Options & Derivatives

The last chapter covered market-making in a stock itself. Options are a different, related product family — and a different way of thinking about risk than anything covered so far. If S&T is a real door you want to keep open, this is one of the biggest concepts to have cold.

What an option actually is

A call option gives you the right (not the obligation) to buy an asset at a fixed price (the strike price) before or at a fixed date (expiration). A put option gives you the right to sell at a fixed price.

You pay a price for that right — the premium — regardless of whether you ever use it. That asymmetry (limited, known downside for the buyer; theoretically much larger potential upside) is the entire appeal of buying options.

Intrinsic value vs. time value

An option's premium splits into two pieces:

• Intrinsic value: how much the option would be worth if exercised right now. A call with a $50 strike on a stock trading at $60 has $10 of intrinsic value. An option with no intrinsic value is "out of the money."
• Time value: everything else in the premium — compensation for the possibility the option becomes more valuable before expiration. Time value shrinks as expiration approaches (theta decay) and hits zero exactly at expiration.

The Greeks — five real, intuitive sensitivities

Each "Greek" answers: how much does the option's price change if one input changes, holding everything else constant?

• Delta: how much the option's price moves for a $1 move in the underlying stock. Also doubles as a rough probability estimate — a delta-0.5 option is roughly a coin-flip to finish in the money.
• Gamma: how much delta itself changes as the stock moves.
• Theta: how much value the option loses purely from one day passing, all else equal.
• Vega: how much the option's price changes if implied volatility changes by 1 percentage point. Higher expected volatility makes an option more valuable, so vega is always positive for both calls and puts, for the buyer.
• Rho: sensitivity to interest rate changes — real, but the least important of the five for day-to-day trading intuition.

Volatility is the product, not just an input

An option's price is fundamentally a bet on volatility, not just direction. You can be right about a stock going up and still lose money on a call option, if implied volatility collapses enough after you buy it — a real, common pattern around earnings announcements. This is why options traders talk about implied volatility (IV) as much as direction — and trading IV itself is a huge part of what an options desk actually does.

A concrete example

Stock trades at $100. You buy a call with a $105 strike, expiring in one month, for a $3 premium.

• Break-even at expiration: $105 + $3 = $108.
• If the stock finishes at $103: option worth $0 — you lose the full $3 premium.
• If the stock finishes at $112: intrinsic value is $7. Profit: $4 — a large percentage return on the $3 risked.

What this connects back to

The market-making chapter showed a desk managing risk from client flow in the underlying stock. Options desks manage a wider set of risks simultaneously (delta, gamma, vega all at once) — the same core skill of continuously hedging and re-pricing risk, just across more dimensions.