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ically the case when the yield curve has a positive or upward-sloping shape,
as it usually does.
Repeating the formula for a call option, we have

Oc F K V.

If investors believe volatility will soon move much higher than anyone
expects, they may want to create a strategy that isolates volatility and ben-
efits from its anticipated change as suggested by Figure 4.10. Why isolate
volatility? Because our investors are not interested in F (or even X, but X is



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Sell
= Volatility trade
Future
Currency


Option
Currency

Buy
FIGURE 4.10 Use of futures and options to create a volatility strategy.

a constant); they are interested in V. How can volatility be isolated? If
investors wish to buy volatility via an option, they will need to strip away
the extraneous variables, namely F.
F is equal to S (1 R) where S is spot and R is the risk-free rate.
Therefore, to isolate volatility, we simply need to go short (sell short) an
appropriate amount of S and R. Mathematically, we want to accomplish:

F K
V S (1 R)
S SR K V S SR
KV

In words, by going short some S and R, we can reduce a call option™s
value to K and V. We are not too concerned about K since it is a constant
and does not change. The objective is to isolate V, and this can be done. Just
how much of S and R do we need to go short? It depends on how far in- or
out-of-the-money the option happens to be. A call option is said to be in-
the-money if S is greater than K, at-the-money if S is equal to K, and out-
of-the-money if S is less than K. For a put option, the formula is written as
Op K S V, and the put option is in-the-money if S is less than K, at-
the-money if S is equal to K, and out-of-the-money if S is greater than K.
When a call or put option is at-the-money, the option has no intrinsic
value; that is, there is no value to the difference between S and K since sub-
tracting one from the other is zero. The only value to an at-the-money option
is its time value RT and its volatility value V. If an at-the-money call option™s
spot value (S) moves just one dollar higher, then it immediately becomes an
in-the-money option. And if it moves just one dollar lower, it immediately
becomes an out-of-the-money option. Theoretically speaking, an option that
is at-the-money has a 50/50 chance of moving higher or lower. It is just as
likely to move up in price as it is likely to move down in price. When
investors purchase an at-the-money option, they obviously believe that
there is a greater than 50 percent chance that the market will go higher, but



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this is entirely their opinion. They may be right and they may be wrong.
From a purely theoretical standpoint, it is always a 50/50 proposition for
an at-the-money option.
The preceding discussion bears a clue for answering the question of how
much of S and R we need to short to neutralize F and isolate V. The answer
is approximately 50 percent. Under standard Black-Scholes assumption of
log-normality, the delta of a call is greater than 50 percent and that of a put
is less than 50 percent.
When an option contract is purchased, it is always in relation to some
underlying reference (or notional) amount of spot. For example, a single option
on the Standard & Poor™s (S&P) 500 trades to an underlying S&P 500 futures
contract with a reference amount of $250 times the current spot value of the
index. In this instance, spot refers to a particular cash value of 500 stocks in
the S&P index. However, when an investor purchases this option, she does
not pay anything close to $250 times the current spot value of the index.
Because the option has a strike price, the cost of a call option is S (1 R)
K V, not S (1 R) V. Therefore, if the S&P is at a level of 800 and an
at-the-money option is being purchased, then the price to be paid is

$250 800 $250 800 R $250 800 V,

which is considerably less than

$250 800 $250 800 R V.

This latter lower price is what many investors are referring to when they
cite the leveraging features of derivatives.
The amount of S that our investor would go short would be the notional
amount of the contract times 50 percent, or

$250 800 50%.

Since the short position is financed at some rate R, both S and R are
neutralized or hedged (effectively offset) by going short. As is discussed more
in Chapter 5 this type of hedge is commonly referred to as a delta hedge.
Delta is the name given to hedging changes in S, as when an investor wants
to isolate some other financial variable, such as V.
Going delta-neutral is not a strategy whereby investors can hedge it and
forget it. Delta changes as spot changes, so a delta-neutral strategy requires
investors to stay abreast of what delta is at all times to ensure proper hedge
relationships between the option and spot. In point of fact, it can be very
difficult indeed to dynamically hedge an option, a lesson many investors
learned the hard way in the stock market crash of 1987.



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Investors who truly want to speculate that volatility will rise typically
will not buy a call or put option and delta hedge it, but will instead buy both
a call option and a put option with at-the-money strike prices. Because both
the call and put are at-the-money, the initial delta of the call at 0.5 offsets
the initial delta of the put at 0.5. The delta of the call is positive because
a call connotes a long position in the underlying; the delta of the put is neg-
ative because a put connotes a short position in the underlying. With the
one delta canceling out the other, the initial position is delta neutral.
Note that it is the initial position that is delta neutral, since a market
rally (sell off) would likely cause the delta of the call (put) to increase and
thus create a mismatch between delta positions that will need to be adjusted
via offsetting positions in spot.
Parenthetically, investors also can hedge R in an option trade. Just as
the risk of a move in S is called delta risk, the risk of a move in R is called
rho risk. One way that rho risk can be hedged is with Eurodollar futures.
The incentive for hedging the rho risk may be to better expose the other
remaining variables embedded in an options structure. For example, if
investors believe that S will rise over the short term but that monetary pol-
icy also might become easier (and with concomitant pressures for lower inter-
est rates), then eliminating or at least reducing the contribution of rho to an
option™s value could very well help. This could be achieved by shorting some
Eurodollar futures (so as to benefit from a drop in R) of an amount equal
to a delta-adjusted amount of the underlying notional value.
Though while the above methods allow for a way to capture volatility,
they can prove to be quite difficult to implement successfully. Of good news
to the investor desiring to isolate volatility is the advent of the volatility or
variance swap. With a volatility swap, an investor gains if the benchmark
rate of volatility is exceeded by the actual rate of volatility at a prespecified
point in time. The payoff profile at expiration of the swap is simply

1sa si 2 N

where
= is the actual volatility of the index over the life of the swap
a
i = is the volatility referenced by the swap
N = the notional amount of the swap (in dollars or another currency)
per unit of volatility

The above formula can also be modified to describe a variance swap,
where variance is the square of volatility and we have

1s2a si 2 N




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Payoff
0.20



0.10


0.00
0.10 0.20 0.30 0.40 Sigma/variance


0.10



0.20

Sigma

Variance

FIGURE 4.11 Payoff profiles for sigma (volatility) and variance.


A buyer of this swap receives N amount of payout for every unit
increase in volatility (variance) above the volatility (variance) referenced by
the swap ( vol or var). vol or var is usually quoted as a percentage and N
as an amount per 1 percent increase in volatility (for example, $1,000/0.5%
volatility change, or $1,000 per 0.5% increase in volatility above the swap
reference rate of volatility).
In Figure 4.11 we present an illustration of the difference in payoff pro-
files between a volatility and variance swap.




Product
creation




This section presents three instances of product creation that involve mix-
ing and matching bonds, equities, and currencies with various cash flows:




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Financial Engineering



1. Callable structures in the bond market (see Figure 4.12)
2. Preferred stock in the equity market
3. Currency-enhanced securities

First, a story.
A happy homeowner has just signed on the dotted line to take out a
mortgage on her dream home. Although the bank probably did not say
“Congratulations, you are now the owner of a new home, a mortgage, and
a call option,” our homeowner is, in fact, long a call option.
How? Well, if interest rates fall, our homeowner may have a rather pow-
erful incentive to refinance her mortgage. That is, she can pay off (prepay)
her existing mortgage with the proceeds generated by securing a new loan
at a lower rate of interest. This lower rate of interest means lower monthly
mortgage payments, and it is this consideration that gives rise to the value
of the call option embedded within the mortgage agreement.
Now then, if our homeowner is long the call option, who is short the
call option? After all, for every buyer there is a seller. Well, here the mort-
gage bank is short the call option. The mortgage bank is short the call option
because it is not the entity who has the right to exercise (trade in) the option
” it is our homeowner who took out the mortgage and who has the right
to trade it in for a more favorable mortgage sometime in the future.
Now, let us assume that our mortgage bank decides, for whatever rea-
son, that it no longer want to hold a large number of home mortgages. One
option it has is to bundle together a pool (collection) of mortgages and sell
them off to a federal agency, such as Fannie Mae or Freddie Mac. These fed-
eral agencies are in the business of helping people have access to affordable
housing. When the mortgage bank bundles up these mortgages and sells them
off, it is transferring over the short call options as well. Once received, Fannie
Mae or Freddie Mac (or whatever entity purchased the mortgage bank™s
loans) has three choices of what to do with the loans.




Spot
Bond
= Callable structure
Buy

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