LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A multi-greek hedge cancels several sensitivities at once. Shares have delta, and they have zero gamma and zero vega, so they can flatten a book's spot exposure and nothing else. The lesson hedges a short Nvidia call by solving for holdings in two other calls that cancel gamma and vega, then buying the shares that cancel the delta those calls leave behind. The solved holdings are about 8,641 of the 550-strike call, about 8,006 short of the 555-strike call, and 46 shares long.
Why it matters
The Black-Scholes European model prices an option from the spot, volatility, the riskless rate, time to expiration, and a strike. The strike stays fixed, so the lesson differentiates with respect to the other four inputs. Spot gives delta. The rate gives rho. Time gives theta. Volatility gives vega. A second spot derivative gives gamma, the sensitivity of delta itself. A book that is only delta-flat still moves when volatility moves, when the spot move is large enough to change delta, and when rates or time move. Neutralizing gamma or vega means trading other options, because the underlying cannot supply those greeks.
The idea in plain words
The starting book is short 1,000 European calls. The lesson's inputs, taken from the market snapshot it uses, are a spot of 543, an implied volatility it records as 53, a strike of 545, maturity of 30/365 of a year, and a 1.5 percent rate used as a LIBOR stand-in that it calls high for a one-month option. The short book prints about minus 523.88 delta, minus 6.35 gamma, and minus 815.54 vega. Two other calls share the same spot, volatility, maturity, and rate, with strikes 550 and 555. Their gammas and vegas are known from the same model.
Canceling two numbers takes two weights. Put the two gammas in the first row of a matrix and the two vegas in the second row. The target vector is the opposite of the short book's gamma and vega, which the lesson builds as 1,000 times the original call's gamma and vega. The weights are the inverse of that matrix times the target. Round the matrix to two decimal places before inverting. The lesson does this to avoid a failure on a matrix that floating-point noise has made uninvertible. The product is about 8,641 contracts in the first extra call and about minus 8,006 in the second.
A check multiplies the rounded greek matrix by those weights and subtracts the target. Gamma and vega both come back as zero in the lesson's print.
How to use it
Price the three calls with the European model and read delta, gamma, and vega. Form the 2 by 2 matrix and the target column in NumPy. Invert the rounded matrix, multiply by the target, and print the weights. Run the dot-product check.
Delta is still open. Rebuild a three-row picture: deltas of the two extra calls on the first row, then gammas, then vegas. Multiply by the same weights and add the original short book's greeks. The lesson's print is about minus 46 delta, with gamma and vega at zero. The shares contribute delta equal to the number of shares and contribute neither gamma nor vega. Buying 46 shares brings the printed delta to zero as well. The finished book is short 1,000 of the 545-strike calls, long about 8,641 of the 550s, short about 8,006 of the 555s, and long 46 shares.
The lesson states the counting rule in one line. Each greek other than delta needs its own extra tradable option. Two extra greeks, gamma and vega, meant two extra strikes. Rho and theta are named and not hedged in the numbers.
What usually goes wrong
Inverting an unrounded greek matrix can fail even when the strikes differ, because the rows are close. The two-decimal rounding is part of the procedure, and the zero check uses that same rounding.
The share trade has to use the residual delta after the option weights, about minus 46, not the original minus 523.88. The weights are large beside the original 1,000 contracts because the extra strikes have similar greeks. They belong to strikes 550 and 555 and to this snapshot. A new spot means a new inverse. The lesson computes the premium on the short 1,000 and then sets that print aside.
Questions people ask
Why can shares not hedge vega?
The lesson treats the underlying as delta one, with zero gamma and zero vega. An offsetting vega has to come from another option.
What weights did the inverse produce?
About 8,641 of the 550-strike call and about 8,006 short of the 555-strike call, after rounding the greek matrix to two decimals.
How many shares finish the hedge?
The residual delta after those option weights is about minus 46. The lesson buys 46 shares and prints zeros for delta, gamma, and vega together.