Sep

28

An interesting approximate relationship, worth knowing, between Gamma and Theta is:

0.5 * S^2 * sigma^2 * Gamma = - Theta

To derive this relationship you can start with the B-S PDE and set the interest rate r equal to zero. Alternatively you can start with the expressions for Gamma and Theta and again setting r=0 show that the left hand side of the above is equal to the right hand side. [read more]

When r is not zero the relationship is only approximate, but in practice reasonably close. So we can write it with a "squiggly" equal sign.

When one hears that "good theta comes with bad gamma and vice versa" this is what is meant. A positive theta (i.e. you earn income from time decay) is associated with a negative gamma (you suffer losses from dynamic hedging) and vice versa.


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