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has a single-A credit rating instead of the triple-A rating associated with
Treasury instruments? The simplest answer is that we do not want to con-
fuse the risks embedded within the underlying spot (e.g., an ounce of gold)
with the risks associated with the underlying spot™s cost-of-carry. In other
words, within a forward transaction, cost-of-carry should be a sideshow to
the main event. The best way to accomplish this is to reserve the cost-of-
carry component for as risk free an investment vehicle as possible.
Why is R also referred to as a financing rate? Recall the discussion of the
mechanics behind securities lending in Chapter 4. With such strategies (inclu-
sive of repurchase agreements and reverse repos), securities are lent and bor-
rowed at rates determined by the forces of supply and demand in their
respective markets. Accordingly, these rates are financing rates. Moreover, they
often are preferable to Treasury securities since the terms of securities lending
strategies can be tailor-made to whatever the parties involved desire. If the
desired trading horizon is precisely 26 days, then the agreement is structured
to last 26 days and there is no need to find a Treasury bill with exactly 26
days to maturity. Are these types of financing rates also risk free? The mar-
ketplace generally regards them as such since these transactions are collater-
alized (supported) by actual securities. Refer again to Chapter 4 for a refresher.
Let us now peel away a few more layers to the R onion. When a financ-
ing strategy is used as with securities lending or repurchase agreements, the
term of financing is obviously of interest. Sometimes an investor knows
exactly how long the financing is for, and sometimes it is ambiguous. Open
financing means that the financing will continue to be rolled over on a daily
basis until the investor closes the trade. Accordingly, it is possible that each
day™s value for R will be different from the previous day™s value. Term financ-
ing means that financing is for a set period of time (and may or may not be
rolled over). In this case, R™s value is set at the time of trade and remains
constant over the agreed-on period of time. In some instances, an investor



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who knows that a strategy is for a fixed period of time may elect to leave
the financing open rather than commit to a single term rate. Why? The
investor may believe that the benefit of a daily compounding of interest from
an open financing will be superior to a single term rate.
In the repurchase market, there is a benchmark financing rate referred
to as general collateral (GC). General collateral is the financing rate that
applies to most Treasuries at any one point in time when a forward compo-
nent of a trade comes into play. It is relevant for most off-the-run Treasuries,
but it may not be most relevant for on-the-run Treasuries. On-the-run
Treasuries tend to be traded more aggressively than off-the-run issues, and
they are the most recent securities to come to market. One implication of
this can be that they can be financed at rates appreciably lower than GC.
When this happens, whether the issue is on-the-run or off-the-run, it is said
to be on special, (or simply special). The issue is in such strong demand that
investors are willing to lend cash at an extremely low rate of interest in
exchange for a loan of the special security. As we saw, this low rate of inter-
est on the cash portion of this exchange means that the investor being lent
the cash can invest it in a higher-yielding risk-free security, such as a
Treasury bill (and pocket the difference between the two rates).
Parenthetically, it is entirely possible to price a forward on a forward
basis and price an option on a forward basis. For example, investors might
be interested in purchasing a one-year forward contract on a five-year
Treasury; however, they might not be interested in making that purchase
today; they may not want the one-year forward contract until three months
from now. Thus a forward-forward arrangement can be made. Similarly,
investors might be interested in purchasing a six-month option on a five-year
Treasury, but may not want the option to start until three months from now.
Thus, a forward-option arrangement may be made. In sum, once one under-
stands the principles underlying the triangles, any number of combinations
and permutations can be considered.




Quantifying
risk
Options




As explained in Chapter 2, there are five variables typically required to solve
for an option™s value: price of the underlying security, the risk-free rate, time



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Risk Management



to expiration, volatility, and the strike price. Except for strike price (since it
typically does not vary), each of these variables has a risk measure associ-
ated with it. These risk measures are referred to as delta, rho, theta, and vega
(sometimes collectively referred to as the Greeks), corresponding to changes
in the price of the underlying, the risk-free rate, time to expiration, and
volatility, respectively. Here we discuss these measures.
Chapter 4 introduced delta and rho as option-related variables that can
be used for creating a strategy to capture and isolate changes in volatility.
Delta and rho are also very helpful tools for understanding an option™s price
volatility. By slicing up the respective risks of an option into various cate-
gories, it is possible to better appreciate why an option behaves the way it
does.
Again an option™s five fundamental components are spot, time, risk-free
rate, strike price, and volatility. Let us now examine each of these in the con-
text of risk parameters.
From a risk management perspective, how the value of a financial vari-
able changes in response to market dynamics is of great interest. For exam-
ple, we know that the measure of an option™s exposure to changes in spot
is captured by delta and that changes in the risk-free rate are captured by
rho. To complete the list, changes in time are captured by theta, and vega
captures changes in volatility. Again, the value of a call option prior to expi-
ration may be written as Oc S(1 RT) K V. There is no risk para-
meter associated with K since it remains constant over the life of the option.
Since every term shown has a positive value associated with it, any increase
in S, R, or V (noting that T can only shrink in value once the option is pur-
chased) is thus associated with an increase in Oc.
For a put option, Op K S(1 RT) V, so now it is only a posi-
tive change in V that can increase the value of Op.
To see more precisely how delta, theta, and vega evolve in relation to
their underlying risk variable, consider Figure 5.12.
As shown in Figure 5.12, appreciating the dynamics of option risk-
characteristics can greatly facilitate understanding of strategy development.
We complete this section on option risk dynamics with a pictorial of gamma
risk (also known as convexity risk), which many option professionals view
as being equally important to delta and vega and more important that theta
or rho (see Figure 5.13).
The previous chapter discussed how these risks can be hedged for main-
stream options. Before leaving this section let™s discuss options embedded
within products. Options can be embedded within products as with callable
bonds and convertibles. By virtue of these options being embedded, they can-
not be detached and traded separately. However, just because they cannot
be detached does not mean that they cannot be hedged.




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Delta

Delta of call Delta of put Delta of call
Stock price
0
1.0
In-the-money

At-the-money

Out-of-the-money




0 “1.0
Stock price Time to expiration
K


Theta


Stock price
Theta of call Theta of call
K Time to expiration
0
Out-of-the-money




In-the-money


At-the-money




Vega

Vega




K Stock price


FIGURE 5.12 Price sensitivities of delta, theta, and vega.




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Risk Management




Gamma
At-the-money




Out-of-the-money




In-the-money

Time to maturity

FIGURE 5.13 Gamma™s relation to time for various price and strike combinations.



Remember that the price of a callable bond can be defined as

Pc Pb Oc,

where Pc Price of the callable
Pb Price of a noncallable bond
Oc Price of the short call option embedded in the callable

Since callable bonds traditionally come with a lockout period, the
option is in fact a deferred option or forward option. That is, the option
does not become exercisable until some time has passed after initial trading.
As an independent market exists for purchasing forward-dated options, it
is entirely possible to purchase a forward option and cancel out the effect
of a short option in a given callable. That market is the swaps market, and
the purchase of a forward-dated option gives us

Pc Pb Oc Oc Pb

While investors do not often go through the various machinations of
purchasing a callable along with a forward-dated call option to create a syn-
thetic noncallable security, sometimes they go through the exercise on paper




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200 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT



to help determine if a given callable is priced fairly in the market. They sim-
ply compare the synthetic bullet bond in price and credit terms with a true
bullet bond.
As a final comment on callables and risk management, consider the rela-
tionship between OAS and volatility. We already know that an increase in
volatility has the effect of increasing an option™s value. In the case of a
callable, a larger value of Oc translates into a smaller value for Pc. A smaller
value for Pc presumably means a higher yield for Pc, given the inverse rela-
tionship between price and yield. However, when a higher (lower) volatility
assumption is used with an OAS pricing model, a narrower (wider) OAS
value results. When many investors hear this for the first time, they do a dou-
ble take. After all, if an increase in volatility makes an option™s price
increase, why doesn™t a callable bond™s option-adjusted spread (as a yield-
based measure) increase in tandem with the callable bond™s decrease in price?
The answer is found within the question. As a callable bond™s price decreases,
it is less likely to be called away (assigned maturity prior to the final stated
maturity date) by the issuer since the callable is trading farther away from
being in-the-money. Since the strike price of most callables is par (where the
issuer has the incentive to call away the security when it trades above par,

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