Every concept, endless questions
Each concept is a small financial model. Read how it works, then draw a real interview question from it: forwards, backwards or what-if, with fresh numbers and a worked solution every time.
Forward runs a concept the usual way. Inverse runs it backwards, which is what separates understanding from memorizing.
Options & Greeks
6 conceptsPut-call parity
Owning a call and selling a put at the same strike gives you exactly the payoff of owning the stock and owing the strike at expiry: above the strike the call pays, below it the put costs you, and either way you end up paying K for the share. Two positions with identical payoffs must cost the same today, or someone buys the cheap one, sells the dear one and pockets the difference with no risk. So C − P = S − PV(K), with no model and no volatility anywhere in it. That is why parity is the first thing an options desk checks: it holds whatever anyone thinks about the future, and when it breaks it is either a stale quote, a dividend or borrow cost nobody mentioned, or free money.
Delta hedging
An option’s delta is how many shares it behaves like for a small move in the stock. A dealer who sells calls is short that many shares of exposure, and buys exactly that many shares to cancel it. The hedge is only right for an instant: as the stock rises a call behaves more like stock, its delta climbs, and the dealer must buy more — and sell again when it falls. Buying high and selling low is the cost of being short an option, and it is exactly what the option premium pays for. Delta hedging does not remove risk; it swaps directional risk for the risk that the stock moves more than the premium assumed.
Gamma P&L
A delta-hedged long option makes money whenever the stock moves, in either direction, because the option’s delta changes in your favour: it gains more on the way up than the hedge loses, and loses less on the way down. That profit grows with the square of the move — half of gamma times the move squared. Nothing is free, though: the option loses a little value every day through theta. The trade reduces to one comparison. If the stock moves more than the market priced in, gamma beats theta; if it moves less, theta wins. That is what it means to be long volatility, and the break-even daily move is just the implied volatility restated in dollars.
Implied versus realized volatility
Implied volatility is a price: it is the volatility you have to plug into the model to get the option’s market price back out. Realized volatility is a measurement: how much the stock actually moved. An options trader’s whole job is the gap between the two. Buy a straddle when you think the stock will move more than the implied volatility is charging for, sell it when you think it will move less. The at-the-money straddle makes the link concrete — its price is roughly 0.8 × stock × vol × √time, which is also roughly the average size of the move the market expects by expiry.
Theta decay
An at-the-money option is worth roughly its expected move, and the expected move grows with the square root of time, not with time itself. So an option’s value falls like √T as expiry approaches — slowly at first and then very fast. Halve the time left and the option keeps about 71% of its value; quarter it and half the value is gone. That is why theta, the daily decay, accelerates into expiry for at-the-money options: the last month costs far more per day than the first. Option sellers love the final weeks; option buyers pay dearly for holding through them.
Spread payoffs at expiry
Buying a call and selling a higher-strike call gives up everything above the upper strike in exchange for a cheaper position. The sold call pays for part of the bought one, so the spread breaks even sooner than the outright call and loses less if the stock goes nowhere — but its profit is capped at the distance between the strikes. The payoff is three straight lines: flat at a loss of the premium below the lower strike, rising dollar for dollar between the strikes, flat again at the maximum above the upper one. Every vertical spread, collar and risk reversal is read the same way: add up the hockey sticks.