Greeks

[wrongsaidfred]

Hi David,

This may be another very silly question, but when considering either the Taylor series approximation or the Delta-Theta-Gamma relationship, if we want to extend these to the value of a portfolio with more than one option, how would we do this?

Would each of our greeks just be multiplied linearly, like delta? In other words is the theta of 25 call options just 25 * theta of 1 option? Does the same work for Gamma?

Thanks,
Mike

 

[David Harper CFA FRM]

Hi Mike,

Right, the answer is yes and this is where I prefer to follow Carol Alexander's approach than Hull's in which we think about working with Position Greeks:
position greek = percentage greek * quantity;
e.g., position delta = percentage delta * quantity --> 600 = 0.6 percentage (unitless) delta * 100 #options
... because hedging is largely a matter of adding positions that reduce a position Greek to zero; in the same way that "neutralizing" duration is a matter of bringing the sum of dollar durations to zero.

And now I quote from Carol herself on this (p 164, Vol III):

"III.3.4.4 Position Greeks
To hedge an option on a single underlying it convenient to work with position Greeks. Given an option on S, we define the option’s position Greek as Position Greek = Percentage Greek × N
where the percentage Greek is the partial derivative of the option price as defined in (III.3.40) and N is the position , i.e. number of units of the underlying that the option contracts to buy or sell. Note that N is negative if we are short the option. We use the notation delta(P) for the position delta, so for a single option with percentage delta, the position delta is
delta(P) = percentage delta × N

Similar definitions apply to the option’s position gamma , position vega , position theta , position rho , etc. That is, the position Greek is just the product of the partial derivative and the position on the underlying. Now consider a portfolio of options on a single underlying S.
The net position delta, gamma and theta are the sum of the individual position Greeks. If all the options have the same maturity then the net position rho may also be obtained by summing the position rho of each option. More generally, the position rho is a vector of sensitivities with respect to a term structure of interest rates with maturities equal to the maturities of the options. Net
position vega, vanna and volga are difficult to define. See Section III.5.6 for further details. We often work with position Greeks because:

• They tell us how much to buy or sell of the underlying asset and/or of other options to construct a hedged portfolio (see below);
• Position Greeks are additive , but only for a single underlying"

Hope that helps, David