7.2
Comparative Statics

ecall that the call option pricing formula is
where
The
variables that determine the option's value are:
S
the stock price
r the interest rate
s the
volatility
X
the strike price
T
the time to maturity
To
calculate how the variable affects the option price, we simply take the
derivative of the call price with respect to that variable.
These derivatives are called the hedge
parameters, and measure the sensitivity
of the option price to the variable. This sensitivity tells you how the option
value changes with a small change in the variable. If you view these variables as risky, then this is called
your (local) exposure to the variable.
There
are several limitations to this analysis.
First,
the sensitivity you obtain is relevant only for small
changes in the variable, and can be quite misleading if there is a large change.
Second,
if you use this model to hedge a position, you should understand that the
numbers you get are only as good as the option pricing model.
If the BlackScholes model is an accurate model of realworld option
prices, the numbers will serve you well;
otherwise, they will not.
Third,
you should recognize that the model assumes that the stock price evolves
continuously. This means that the
option price also evolves continuously, and therefore so do the hedge
parameters.
Finally,
note that even though we are going to measure the sensitivity to, say, the
riskfree interest rate, the model actually assumes that the interest rate is
constant. The same applies to
volatility. In spite of this
limitation, the BlackScholes hedge parameters are widely applied to measure and
hedge the risk from these factors.
Because
the hedge parameters are denoted by Greek letters they are sometimes called the
"option Greeks."
Delta
The
partial derivative of the option value with respect to the underlying stock
price is called the delta of the option.
For
a call option it is given by
and
for a put option it is given by
.
Thus,
N(d1) is the delta of the call; it tells us how the value of the
call will change if there is a small change in the underlying stock
price.
The
formal derivation of delta is provided in the technical topic 7.3, Derivation
of Delta.
To
show how the delta is used to hedge the risk of stock price changes, suppose you
own one call option. You want to
ensure that a small change in the stock price does not affect the value of your
call position. One way to do this
is to sell delta (D) stocks. If you do this, your position is
D
stocks, +1 call
and
the value of the position is
Then,
Now,
you are completely hedged against price risk.
Of course, this would not be true if there were a large change in the
stock price. Note also that over
time, you would have to hedge your position continuously to maintain the initial
value. This is called
"rebalancing." You
may see the similarities with the delta
hedging that
you studied for the binomial model in Chapter 4, topic 4.3.
For
example, suppose that you have written (or sold) 1,000 call options, and the
delta of each option is 0.7. Then,
you would need to be long 700 stocks to be hedged, or to be "delta
neutral." If the delta changes to 0.71, you would need to purchase an
additional 10 stocks to maintain neutrality.
The phrase, "deltaneutral," means that the delta of your position
(i.e., the options and the stocks) is zero.
More
generally, the delta of your position measures the exposure of your position to
price risk. In layman's terms, it
measures the "bet" that you have taken that prices will rise or fall.
For example, consider the portfolio:
The
position delta is:
If
this is positive, say, it equals 1,000, then every $1 increase in S
translates into a $1,000 increase
in the value of your position. With
this position, you have taken a bet that the stock price will rise.
Of course, you lose $1,000 if the stock price falls by a dollar!
Similarly,
a negative delta is a bet that the stock price will fall.
Gamma
The
gamma of an option measures how the delta changes with S.
This is useful in that delta
hedging requires continual rebalancing of your portfolio.
The transaction costs associated with trade make it prohibitively
expensive to trade so frequently. If
you rebalance less frequently than required, you need to know how far you are
from a hedged position.
The
gamma tells you how far you are from a hedged position.
For both put and call options, gamma is given by
Online,
in the subject Option Sensitivities, you can see that an option is most
sensitive to changes in delta when it is "at the money" (i.e., the
strike and underlying asset price are about the same).
You will also see that this sensitivity increases as the option gets
closer to maturity.
Suppose
you want your call option position to be both delta and gammahedged against
small changes in the stock price. You
cannot do this with just the stock, as we did before.
Instead you need at least two securities (e.g., a stock and, say, a put
option).
To
see this, suppose that you have written (sold) n2 call
options, but wish to maintain a position that is both delta and gammaneutral.
Your position delta is:
and
the position gamma is:
You
have to choose n2
and n3
to make the lefthand side of both equations zero. You can do this since you have two equations and two
unknowns. This also shows you why
you need two securities to create a delta and gammaneutral position for your
option writing activities.
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. The vega
is the same for puts and calls. The
calculation for a call is
and
thus
Suppose
you hold a deltaneutral position where you are long one call and short delta
stocks. This position is not hedged against shifts in the stock's volatility;
the vega of the position equals the vega of the call, and is positive.
As a result, a trader holding a deltaneutral position is effectively
betting on a volatility shift. If
volatility increases, then the value of the position is expected to increase.
On the other hand, if volatility declines then the position value will
fall.
If
you desire to be delta, gamma, and vegahedged, then you need at least
three securities in addition to your existing position.
In hedging volatility risk using vega, you need to remember that in the
option pricing model, volatility was assumed to be constant over the life of
the option.
Also
note that the sign of vega is positive. This
implies that options defined on more volatile stocks are more valuable than
options defined on less volatile ones. You
might like to recall the example we considered when applying the certainty model
to pricing IBM American call options in Chapter 6, topic 6.2, Option
Valuation under Certainty. In
this example, the certainty model systematically underpriced the call option.
This is consistent with the BlackScholes model, since volatility was
ignored.
Rho
Rho measures
the change in the option price with respect to a change in the riskless interest
rate. For a call option, rho is
For
a put option, rho is
Theta
Theta
measures the change in an option price with respect to the option's maturity.
This is not a "risk" that we are interested in hedging, but it
is useful for understanding the behavior of an option over time. For a call option, theta is
For
a put option, the theta is
Strike Price
Finally,
for completeness, we show you how the option price changes with the strike
price. For a call option, the
partial derivative with respect to the strike price is:
For
a put option, the strike price is:
This
completes our discussion of the comparative statics of the BlackScholes model. In the next chapter we look at some common option trading
strategies. The BlackScholes model
lets you value calendar spreads at a future point in time.
Chapter 9 presents an important extension of the BlackScholes model
which is to relax the assumption that the stock pays no dividends.
This extension is important for virtually every practical application of
the model. In Chapter 9, topic 9.2,
Constant Continuous Dividend Yield Option
Pricing Model, the pricing model is extended to the case where the stock
pays a continuous constant dividend yield.
Later you will see, this simple extension lets you apply the model not
only to such stocks, but also to futures, currencies, and stock indices.