**9.3
Comparative Statics
**

A |

n important application
of option pricing models is to manage risk.
The option pricing model identifies the value of the option as a function
of different variables such as the underlying asset’s price and volatility.
Therefore, the change in the option’s
value that results from a small change in valuation variable can be calculated. Recall from Chapter 7 that this number, referred to as a
comparative static, is used to identify portfolios that can hedge risk.

The comparative statics
derived in this section allow the comparative statics of any application of this
continuous constant dividend yield model to be derived by using an appropriate
substitution for *q*, the constant
continuous dividend yield.

The call option pricing
formula for the constant dividend yield model is

and since ln(*S**'/X*)
= ln(*S/X*) - *qT*

The underlying variables
that determine the option's value are as before, except for the addition of a
constant dividend yield term, q:

S =
the stock price

*r*
= the interest rate

*q*
= constant continuous dividend yield

s
= the volatility

X
= the strike price

*T*
= the time to maturity

Once again, the partial
derivative of the call price with respect to each variable is computed.
This characterizes the effect of a small change in the variable on the
value of the call, holding all other variables constant.

**Delta
**

The derivative of an
option with respect to the underlying stock price is called the delta of the
option. It provides a measure of
the price risk associated with a particular position.

For a call option and a
put option, respectively, it is
given by

and

Delta tells us how the
value of the call will change if there is a *small
change in the price* of the underlying stock.

**Gamma
**

The gamma of an option
measures how the delta changes with *S*.
This is useful in that, in theory, delta hedging requires continual
rebalancing. This of course is
impossible to implement and even very frequent rebalancing would be
prohibitively expensive. The gamma
tells you how far you are from a hedged position.
It is given for a call option by

Online, in the **Option
Sensitivities** subject, you can see that a position is most sensitive to
changes in delta when it is "at-the-money."
That is, the strike and underlying asset price are about the same.

**Vega (Lambda or Kappa)
**

The vega measures the
effect of a change in the volatility of the underlying stock.
It is also known as lambda and kappa.
It is given by

and thus

Suppose you hold a
position that is delta-gamma-hedged. This
position is not hedged against shifts in the underlying security's volatility.
As a result, a trader holding a delta-neutral position is effectively
betting on a volatility shift. If
volatility increases, then from the comparative statics analysis, the value of
the option is expected to increase. On
the other hand, if volatility declines, then the option value will fall.

Again, if you want to be
delta-, gamma-, __and__ vega-hedged for small changes, this requires at least
three securities.

**Rho
**

Rho measures the change
in the call option with respect to a change in the risk-free interest rate.
It is given by

**Theta
**

Theta measures the
change in the call option with respect to a change in time.
This is useful for
understanding the behavior of option prices over time.
It is given by

**Strike Price
**

The change in the call
option with respect to a change in the strike price is not a risk that we are
interested in hedging, but again it is useful for understanding the behavior of
an option price over different contractual specifications.
It is given by

In
this topic, we have reviewed the basic set of option sensitivities with respect
to small changes in the underlying
pricing variables. Next we focus
our attention on some important applications of the continuous dividend yield
model. The first application is
valuing __Currency Options__,
which is presented in Chapter 10.