Lessons · Options
Options and the Greeks
Options questions in trading interviews are less about formulas and more about intuition: what a position pays, how its value moves, and quick estimates you can do in your head.
Payoffs first
A call with strike K pays max(S − K, 0) at expiry; a put pays max(K − S, 0). Almost every strategy is a combination of these. A straddle (call + put at the same strike) pays |S − K|, so it profits from big moves either way. A call spread (long a lower strike, short a higher one) caps both the cost and the payoff.
Put–call parity
A long call plus a short put at the same strike pays S − K in every outcome, exactly like owning the stock and owing K. Positions with identical payoffs must cost the same, so
C − P = S − K·e^(−rT)
With zero rates, C − P = S − K. If a quoted call and put break this, there's an arbitrage.
Pricing with a one-step tree
A $100 stock will be $120 or $90 next period, and rates are zero. Price a $100 call. Find the probability q that makes the stock a fair bet: 120q + 90(1 − q) = 100, so q = 1/3. The call pays $20 in the up state, so it's worth 20 × 1/3 ≈ $6.67. You can replicate it by holding ⅔ of a share, which is the option's delta: (20 − 0)/(120 − 90). This no-arbitrage idea is the heart of Black–Scholes.
The Greeks
- Delta: how much the option price moves per $1 in the stock. Calls range from 0 to 1, puts from −1 to 0. An at-the-money option has delta near 0.5 (−0.5 for a put).
- Gamma: how fast delta changes. It's largest at the money near expiry, where delta can flip from 0 to 1 over a small move. Long options are long gamma.
- Vega: sensitivity to implied volatility. Largest for long-dated at-the-money options.
- Theta: time decay. Long options lose value as time passes, all else equal.
A delta-hedged long option makes about ½ · Γ · (ΔS)² on a move and pays theta for the privilege. That's the core trade-off of options market making.
The mental-math price
For an at-the-money option with low rates:
ATM call ≈ 0.4 × S × σ × √T
The 0.4 is 1/√(2π) ≈ 0.399. A 1-year ATM call on a $100 stock with 20% volatility is about 0.4 × 100 × 0.2 = $8 (exactly $7.97). The price is roughly linear in volatility, so doubling vol roughly doubles the price.
Common traps
- Early exercise. An American call on a stock with no dividends should never be exercised early. It's worth more alive.
- Mixing up gamma and vega. Gamma is about the stock moving; vega is about implied volatility changing.