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a true option is purchased, delta will automatically adjust to price changes
in the underlying security. For example, if a call option is purchased on a
share of General Electric (GE) equity, delta will automatically move closer
to 1 as the share price rises. Conversely, delta will move closer to zero as



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the share price falls. Delta of a synthetic option must be monitored constantly
because it will not automatically adjust itself to price changes in the under-
lying security.
If an initial delta of 0.5 is required for a synthetic call option, then
investors will go long a forward to cover half (0.5) of the underlying secu-
rity™s face value, and Treasury bills will be purchased to cover 100 percent
of the underlying security™s forward value. We cover 100 percent of the secu-
rity™s forward value because this serves to place a “floor” under the strat-
egy™s profit/loss profile. If yields fall and the implied value for delta increases,
a larger forward position will be required. If yields rise and the implied value
for delta decreases, a smaller forward position will be required. The more
volatile the underlying security, the more expensive it will become to man-
age the synthetic option. This is consistent with the fact that an increase in
volatility serves to increase the value of a true option. The term implied delta
means the value delta would be for a traditional option when valued using
the objective strike price and expected volatility. Just how we draw a syn-
thetic option™s profit/loss profile depends on a variety of assumptions. For
example, since the synthetic option is created with Treasury bills and for-
wards, are the Treasury bills financed in the repo market? If yes, this would
serve to lever the synthetic strategy. It is an explicit assumption of traditional
option pricing theory that the risk-free asset (the Treasury bill) is leveraged
(i.e., the Treasury bill is financed in the repo market).
Repo financing on a synthetic option that is structured with a string of
overnight repos is consistent with creating a synthetic American option,
which may be exercised at any time. Conversely, the repo financing structured
with a term repo is consistent with a European option, which may be exer-
cised only at option maturity. Since there is no secondary market for repo
transactions, and since investors may not have the interest or ability to exe-
cute an offsetting repo trade, a string of overnight repos may be the best
strategy with synthetic options.
By going long a forward, we are entering into an agreement to purchase
the underlying security at the forward price. Thus, if the actual market price
lies anywhere above (below) the forward price at the expiration of the for-
ward, then there is a profit (loss). There is a profit (loss) because we pur-
chase the underlying security at a price below (above) the prevailing market
price and in turn sell that underlying security at the higher (lower) market
price. Of course, once the underlying security is purchased, investors may
decide to hang onto the security rather than sell it immediately and realize
any gains (losses). Investors may choose to hold onto the security for a while
in hopes of improving returns.
A long option embodies the right to purchase the underlying security. This
is in contrast to a long forward (or a long future) that embodies the obliga-
tion to purchase the underlying security. Thus, an important distinction to



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be made between a true option and an option created with Treasury bills and
forwards is that the former does not commit investors to a forward purchase.
Although secondary markets (markets where securities may be bought
or sold long after they are initially launched) may not be well developed for
all types of forward transactions, an offsetting trade may be made easily if
investors want to reverse the synthetic option strategy prior to expiration.
For example, one month after entering into a three-month forward to pur-
chase a 10-year Treasury, investors may decide to reverse the trade. To do
this, investors would simply enter into a two-month forward to sell the 10-
year Treasury. In short, these forward transactions would still require
investors to buy and sell the 10-year Treasury at some future date. However,
these offsetting transactions allow investors to “close out” the trade prior
to the maturity of the original forward transaction. “Close out” appears in
quotes because the term conveys a sense of finality. Although an offsetting
trade is indeed executed for purposes of completing the strategy, the strat-
egy is not really dead until the forwards mature in two months™ time. And
when we say that an offsetting forward transaction is executed, we mean
only that an opposite trade is made on the same underlying security and for
the same face value. The forward price of an offsetting trade could be higher,
lower, or the same as the forward price of the original forward trade. The
factor that determines the price on the offsetting forward is the same factor
that determines the price on the original forward contract: cost-of-carry.
Figure 5.22 shows how combining forwards and Treasury bills creates
a synthetic option profile. The profile shown is at the expiration of the syn-
thetic option.
If the synthetic call option originally were designed to have a delta of 0.5,
then the investors would go long a forward to cover half of the underlying
security™s face value and would purchase Treasury bills equal to 100 percent
of the underlying security™s forward value. One half of the underlying security™s
face value is the benchmark for the forward position because the target delta
is 0.5. If the target delta were 0.75, then three quarters of the underlying
security™s face value would be the benchmark. If the price of the underly-
ing security were to rise (fall), then the forward position would be increased
(decreased) to increase (decrease) the implied delta. The term implied delta
means the value for delta if our synthetic option were a true option.
The preceding example assumes that the synthetic option is intended to
underwrite 100 percent of the underlying asset. For this reason our at-the-
money synthetic option requires holding 50 percent of the underlying face
value in our forward position. If our synthetic option were to move in-the-
money with delta going from 0.5 to close to 1.0, we would progressively hold
up to 100 percent of the underlying™s face value in our forward position.




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Treasury bill Treasury forward
Total return Total return




At maturity of At maturity of
the Treasury bill the Treasury bill




Synthetic option
Total return

This distance below a zero total
return represents the
transaction costs associated
with the constant fine-tuning
required for a synthetic option.
At maturity of the
In short, the floor return
synthetic option
(generated by the fixed and
known return on the Treasury
bill) is lowered by the costs of
delta hedging.

FIGURE 5.22 Synthetic option profile.


It is a simple matter to determine the appropriate size of the forward
position for underwriting anything other than 100 percent of the underly-
ing asset. For example, let us assume that we want to underwrite 50 per-
cent of the underlying asset. In this instance, we would want to own 50
percent of the underlying™s face value in Treasury bills and 25 percent of the
underlying™s forward value for an at-the-money option. The delta for an at-
the-money option is 0.5, and 50 percent times 0.5 is equal to 25 percent.
Thus, we want to own 25 percent of the underlying™s forward value in our
forward position.
Again, the delta of a synthetic option will not adjust itself continuously
to price changes in the underlying security. Forward positions must be man-
aged actively, and the transaction costs implied by bid/offer spreads on suc-
cessive forward transactions are an important consideration. Thus, how well
the synthetic option performs relative to the true option depends greatly on
market volatility. The more transactions required to manage the synthetic
option, the greater its cost. The horizontal piece of the profit/loss profile is
drawn below zero to reflect expected cumulative transactions costs at expi-
ration. Thus, expected volatility may very well be the most important crite-
rion for investors to consider when evaluating a synthetic versus a true option



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strategy. That is, if investors believe that the true option is priced rich on a
volatility basis, they may wish to create a synthetic option. If the realized
volatility happens to be less than that implied by the true option, then the
synthetic option may well have been the more appropriate vehicle for exe-
cuting the option strategy.
Finally, the nature of discrete changes in delta may pose special chal-
lenges when investors want to achieve a delta of zero. For example, there
may be a market level where investors would like to close out the synthetic
option. Since it is unlikely investors can monitor the market constantly, they
probably would leave market orders of where to buy or sell predetermined
amounts of forwards or Treasury bills. However, just leaving a market order
to be executed at a given level does not guarantee that the order will be filled
at the prices specified. In a fast-moving market, it may well be impossible
to fill a large order at the desired price. An implication is that a synthetic
option may be closed out, yet at an undesirable forward price. Accordingly,
the synthetic option may prove to be a less efficient investment vehicle than
a true option. Thus, creating synthetic options may be a worthwhile con-
sideration only when replicating option markets that are less efficient. That
is, a synthetic strategy may prove to be more successful when structured
against a specialized option-type product with a wide bid/ask spread as
opposed to replicating an exchange-traded option.
Aside from using Treasury bills and forwards to create options, Treasury
bills may be combined with Treasury note or bond futures, and Treasury bill
futures may be combined with Treasury note or bond futures and/or for-
wards. However, investors need to consider the nuances of trading in these
other products. For example, a Treasury bill future expires into a three-
month cash bill; it does not expire at par. Further, Treasury futures have
embedded delivery options.
Let us now take a step back for a moment and consider what has been
presented thus far. Individual investors are capable of knowing the products
and cash flows in their portfolio at any point in time. However, at the com-
pany level of investing (as with a large institutional fund management com-
pany or even an investment bank), it would be unusual for any single trader
to have full knowledge of the products and cash flows held by other traders.
Generally speaking, only the high-level managers of firms have full access
to individual trading records. Something that clearly is of interest to high-
level managers is how the firm™s risk profile appears on an aggregated basis
as well as on a trader-by-trader basis. In other words, assume for a moment
that there is just one single firm-wide portfolio that is composed of dozens
(or even hundreds) of individual portfolios. What would be the risk profile
of that single firm-wide portfolio? In point of fact, it may not be as large as
you might think. Why not? Because every portfolio manager may not be fol-
lowing the same trading strategies as everyone else, and/or the various strate-



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gies may be constructed with varying cash flows. Let us consider an exam-
ple involving multiple traders, where each trader is limited to having one
strategy in the portfolio at any given time.
Say that trader A has a volatility trade in her portfolio that was created
by going long an at-the-money call option and an at-the-money put option.
Trader A simply believes that volatility is going to increase more than gen-
erally expected. Say trader B has a future in his portfolio and believes that
the underlying security will appreciate in price. Note that these trades may
not at all appear to be contradictory on the surface. Volatility can increase
even without a change in pattern of the underlying asset™s price (as with a
surprise announcement affecting all stocks, such as the sudden news that the
federal government will shut down over an indefinite period owing to a dead-
lock with the Congress over certain key budget negotiations). Such a risk
type is sometime referred to as event risk. The whole idea behind isolating
volatility is to be indifferent to such asset price moves. From the presenta-
tions above, we know that a future can be created with a long at-the-money
call option and a short at-the-money put option. Accordingly, when we sum
across the portfolios of traders A and B we have

Oc Op Oc Op 2 Oc.

By combining one strategy that is indifferent to price moves with
another that expects higher prices, the net effect is a strong bias to upward-
moving prices. It should now be easy to appreciate how an aggregation of
individual strategies can be a necessary and insightful exercise for firms with
large trading operations.
Let us now take this entire discussion a step further. Assume that all of
a firm™s cash flows have been distilled into one of three categories: spot, for-
ward and futures, and options. The aggregate spot position may reflect a
net positive outlook for market prices; the net forward and future position
also may reflect a net positive outlook though on a smaller scale; and the
net option position may reflect a negative outlook on volatility. Could all of
these net cash flows be melted into a single dollar (or other currency) value?
Yes, if we can be permitted to make some assumptions to simplify the issue.
For example, we already know from our various tours around the triangle
that with some pretty basic assumptions, we can bring a forward /future or
option back to spot. By doing this we could distill an entire firm™s trading
operation into a single number. Would such a number have limitations to
meaningful interpretation? Absolutely yes. The fact that we could distill myr-
iad products and cash flows into a single value does not mean that we can
or should rely on it as a daily gauge of capital at risk. We can think of quan-
tifying risk as an exercise that can fall along a continuum. At one end of the



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