day.
Your turn
Open NVDA in the Terminal, choose the monthly expiration, and read the Δ column. Start at the money, then move up the call side to the strike whose delta is nearest 0.25, and work out how many shares each one needs. Then read the at-the-money Γ to see how quickly the first hedge will drift.