Three ideas, no prior calculus needed
A call option is the right to buy a stock at a fixed price K (the strike) on a future date. A put is the right to sell at K. Its price today depends on the inputs in the panel.
A derivative is a rate of change
Your car's speedometer shows a derivative: how fast your position is changing right now. Delta is the option's speedometer for stock moves: dollars of option per dollar of stock.
"Partial" means one knob at a time
Think of a mixing desk. Turn one knob, leave the others alone, and listen to what changes. Each Greek turns exactly one knob: stock price, volatility, time, or rates.
A second derivative is the change of the change
Acceleration is how fast your speed changes. Gamma is how fast delta changes when the stock moves.
A car that pulls away, then cruises
Press play, or drag sideways on the charts to scrub through time. The faint dots on the road mark where the car was at each second: the gaps grow while it speeds up, then stay equal once it cruises.
| Car | Option | |
|---|---|---|
| the function | position x | option price V |
| 1st derivative | velocity v = dx/dt | delta Δ = ∂V/∂S |
| 2nd derivative | acceleration a = d²x/dt² | gamma Γ = ∂²V/∂S² |
| 3rd derivative | jerk | speed (yes, really) |
| what's changing | time t | stock price S |
The key twist: for a car the thing that changes is time. For delta and gamma it is the stock price, so think of the stock price as the option's clock. The option also has a real "velocity through time": that's theta, Θ = ∂V/∂t. And because the option's price depends on several inputs at once, we write the curly ∂ (partial) to say which one is moving.
Five questions about your option
Each card asks one "what if" question and answers it for the option in the panel. The small formula underneath is the same question written in calculus.
Go deeper: the Black–Scholes formula behind these numbers
All of the numbers come from one formula. You don't need to memorise it; the point is that it is an ordinary function, so we can take its derivatives.
\[ C = S e^{-q\tau} N(d_1) - K e^{-r\tau} N(d_2), \qquad d_1 = \frac{\ln(S/K) + (r - q + \tfrac12\sigma^2)\,\tau}{\sigma\sqrt{\tau}}, \quad d_2 = d_1 - \sigma\sqrt{\tau} \]S stock price, K strike, σ volatility, τ years to expiry, r interest rate, q dividend yield. N is the bell-curve probability function. A partial derivative nudges one input by a tiny amount h and keeps the rest fixed:
\[ \frac{\partial V}{\partial S} = \lim_{h \to 0} \frac{V(S + h,\ \sigma,\ \tau,\ r) - V(S,\ \sigma,\ \tau,\ r)}{h} \]Delta is how steep the price curve is
The top chart shows what the option is worth at every stock price. Delta is how steep that curve is where you're standing, shown by the blue line that just touches it. A steep curve means the option moves almost dollar-for-dollar with the stock. A flat curve means it barely reacts.
Now plot that steepness for every stock price: that's the middle chart, delta. How steep is that curve? That's gamma, at the bottom. Gamma is biggest near the strike, where the price curve bends the most. Drag sideways on any chart to move the stock price.
Measure the slope yourself
A computer measures slope the way you would with a ruler. Check the price a step h to the right and a step to the left, then divide the rise by the run. The smaller the step, the closer you get to the exact Δ.
In car terms: it's like working out speed from two mile markers, the distance between them divided by the time it took. Time a shorter stretch and you get closer to what the speedometer says.
| Method | Value | Off by |
|---|
Go deeper: is a smaller step always better?
Mostly, yes. Looking both ways (central) shrinks the error much faster than looking one way (forward). But below a step of about 0.00001, the computer's rounding errors start to win and the estimate gets worse again. That makes the V shape below.
Walking on a price landscape
Let two inputs change instead of one, and the option's price becomes a landscape. Stock price runs one way, volatility the other, and the height is the option's value.
Stand on the black dot. Walk along the blue path, where only the stock price changes, and the steepness under your feet is delta. Walk along the orange path, where only volatility changes, and the steepness is vega. You only ever walk in one direction at a time, and that's what partial means.
Try this: set the height to Δ. Walking along blue now measures how delta changes with the stock, and that is gamma.
Take the slope again, and again
Every Greek is itself a number that changes, so you can ask how it moves. Gamma is the delta of delta. Vanna asks how delta changes when volatility moves. Each arrow below means "how does this change when ___ moves?"
Here's a neat fact. Where two arrows meet, the order doesn't matter. "How does delta change with volatility?" and "how does vega change with the stock price?" give exactly the same number. Click Vanna to check.
Predict the new price without a pricing model
Say the stock jumps $5, volatility rises 2 points and a week goes by. Each Greek predicts its own share of the change: delta covers the stock move, vega the volatility move, theta the passing time. Add the shares up and you have a quick estimate.
For small moves that's nearly perfect. For big moves the curves bend, and the second-order terms (gamma and friends) correct for it. The bottom rows compare both estimates with the true new price.
Go deeper: the formula (a Taylor expansion)
Theta pays for gamma
Owning an option is like holding a ticket that pays off when the stock moves around. Gamma is how much you gain from those moves. Theta is the rent you pay every day to keep holding the ticket.
Black–Scholes sets the rent so that, for a typical day's move, the two cancel out exactly. The bars below show the balance for your option: the four pieces always add up to zero.
Go deeper: the Black–Scholes equation
The balance above is a partial differential equation that every Black–Scholes price satisfies:
\[ \underbrace{\frac{\partial V}{\partial t}}_{\Theta} + \tfrac12\sigma^2 S^2 \underbrace{\frac{\partial^2 V}{\partial S^2}}_{\Gamma} + (r - q)\,S\,\underbrace{\frac{\partial V}{\partial S}}_{\Delta} - rV = 0 \]