Input one: the price of the underlying

The first of the model's five inputs is the one everybody watches. It is worth being precise about how it acts, because the relationship is not proportional.

For a call, a higher underlying price means a higher option price, always. For a put, the reverse. That much is obvious. What is not obvious is that the size of the response depends on where the price currently sits relative to the strike, and that the response changes as the price moves.

Using the base case — underlying 100, strike 100, 30 days, 4% rate, 25% volatility — the call is worth $3.02. Move the underlying to $101 and it is worth $3.58, a gain of $0.56 for a $1.00 move. Move it to $102 and it is worth $4.20, so the second dollar was worth $0.61. Move down to $99 instead and it is worth $2.52, so the first dollar down cost only $0.50. The same one-dollar step produces a different result depending on where you take it from.

That variation is the whole reason two separate Greeks exist for this one input. Delta measures the current rate of response; gamma measures how fast the rate itself is changing. An option is a curved instrument, and one number cannot describe a curve.

One practical note on the input itself: which price. The model wants the price at which you could actually transact, so a quoted mid-price on a thinly traded underlying introduces error before any of the harder assumptions do.