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. It 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, and 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.