Random World: Book - The Complete Guide to Option Pricing Formulas

The Complete Guide to Option Pricing Formulas - Chapter 2

Option Greeks are partial derivatives of BSM. The partial derivatives is a measure of sensitivity of the optio price to a small change in a parameter of the formula.

Delta - is the option's sensitivity to small changes in the underlying asset price

$\Delta _{call} = \frac{\partial p}{\partial s} = e^{(b-r)T} [N(d1)] > 0$

$\Delta _{put} = \frac{\partial p}{\partial s} = e^{(b-r)T} [N(d1) -1] < 0$

The Behavior of Delta
As a Call option gets deep-in-the-money, N(d1) approaches 1, but never exceeds 1 (since it's a cumulative distribution function). For a European call option on a non dividend-paying stock, N(d1) is moreover equal to the option's delta. Delta can therefore never exceed 1 for this option. For general European call options, delta is given by $_{e}^{(b - r)T} N(d1)$ . If the term $_{e}^{(b - r)T} N(d1)$ is larger than 1 and the option is deep-in-the-money, the delta can thus become considerably larger than 1. This occurs if the cost-of

Delta Mirror Strikes and Assets
The following strikes equalize the absolute values of deltas of put and call options

$X_{put} = \frac{S^{2}}{Xcall} e^{(2b+\sigma^2)T}$
$X_{put} = \frac{S^{2}}{Xcall} e^{(2b+\sigma^2)T}$

Stike from Delta
Options are quoted by delta rather than strike in several OTC markets. In this type of quotation method one typically asks for delta and expects the salesperson to return a price as well as the strike given a spot reference.

Futures Delta from Spot Delta

DdeltaDvol and DvegaDspot
DdeltaDvol, defined as $\frac{\partial \Delta }{\partial \sigma}$ , is mathematically the same as DvegaDspot, defined as $\frac{\partial vega }{\partial S}$
where n(x)

Book - The Complete Guide to Option Pricing Formulas - Option Greeks