Options & Greeks
From the price that needs no model to the contest a hedged option is really about: parity, payoffs at expiry, the square root of time, implied against realized volatility, delta hedging, and gamma against theta.
Options questions in quant interviews follow one path: what the position pays, what must be true of its price without any model, how it moves when something changes, how you would hedge that, and what the hedge leaves behind. This lesson walks that path with live numbers. It starts with put-call parity, because it is the one option price that holds whatever anyone believes, and ends with gamma against theta, because that is what an options trader is actually paid for.
The framework from the old lesson survives as the order of the chapters: payoff, then replication and no-arbitrage, then sensitivity, then hedge, then the caveat. Two approximations do most of the arithmetic: an at-the-money option is worth about 0.4 × S × σ × √T, and a straddle about twice that. Learn them and most phone-screen options questions become one line.
- Put-call parity. C − P = S − PV(K); the put from the call; the stock and the rate the options imply; the arbitrage package; why rates favour calls.
- Call spreads. Debit, maximum profit and loss, break-even, the three lines, the spread against the outright call.
- Theta decay. 0.4 S σ √T, value scaling with √(days left), theta ≈ C ÷ (2 × days), days from a price ratio.
- Implied against realized. The straddle as an expected move, implied vol from a price, √252 for realized, vega, which way to trade a gap.
- Delta hedging. The share hedge, the rebalance, why delta-neutral still loses on big moves, short puts.
- Gamma against theta. ½ Γ (ΔS)² − θ, the break-even move, implied vol in dollars.
Option time is calendar days ÷ 365, because an option lives through weekends. Realized volatility annualizes daily returns with √252, because returns are only measured on trading days. Every recipe in this topic follows that split, and mixing them up is the most common arithmetic slip in the topic.
An options question, start to finish
Each step points to the chapter that practices it.
Chapters
Put-call parity: the price that needs no model
11 minA call minus a put at the same strike is the stock minus the discounted strike. No volatility, no view, no model. When it breaks, it is a stale quote, a dividend nobody mentioned, or free money.
- Compute a put from a call, or the other way round, with the discounted strike
- Back out the stock price, or the interest rate, the options imply
- Size and describe the arbitrage when a put is quoted away from parity
- Say what rates do to calls against puts, and why volatility never appears
Spreads at expiry: add up the hockey sticks
10 minBuy a call and sell a higher-strike call: cheaper, an earlier break-even, and a capped profit. The payoff is three straight lines, and every vertical spread and collar is read the same way.
- Compute a bull call spread's P&L at any expiry price
- State its maximum profit, maximum loss and break-even
- Compare it with the outright call and find where the two cross
- Recover the upper strike from the maximum profit
An at-the-money option and the square root of time
11 minAn at-the-money option is worth roughly its expected move, and the expected move grows with the square root of time. So its value decays slowly at first and very fast at the end.
- Price an at-the-money call with 0.4 × S × σ × √T, and read implied vol back out of a price
- Scale a price forward in time with √(days left ÷ days at the start)
- Compute theta per day and show why it accelerates into expiry
- Recover the days left from a price ratio
Implied against realized: a price against a measurement
11 minImplied volatility is the number that makes the model give the market price. Realized volatility is how much the stock actually moved. An options trader's job is the gap between them.
- Price a straddle with 0.8 × S × σ × √T and read the one-sigma move off it
- Back implied volatility out of a straddle price
- Annualize a daily standard deviation and compare it with implied
- Compute a vega P&L and say which way to trade a gap
Delta hedging: right for an instant
12 minAn option's delta is how many shares it behaves like. Hedge it and you are flat, until the stock moves and the delta changes. A short-option hedger buys after rallies and sells after declines, and that is the cost the premium pays for.
- Size the share hedge for a short call position from its delta
- Compute the rebalancing trade after a move, and its direction
- Explain why a delta-neutral short position still loses on a large move
- Hedge short puts from the call delta, and recover the contracts from the hedge
Gamma against theta: the contest a hedged option is about
12 minA delta-hedged long option makes half of gamma times the move squared every day, and pays theta for the privilege. The break-even daily move is the implied volatility restated in dollars.
- Compute a day's gamma P&L from gamma and the move
- Net it against theta and say who won the day
- Find the break-even daily move, and the implied volatility it corresponds to
- Explain what a long-volatility trader is really betting on