LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Delta is the partial derivative of a model option price with respect to the underlying price, with every other input held fixed. On a chart of price against spot it is the slope of the tangent. The lesson draws that tangent for a Black-Scholes call, the closed-form European price under geometric Brownian motion, using the qfin library. At a spot of 85 the call is worth about 5.0931 and the delta is about 0.3476. A one-dollar move stays close to the tangent. A move from 85 to 100 opens a wide gap. That gap is the picture of a linear hedge on a curved price.
Why it matters
A market maker who owns the call wants a number for how the value changes when the share moves. Delta is that number, and it is local. The lesson sets aside model-free pricing, including volatility-feature fits and neural nets trained across models, and stays inside one model so the slice is easy to plot. The same slice, taken with respect to volatility, time, or the riskless rate, is vega, theta, or rho. The strike has no greek in this session because it does not move. Gamma is the next slope: the sensitivity of delta to the spot, visible only after you plot delta itself.
The idea in plain words
The call object is built from maturity, a riskless rate, a volatility number, the spot, and the strike. The lesson's base contract uses spot 100, volatility 30 percent, strike 100, one year, and a rate of zero, then reads the price field. To see the price as a function of spot alone, it builds a grid from 60 to 110 with 1,000 points and prices a call at each point, leaving the other four inputs unchanged. The plot is one slice through a price that depends on five numbers. Change volatility, and the curve on the same spot grid moves.
The concrete point is spot 85, same volatility, strike, year, and zero rate. The price field returns 5.0931, which sits on the curve. Delta at that point returns 0.3476 at four decimal places. The tangent is the point-slope line: the vertical gap from 5.0931 equals 0.3476 times the horizontal gap from 85. Drawn from 60 to 110, the line touches the curve at 85 and separates as the spot moves away.
A new call at spot 86, same fixed inputs, is the true price one dollar higher. The line's prediction is the old price plus delta times one. The lesson subtracts them, takes the absolute value, and calls the gap a small hedging error. Repeating the comparison at spot 100, using delta times 15 plus 5.0931 as the line, produces a gap the lesson describes as vastly larger. No numeric error is read out. The chart is the evidence: near 86 the line and the curve are hard to tell apart, and at 100 a clear space has opened.
How to use it
Install qfin and construct the call at the base inputs to confirm the object returns a price. Build the spot grid and the list of prices and plot them. Construct the spot-85 call, print the price and the delta, and overlay the point-slope line. Then price spot 86 and spot 100 with the same volatility, strike, maturity, and rate. Compare each model price with the line. Expect a tight gap at 86 and a wide gap at 100.
To see another greek, repeat the slice along that input. Vega is the slope against volatility. Theta is the slope against time to maturity. Rho is the slope against the riskless rate. Gamma needs a plot of delta against spot, and then a tangent to that plot.
What usually goes wrong
Quoting delta far from the spot where it was computed treats the tangent as the curve. The move from 85 to 100 is the counterexample. Letting another input change mixes slices. This curve is drawn at 30 percent volatility, strike 100, one year, and a zero rate.
The strike is fixed, so this session has no strike greek. The gap between price and tangent is the remainder of the first-order approximation. Gamma is the slope of delta, drawn on a later plot.
Questions people ask
What are the lesson's price and delta at spot 85?
The call price is about 5.0931 and the delta is about 0.3476, with volatility 30 percent, strike 100, one year, and a zero rate.
What does the tangent get wrong?
It tracks the curve closely one dollar away, near spot 86, and it misses by a wide margin at spot 100. The lesson describes that second error as far larger and does not print the size.
Which inputs have greeks here?
Spot gives delta, volatility gives vega, time gives theta, and the riskless rate gives rho. Gamma is the spot sensitivity of delta. The strike is held fixed.