Chapter 2: FAQs 113
parameter and may, for example, have been assumed to
be constant. That would be internally inconsistent.
As with gamma hedging, one can vega hedge to reduce
sensitivity to the volatility. This is a major step towards
eliminating some model risk, since it reduces depen-
dence on a quantity that is not known very accurately.
There is a downside to the measurement of vega. It is
only really meaningful for options having single-signed
gamma everywhere. For example it makes sense to mea-
sure vega for calls and puts but not binary calls and
binary puts. The reason for this is that call and put
values (and options with single-signed gamma) have
values that are monotonic in the volatility: increase the
volatility in a call and its value increases everywhere.
Contracts with a gamma that changes sign may have
a vega measured at zero because as we increase the
volatility the price may rise somewhere and fall some-
where else. Such a contract is very exposed to volatility
risk but that risk is not measured by the vega.
Rho ρ, is the sensitivity of the option value to the inter-
est rate used in the Black--Scholes formulæ:
ρ=∂V
∂r.In practice one often uses a whole term structure of
interest rates, meaning a time-dependent rater(t). Rho
would then be the sensitivity to the level of the rates
assuming a parallel shift in rates at all times.
Rho can also be sensitivity to dividend yield, or foreign
interest rate in a foreign exchange option.
Charm The charm is the sensitivity of delta to time.
∂^2 V
∂S∂t.