Delta and Theta: Sensitivity and Decay

Delta as exposure and as a rough probability

Delta measures how much the option’s value changes when the stock moves a dollar, and because the contract covers a hundred shares it doubles as the exchange rate between contracts and share exposure: ten contracts of a 0.35-delta call behave like 350 shares for small moves. That is the number to size from, because two instruments with the same contract count can have completely different exposure — a 0.20-delta contract is a third of the position that a 0.60-delta contract is. Delta also carries a rough probability reading, since an option that is far out of the money has a small delta and a small chance of finishing in the money; the approximation is close enough to be useful and too loose to be a risk measure, because it prices nothing about what happens before expiry. Delta is not constant, and the rate at which it changes is what makes short-dated positions explosive. As a contract moves into the money its delta rises toward one, so an at-the-money contract with a month left can be worth twice as much of a move by the end of a week if the stock trends — which is the convexity buyers pay for. Near expiry the change becomes violent: a contract that is just out of the money can go from nearly worthless to deep in the money within a session, which is why same-day expiries behave like a different instrument from the ones a month out. From contracts to exposure to cost — Five contracts, delta 0.42: 210 share-equivalents of exposure · A $1.00 move: About $210 on the position ← · Theta −$0.045 a share a day: $22.50 a day on the position · 21 days of decay: $472, needing a $2.25 move just to offset ← Delta is usually quoted per contract, so position delta is contracts × 100 × delta. Summing deltas across a book is the fastest way to see whether one rate move or one stock move dominates everything you own.

Theta: the cost of holding exposure through options

Theta is the decay of extrinsic value per day, and it is concentrated where the extrinsic value is: an at-the-money contract carries the most time value and therefore decays fastest, while a deep in-the-money contract is mostly intrinsic and decays slowly. The decay is also non-linear in time, accelerating in the final weeks as the remaining possibility collapses — which is the arithmetic behind the common cadence of selling thirty to forty-five days out and managing before the last fortnight, where most of the risk lives and least of the premium is left. For a buyer, theta is the daily cost of the leverage and the defined risk, and it is worth translating into the move it requires: if the position burns $22.50 a day against 210 shares of exposure, the stock has to rise about ten cents a day simply to hold value. For a seller, theta is income — but the income is compensation for an uncertain liability, so the question is never whether the decay is pleasant, it is whether the premium was rich relative to the movement that actually arrives. That is O7’s subject, and it is why theta and implied volatility belong in the same sentence. • Position delta = contracts × 100 × delta — size from exposure, not contract count. • Theta is largest at the money and accelerates into expiry. • A buyer’s theta is the daily rent on leverage and capped loss. • A seller’s theta is income paid for an uncertain liability. • Sum deltas across a book to see where the real risk sits. Treating delta as a probability of profit flatters a position. It approximates the chance of finishing in the money, not the chance of making money after the premium; an at-the-money contract bought at a rich price can have a 50% delta and a much smaller chance of being profitable.

Notional exposure is not capital at risk

Delta tells you how much exposure a position carries, and it says nothing about how much money is behind it. Five contracts at a 0.42 delta behave like 210 shares, and on a $50 stock that is **$10,500 of notional** — a real position in every risk sense. The capital committed to get it is not $10,500; it is the premium, about **$1,550**, plus whatever the position will lose if the thesis is wrong. Two numbers, a factor of seven apart, describing the same position. That gap is the whole point of options and the source of most of the damage. Because the capital is small, the *return on capital* is enormous for the same move: a $5 stock move is +$1,050, a 68% gain on the $1,550 committed. The same calculation runs in reverse, and this is the part learners skip: the notional is what you are exposed to, so the **max loss is bounded by the premium but the percentage of the position is not**. Two losing trades at that size cost more than the account can replace. The practical consequence is a sizing rule that does not depend on the premium at all. Size from delta-equivalent shares — as though you were buying the stock — and then ask whether the account can carry the exposure if the position goes to zero. Sizing from premium instead of delta is how a learner ends up with twenty small positions that sum to several times their account in notional, all of them down on the same day, which is the mistake this lesson exists to prevent. The same position, two numbers ($50 stock) — Five contracts, delta 0.42: 210 share-equivalents of exposure · Notional exposure: $10,500 — the risk that actually moves your account ← · Capital committed: $1,550 of premium · A $5 move higher: +$1,050 = +68% on capital, +10% on notional ← · A $5 move lower: −$1,050, and the notional was the thing that mattered ← Both framings are true and they answer different questions. Notional answers “how much risk do I have?”, capital answers “how much did it cost?”. Confusing the two is why an option portfolio can look cheap and behave large.

A hedge ratio that drifts

Delta is defined as the change in the option’s price for a one-dollar change in the stock, and the two uses of that number are the same fact seen from two sides. As a **hedge ratio** it tells you how many shares offset the position: if you are short twelve 0.42-delta calls, buying 504 shares leaves you flat to small moves. As an **exposure** it tells you how much stock risk you are effectively carrying, which is why a book of options is summarised by its net delta before anything else. It is also the reason a hedge does not stay a hedge. Delta is a local slope, so it changes as the stock moves — that rate of change is gamma, which the next lesson in this tier takes apart — and it changes as time passes, since a contract’s delta drifts toward zero or one as expiry approaches depending on which side of the strike it finishes. A position that is delta-neutral today is not delta-neutral after the stock moves two percent. Re-hedging is what makes that drift manageable, and it is also why hedging in a trending market costs money while hedging in a choppy one can pay: you are systematically buying what is rising and selling what is falling. There is one piece of precision worth carrying, because it is where most explanations quietly cheat. Delta is not exactly the probability of finishing in the money. The hedge ratio and the probability of exercise are two different quantities from the same pricing model, and the hedge ratio is the larger of the two for any contract with time left. The gap is small for short-dated options and widens with time and volatility, so “delta is roughly the probability” is a useful shorthand near expiry and a misleading one for a two-year contract. Use it to rank exposures, never to price a bet. The same 12 calls at three moments ($50 stock, $52 strike) — Stock at $50, 40 days left, delta 0.42: Exposure 504 shares; hedge drifts as the stock moves · Stock at $53, 40 days left, delta rises: Exposure grows with the position — the hedge must be added to · Stock at $50, 3 days left, delta falls toward zero: Exposure bleeds away even though nothing happened ← Portfolio delta is what risk management watches, not any one leg. It is the sum of each contract’s delta times its multiplier times its sign, and a book that looks balanced leg by leg can still carry a large net exposure once the signs are added up.

Delta hedging is a bet on volatility

Delta tells you the exposure to hold. Acting on it repeatedly — buying and selling the underlying to return to a neutral delta — looks like a way of removing risk, and it is actually a specific trade with a specific counterparty: it is a position, expressed through rebalancing, on whether realised volatility will turn out higher or lower than the volatility implied by the option you own. Here is the mechanism, in the smallest example that shows it. Buy a call and short the delta-equivalent number of shares; the position has no first-order exposure to the stock. If the stock then moves, the call’s delta changes — gamma — and you must trade shares to restore neutrality, and the direction of that trade is always the profitable one: you sell shares after the price has risen and buy after it has fallen. Each rebalance locks in a small gain proportional to the square of the move, and the sum of those gains grows with how much the stock actually moves. What pays for it is time: the option decays, and the decay is exactly the cost of running the hedge. So the position is long realised volatility and short implied volatility, and the comparison decides the outcome. If the stock’s realised movement over the holding period exceeds the implied volatility you paid, the accumulated rebalancing gains exceed the decay and the hedge profits. If the stock sits still, the decay wins and the hedge loses the difference. The hedge removed the directional risk and replaced it with a volatility risk that was always there, sitting underneath the delta number. Three practical consequences follow. First, the hedging interval matters and it trades off against cost: hedging more frequently captures more of the realised movement and pays more in spreads, so the benefit is not monotonic and the optimum depends on the instrument’s liquidity. Second, the result does not depend on forecasting the price at all, which is why the technique is the foundation of the derivative-pricing argument rather than a chart technique. Third, the same arithmetic applies to a market maker’s book read in reverse: a dealer who is short options is short gamma, must trade with the market rather than against it, and therefore loses if the market moves a great deal — which is the mechanical reason dealer positioning shows up in realised volatility. The honest framing returns to something this subject repeats. A position has several exposures at once, and a hedge removes one of them while leaving the others intact. A delta-hedged option position has no price risk and all of the volatility risk, and the number that decides its P&L is not where the stock goes but how much it moves on the way. • Rebalancing to a neutral delta systematically buys weakness and sells strength. • The gains accumulate with realised movement; the option’s decay pays for them. • A delta-hedged long option is long realised volatility and short implied volatility. • The hedging interval is a cost decision, so the benefit is not monotonic in frequency. A useful way to see it on a live position: divide the option’s daily decay by its gamma. The answer is the daily move the stock has to make for the hedge to break even, and comparing it with recent realised movement tells you whether the position is paying for itself.

Sources

• The Greeks: construction and interpretationHull, “Options, Futures, and Other Derivatives”
• Delta as a probability approximationStandard options theory; risk-neutral density literature
• Theta decay profiles by moneynessCBOE educational materials on time decay