the yield table, consideration of transaction costs would make it difficult to structure a worthwhile arbitrage around the 3.5 bp differential.
Finally, note that the return of 5.670 percent is 15 bps above the return that could be earned on the three-month U.S. Treasury bill. Therefore, given a choice between a three-month Canadian Treasury bill fully hedged into U.S dollars earning 5.670 percent and a three-month U.S. Treasury bill earn- ing 5.520 percent, the fully hedged Canadian Treasury bill appears to be the better investment.
Rather than compare returns of the above strategy with U.S Treasury bills, many investors will do the trade only if returns exceed the relevant Eurodollar rate. In this instance, the fully hedged return would have had to exceed the three-month Eurodollar rate. Why? Investors who purchase a Canadian Treasury bill accept a sovereign credit risk, that is, the risk the gov- ernment of Canada may default on its debt. However, when the three-month Canadian Treasury bill is combined with a forward contract, another credit risk appears. In particular, if investors learn in three months that the coun- terparty to the forward contract will not honor the forward contract, investors may or may not be concerned. If the Canadian dollar appreciates over three months, then investors probably would welcome the fact that they were not locked in at the forward rate. However, if the Canadian dollar depre- ciates over the three months, then investors could well suffer a dramatic loss.
The counterparty risk of a forward contract is not a sovereign credit risk.
Forward contract risks generally are viewed as a counterparty credit risk. We can accept this view since banks are the most active players in the currency forwards marketplace. Though perhaps obvious, an intermediate step between an unhedged position and a fully hedged strategy is a partially hedged investment. With a partial hedge, investors are exposed to at least some upside potential with a trade yet with some downside protection as well.
interrelationship between forwards and futures. Parenthetically, there is a scenario where the marginal differences between a forward and future actu- ally could allow for a material preference to be expressed for one over the other. Namely, since futures necessitate a daily marking-to-market with a margin account set aside expressly for this purpose, investors who short bond futures contracts (or contracts that enjoy a strong correlation with interest rates) versus bond forward contracts can benefit in an environment of rising interest rates. In particular, as rates rise, the short futures posi- tion will receive margin since the future’s price is decreasing, and this greater margin can be reinvested at the higher levels of interest. And if rates fall, the short futures position will have to post margin, but this financing can be done at a lower relative cost due to lower levels of interest. Thus, investors who go long bond futures contracts versus forward contracts are similarly at a disadvantage.
There can be any number of incentives for doing a trade with a partic- ular preference for doing it with a forward or future. Some reasons might include:
䡲 Investors’ desire to leapfrog over what may be perceived to be a near- term period of market choppiness into a predetermined forward trade date and price
䡲 Investors’ belief that current market prices generally look attractive now, but they may have no immediate cash on hand (or perhaps may expect cash to be on hand soon) to commit right away to a purchase
䡲 Investors’ hope to gain a few extra basis points of total return by actively exploiting opportunities presented by the repo market via the lending of particular securities. This is discussed further in Chapter 4.
Table 2.5 presents forward formulas for each of the big three.
Options
We now move to the third leg of the cash flow triangle, options.
Continuing with the idea that each leg of the triangle builds on the other, the options leg builds on the forward market (which, in turn, was built on
the spot market). Therefore, of the five variables generally used to price an option, we already know three: spot (S), a financing rate (R), and time (T).
The two additional variables needed are strike price and volatility. Strike price is the reference price of profitability for an option, and an option is said to have intrinsic value when the difference between a strike price and an actual market price is a favorable one. Volatility is a statistical measure of a stock price’s dispersion.
Let us begin our explanation with an option that has just expired. If our option has expired, several of the five variables cited simply fall away. For example, time is no longer a relevant variable. Moreover, since there is no time, there is nothing to be financed over time, so the finance rate variable is also zero. And finally, there is no volatility to be concerned about because, again, the game is over. Accordingly, the value of the option is now:
Call optionvalue is equal to SK where
Sthe spot value of the underlying security Kthe option’s strike price
The call option value increases as Sbecomes larger relative to K. Thus, investors purchase call options when they believe the value of the underly- ing spot will increase. Accordingly, if the value of Shappens to be 102 at expiration with the strike price set at 100 at the time the option was pur- chased, then the call’s value is 102 minus 100 2.
Aput optionvalue is equal to K S. Notice the reversal of positions ofSandKrelative to a call option’s value. The put option value increases as S becomes smaller relative to K. Thus, investors purchase put options when they believe that the value of the underlying spot will decrease.
Now let us look at a scenario for a call’s value prior to expiration. In this instance, all five variables cited come into play.
The first thing to do is make a substitution. Namely, we need to replace theSin the equation with an F.T, time, now has value. And since Tis rel- evant, so too is the cost to finance Sover a period of time; this is reflected
TABLE 2.5 Forward Formulas for Each of the Big Three
Product Formula
No Cash Flows Cash Flows
Bonds S(1RT) S(1T(RYc))
Equities S(1RT) S(1T(RYc))
Currencies S (1 T(RhRo))
byRand is embedded along with TwithinF. And finally, a value for volatil- ity, V, is also a vital consideration now. Thus, we might now write an equa- tion for a call’s value to be
Just to be absolutely clear on this point, when we write Vas in the last equation, this variable is to be interpreted as the valueof volatility in price terms (not as a volatility measure expressed as an annualized standard devi- ation).12 Since there is a number of option pricing formulas in existence today, we need not define a price value of volatility in terms of each and every one of those option valuation calculations. Quite simply, for our pur- poses, it is sufficient to note that the variables required to calculate a price value for volatility include R,T, and , where is the annualized standard deviation of S.13
On an intuitive level, it would be logical to accept that the price value of volatility is zero when T0, because Tbeing zero means that the option’s life has come to an end; variability in price (via ) has no meaning. However, ifRis zero, it is still possible for volatility to have a price value. The fact that there may be no value to borrowing or lending money does not automati- cally translate into a spot having no volatility (unless, of course, the under- lying spot happens to be Ritself, where Rmay be the rate on a Treasury bill).14 Accordingly, a key difference between a forward and an option is the role of R;Rbeing zero immediately transforms a forward into spot, but an option remains an option. Rather, the Achilles’ heel of an option is ;being zero immediately transforms an option into a forward. With 0 there is no volatility, hence there is no meaning to a price value of volatility.
Finally, saying that one cash flow type becomes another cash flow type under various scenarios (i.e., T0, or 0), does not mean that they some- how magically transform instantaneously into a new product; it simply high- lights how their new price behavior ought to be expected to reflect the cash flow profile of the product that shares the same inputs.
Call value F K V.
12It is common in some over-the-counter options markets actually to quote options by their price as expressed in terms of volatility, for example, quoting a given currency option with a standard three-month maturity at 12 percent.
13The appendix of this chapter provides a full explanation of volatility definitions, including volatility’s calculation as an annualized standard deviation of S.
14Perhaps the most recent real-world example of Rbeing close to (or even below) zero would be Japan, where short-term rates traded to just under zero percent in January 2003.
Rewriting the above equation for a call option knowing that F S SRT, we have
Call value = S+SRTK+V.
If only to help us reinforce the notions discussed thus far as they relate to the interrelationships of the triangle, let us consider a couple of what-if?
scenarios. For example, what if volatility for whatever reason were to go to zero? In this instance, the last equation shrinks to
Call value = S+SRTK.
And since we know that F S SRT, we can rewrite that equation into an even simpler form as:
Call value = FK.
But sinceKis a fixed value that does not change from the time the option is first purchased, what the above expression really boils down into is a value forF. We are now back to the second leg of the triangle. To put this another way, a key difference between a forward and an option is that prior to expi- ration, the option requires a price value for V.
For our second what-if? scenario, let us assume that in addition to volatility being zero, for whatever reason there is also zero cost to borrow or lend (financing rates are zero). In this instance, call value SSRT KVnow shrinks to
Call value = SK.
This is because with TandRequal to zero, the entire SRTterm drops out, and of course Vdrops out because it is zero as well. With the recogni- tion, once again, that Kis a fixed value and does not do very much except provide us with a reference point relative to S, we now find ourselves back to the first leg of the triangle. Figure 2.18 presents these interrelationships graphically.
As another way to evaluate the progressive differences among spot, for- wards, and options, consider the layering approach shown in Figure 2.19.
The first or bottom layer is spot. If we then add a second layer called cost of carry, the combination of the first and second layers is a forward. And if we add a third layer called volatility (with strike price included, though “on the side,” since it is a constant), the combination of the first, second, and third layers is an option.
As part and parcel of the building-block approach to spot, forwards, and options, unless there is some unique consideration to be made, the pre- sumption is that with an efficient marketplace, investors presumably would be indifferent across these three structures relative to a particular underly- ing security. In the context of spot versus forwards and futures, the decision to invest in forwards and futures rather than cash would perhaps be influ- enced by four things:
Spot
Options
Forwards F = S(1 + RT) S
Call value = F–K + V = S(1 + RT) – K + V
WhenT is zero (as at the expiration of an option), the call option value becomes
S – K. This happens because F becomes S (see formula for F) and V drops away;
volatility has no value for a security that has ceased to trade (as at expiration). In sum, sinceK is a constant, S is the last remaining variable. If just V is zero, then the call option value prior to expiration is F – K.
Therefore, F is differentiated from an option by K and V, and S is differentiated from an option by K, V, and RT.
When either R or T is zero (as with a zero cost to financing, or when there is immediate settlement),F = S.
Therefore, F is differentiated fromS by cost of carry (SRT)
Special Note
Some market participants state that the value of an option is really composed of two parts:
an intrinsic value and a time value. Intrinsic value is defined asF K prior to expiration (for a call option) and as S K at expiration; all else is time value, which, by definition, is zero whenT = 0 (as at expiration).
FIGURE 2.18 Key interrelationships among spot, forwards, and options.
V
SRT
S
Volatility
Cost of carry
Forwards
Options
Spot
FIGURE 2.19 Layers of distinguishing characteristics among spot, forwards, and options.
1. The notion that the forward or future is undervalued or overvalued rela- tive to cash; that in the eyes of a particular investor, there is a material dif- ference between the market value of the forward and its actual worth 2. Some kind of investor-specific cash flow or asset consideration where
immediate funds are not desired to be committed; that the deferred exchange of cash for product provided by the forward or future is desirable 3. The view that something related to SRTis not being priced by the mar-
ket in a way that is consistent with the investor’s view of worth; again, a material difference between market value and actual worth
4. Some kind of institutional, regulatory, tax, or other extra-market incen- tive to trade in futures or forwards instead of cash
In the case of investing in an option rather than forwards and futures or cash, this decision would perhaps be influenced by four things:
1. The notion that the option is undervalued or overvalued relative to for- wards or futures or cash; that in the eyes of a particular investor, there is a material difference between the market value of the forward and its actual worth
2. Some kind of investor-specific cash flow or asset consideration where the cash outlay of a strategy is desirable; note the difference between payingSversusSK
3. The view that something related to Vis not being priced by the market in a way that is consistent with the investor’s view of worth; again, a material difference between market value and actual worth
4. Some kind of institutional, regulatory, tax, or other extra-market incen- tive to trade in options instead of futures or forwards or cash
It is hoped that these illustrations have helped to reinforce the idea of inter- locking relationships around the cash flow triangle. Often people believe that these different cash flow types somehow trade within their own unique orbits and have lives unto themselves. This does not have to be the case at all.
As the concept of volatility is very important for option valuation, the appendix to this chapter is devoted to the various ways volatility is calcu- lated. In fact, a principal driver of why various option valuation models exist is the objective of wanting to capture the dynamics of volatility in the best possible way. Differences among the various options models that exist today are found in existing texts on the subject.15
15See, for example, John C. Hull, Options, Futures, and Other Derivatives(Saddle- River. NJ: Prentice Hall, 1989).
Because bonds are priced both in terms of dollar price and yield, an overview of various yield types is appropriate. Just as there are nominal yield spreads and forward yield spreads, there are also option-adjusted spreads (OASs).
Refer again to the cash flow triangle and the notion of forwards building on spots, and options, in turn, building on forwards. Recall that a spot spread is defined as being the difference (in basis points) between two spot yield lev- els (and being equivalent to a nominal yield spread when the spot curve is a par bond curve) and that a forward spread is the difference (in bps) between two forward yield levels derived from the entire relevant portion of respective spot curves (and where the forward curve is equivalent to a spot curve when the spot curve is flat). A nominal spread typically reflects a measure of one security’s richness or cheapness relative to another. Thus, it can be of interest to investors as a way of comparing one security against another. Similarly, a forward spread also can be used by investors to compare two securities, par- ticularly when it would be of interest to incorporate the information contained within a more complete yield curve (as a forward yield in fact does).
An OAS can be a helpful valuation tool for investors when a security has optionlike features. Chapter 4 examines such security types in detail.
Here the objective is to introduce an OAS and show how it can be of assis- tance as a valuation tool for fixed income investors.
If a bond has an option embedded within it, a single security has charac- teristics of a spot, a forward, and an option all at the same time. We would expect to pay par for a coupon-bearing bond with an option embedded within it if it is purchased at time of issue; this “pay-in-full at trade date” feature is most certainly characteristic of spot. Yet the forward element of the bond is a “deferred” feature that is characteristic of options. In short, an OAS is intended to incorporate an explicit consideration of the option component within a bond (if it has such a component) and to express this as a yield spread value. The spread is expressed in basis points, as with all types of yield spreads.
Recall the formula for calculating a call option’s value for a bond, equity, or currency.
OcFKV.
Options Bonds
Table 2.6 compares and contrasts how the formula would be modified for calculating an OAS as opposed to a call option on a bond.
Consistent with earlier discussions on the interrelationships among spot, for- wards, and options, if the value of volatility is zero (or if the par bond curve is flat), then an OAS is the same thing as a forward spread. This is the case because a zero volatility value is tantamount to asserting that just one forward curve is of relevance: today’s forward curve. Readers who are familiar with the binom- inal option model’s “tree” can think of the tree collapsing into a single branch when the volatility value is zero; the single branch represents the single prevailing path from today’s spot value to some later forward value. Sometimes investors deliberately calculate a zero volatility spread(orZV spread) to see where a given security sits in relation to its nominal spread, whether the particular security is embedded with any optionality or not. Simply put, a ZV spread is an OAS cal- culated with the assumption of volatility being equal to zero. Similarly, ifT 0 (i.e., there is immediate settlement), then volatility has no purpose, and the OAS and forward spread are both equal to the nominal spread.
An OAS can be calculated for a Treasury bond where the Treasury bond is also the benchmark security. For Treasuries with no optionality, calculating an OAS is the same as calculating a ZV spread. For Treasuries with option- ality, a true OAS is generated. To calculate an OAS for a non-Treasury secu- rity (i.e., a security that is not regarded as risk free in a credit or liquidity context), we have a choice; we can use a Treasury par bond curve as our ref- erence curve for constructing a forward curve, or we can use a par bond curve of the non-Treasury security of interest. Simply put, if we use a Treasury par bond curve, the resulting OAS will embody measures of both the risk-free and non–risk-free components of the future shape in the forward curve as well as a measure of the embedded option’s value. Again, the term “risk free” refers to considerations of credit risk and liquidity risk.
TABLE 2.6
UsingOcFKVto Calculate a Call Option on a Bond versus an OAS (assuming the embedded option is a call option)
For a Bond For an OAS
•Ocis expressed as a dollar value OAS is expressed in basis points.
(or some other currency value).
•Fis a forward price value. Fis a forward yield value (which, via bootstrapping, embodies a forward curve).
•Kis a spot price reference value. Kis expressed as a spot yield value (typically equal to the coupon of the bond).
•Vis the volatility price value. Same.