Greeks: option sensitivities as partial derivatives
The four levers that move every option book. Traders hedge by Greeks daily; interviewers test Greeks in every first-round screen. Know them cold before you walk in.
An option’s price moves when the stock moves, when vol changes, when time passes, or when rates shift. The Greeks tell you how much. Delta measures the stock-move sensitivity; Gamma is its curvature; Vega is the vol sensitivity; Theta is the time-decay. We’ll add them up to estimate one trade’s P&L overnight.
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Worked example
A non-dividend-paying stock trades at $S = \$100$. A 3-month European call with strike $K = \$100$ has Black–Scholes Greeks $\Delta = 0.55$, $\Gamma = 0.04$ (per $\$1$), $\nu = 20$ (per 1.00 unit of volatility, i.e. 100 vol-points), and $\Theta = -10$ (per year). You are long 1 contract on 100 shares. (a) Stock jumps from $\$100$ to $\$102$ overnight. Estimate the call’s P&L using $\Delta$ and $\Gamma$. (b) Implied vol simultaneously rises from 25% to 27%. Add the vega contribution. (c) One day passes ($1/365$ year). Add the theta bleed. Report total P&L in dollars (per contract on 100 shares).
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Practice 1 of 3Type a fraction, decimal, or expression β mathjs parses it.
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More examples
A handful of harder problems on the same theme. Click any prompt to reveal the solution sketch.
For European and American options on a single stock, the standard variable shorthand is: $S$ = current spot, $K$ = strike, $\tau$ = time to expiry, $\sigma$ = volatility, $r$ = risk-free rate, $D$ = present value (at time $t$) of dividends paid between $t$ and expiry. Call price is denoted $c$ (European) or $C$ (American); put price is $p$ or $P$. When each of $S, K, \tau, \sigma, r, D$ rises (other things equal), in which direction do the prices of European and American calls and puts move?
variable rises
European call
European put
American call
American put
$S$
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$K$
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$\tau$
usually β for non-dividend
ambiguous
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$\sigma$
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$r$
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$D$
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Intuitions: calls benefit from spot up, puts from spot down. Volatility always raises option value (more upside for the holder, downside capped at premium). For European calls on dividend-paying stock, longer $\tau$ can decrease value if dividends accelerate. American options never decrease in value as $\tau$ rises because the holder can always exercise sooner.
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Reflection
Why are option Greeks a Taylor expansion of the price function rather than something option-specific? When does the second-order $\Gamma$ term dominate the first-order $\Delta$ term β and what does that tell you about why traders sometimes care more about “gamma” than about delta?