Financial engineering principles phần 8 pot

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 ϪO c translates into a smaller value for P c . A smaller value for P c presumably means a higher yield for P c, 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, and to let the issue simply continue to trade when it is at prices below par), anything that has the effect of pulling the callable away from being in-the- money (as with a larger value of ϪO c ) also has the effect of reducing the call risk. Thus, OAS narrows as volatility rises. 200 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT Quantifying risk Credit Borrowing from the drift and default matrices first presented in Chapter 3, a credit cone (showing hypothetical boundaries of upper and lower levels of potential credit exposures) might be created that would look something like that shown in Figure 5.14. This type of presentation provides a very high-level overview of credit dynamics and may not be as meaningful as a more detailed analysis. For example, we may be interested to know if there are different forward-looking total return characteristics of a single-B company that: 05_200306_CH05/Beaumont 8/15/03 12:52 PM Page 200 Ⅲ Just started business the year before, and as a single-B company, or Ⅲ Has been in business many years as a double-B company and was just recently downgraded to a single-B (a fallen angel), or Ⅲ Has been in business many years as a single-C company and was just recently upgraded to a single-B. In sum, not all single-B companies arrive at single-B by virtue of hav- ing taken identical paths, and for this reason alone it should not be surprising that their actual market performance typically is differentiated. For example, although we might think that a single-B fallen angel is more likely either to be upgraded after a period of time or at least to stay at its new lower notch for some time (especially as company management redoubles efforts to get things back on a good track), in fact the odds are less favorable for a single-B fallen angel to improve a year after a downgrade than a single-B company that was upgraded to a single-B status. However, the story often is different for time horizons beyond one year. For periods beyond one year, many single-B fallen angels successfully reposition them- selves to become higher-rated companies. Again, the statistics available from the rating agencies makes this type of analysis possible. There is another dimension to using credit-related statistical experience. Just as not all single-B companies are created in the same way, neither are all single-B products. A single-A rated company may issue debt that is rated double-B because it is a subordinated structure, just as a single-B rated com- pany may issue debt that is rated double-B because it is a senior structure. Generally speaking, for a particular credit rating, senior structures of lower- Risk Management 201 25 20 15 10 5 0 Single C Single B Initial credit ratings Likelihood of default at end of one year (%) FIGURE 5.14 Credit cones for a generic single-B and single-C security. 05_200306_CH05/Beaumont 8/15/03 12:52 PM Page 201 rated companies do not fare as well as junior structures of higher-rated com- panies. In this context, “structure” refers to the priority of cash flows that are involved. The pattern of cash flows may be identical for both a senior and junior bond (with semiannual coupons and a 10-year maturity), but with very different probabilities assigned to the likelihood of actually receiving the cash flows. The lower likelihood associated with the junior structure means that its coupon and yield should be higher relative to a senior struc- ture. Exactly how much higher will largely depend on investors’ expectations of the additional cash flow risk that is being absorbed. Rating agency sta- tistics can provide a historical or backward-looking perspective of credit risk dynamics. Credit derivatives provide a more forward-looking picture of credit risk expectations. As explained in Chapter 3, a credit derivative is simply a forward, future, or option that trades to an underlying spot credit instrument or variable. While the pricing of the credit spread option certainly takes into consider- ation any historical data of relevance, it also should incorporate reasonable future expectations of the company’s credit outlook. As such, the implied forward credit outlook can be mathematically backed-out (solved for with relevant equations) of this particular type of credit derivative. For example, just as an implied volatility can be derived using a standard options valua- tion formula, an implied credit volatility can be derived in the same way when a credit put or call is referenced and compared with a credit-free instru- ment (as with a comparable Treasury option). Once obtained, this implied credit outlook could be evaluated against personal sentiments or credit agency statistics. In 1973 Black and Scholes published a famous article (which subse- quently was built on by Merton and others) on how to price options, called “The Pricing of Options and Corporate Liabilities.” 6 The reference to “lia- bilities” was to support the notion that a firm’s equity value could be viewed as a call written on the assets of the firm, with the strike price (the point of default) equal to the debt outstanding at expiration. Since a firm’s default risk typically increases as the value of its assets approach the book value (actual value in the marketplace) of the liabilities, there are three elements that go into determining an overall default probability. 1. The market value of the firm’s assets 2. The assets’ volatility or uncertainty of value 3. The capital structure of the firm as regards the nature of its various con- tractual obligations 202 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 6 F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, 81 (May–June 1973): 637–659. 05_200306_CH05/Beaumont 8/15/03 12:52 PM Page 202 Figure 5.15 illustrates these concepts. The dominant profile resembles that of a long call option. Many variations of this methodology are used today, and other method- ologies will be introduced. In many respects the understanding and quan- tification of credit risk remains very much in its early stages of development. Credit risk is quantified every day in the credit premiums that investors assign to the securities they buy and sell. As these security types expand beyond traditional spot and forward cash flows and increasingly make their way into options and various hybrids, the price discovery process for credit generally will improve in clarity and usefulness. Yet the marketplace should most certainly not be the sole or final arbiter for quantifying credit risk. Aside from more obvious considerations pertaining to the market’s own imper- fections (occasions of unbalanced supply and demand, imperfect liquidity, the ever-changing nature of market benchmarks, and the omnipresent pos- sibility of asymmetrical information), the market provides a beneficial though incomplete perspective of real and perceived risk and reward. In sum, credit risk is most certainly a fluid risk and is clearly a consid- eration that will be unique in definition and relevance to the investor con- sidering it. Its relevance is one of time and place, and as such it is incumbent on investors to weigh very carefully the role of credit risk within their over- all approach to investing. Risk Management 203 FIGURE 5.15 Equity as a call option on asset value. Source: “Credit Ratings and Complementary Sources of Credit Quality Information,” Arturo Estrella et al., Basel Committee on Banking Supervision, Bank for International Settlements, Basel, August 2000. 05_200306_CH05/Beaumont 8/15/03 12:52 PM Page 203 [Image not available in this electronic edition.] One explanation might be that heightened volatility emerged among the fewer remaining so-called global reserve currencies (namely the U.S. dollar, the yen, and the euro), and that heightened volatility emerged among interest rates between euro-member countries and the rest of the world. In fact, both of these things occurred following the euro’s launch. As a second example, consider the statistical methods between equities and bonds presented earlier in this chapter, namely, in the discussion of how the concepts of duration and beta can be linked with one another. Hypothetically speaking, once a basket of particular stocks is identified that behaves much like fixed income securities, a valid question becomes which bundle would an investor prefer to own: a basket of synthetic fixed income securities created with stocks or a basket of fixed income securities? The question is deceptively simple. When investors purchase any fixed income security, are they purchasing it because it is a fixed income security or because it embodies the desired characteristics of a fixed income security (i.e., pays periodic coupons, holds capital value etc.)? If it is because they want a fixed income security, then there is nothing more to discuss. Investors will buy the bundle of fixed income securities. However, if they desire the char- acteristics of a fixed income security, there is a great deal more to talk about. Namely, if it is possible to generate fixed income returns with non–fixed income products, why not do so? And if it is possible to outperform tradi- tional fixed income products with non—fixed income securities and for com- parable levels of risk, why ever buy another note or bond? Again, if investors are constrained to hold only fixed income products, then the choice is clear; they hold only the true fixed income portfolio. If they want only to create a fixed income exposure to the marketplace and are indifferent as to how this is achieved, then there are choices to make. How can investors choose between a true and synthetic fixed income port- folio? Perhaps on the basis of historical risk/return profiles. If the synthetic fixed income portfolio can outperform the true fixed income portfolio on a consistent basis at the same or a lower level of risk, then investors might seriously want to consider owning the synthetic port- folio. A compromise would perhaps be to own a mix of the true and syn- thetic portfolios. For our third example, consider the TED spread, or Treasury versus Eurodollar spread. A common way of trading the TED spread is with futures contracts. For example, to buy the TED spread, investors buy three-month Treasury bill futures and sell three-month Eurodollar futures. They would purchase the TED spread if they believed that perceptions of market risk or volatility would increase. In short, buying the TED spread is a bet that the spread will widen. If perceptions of increased market risk become manifest in moves out of risky assets (namely, Eurodollar-denominated securities that are dominated by bank issues) and into safe assets (namely, U.S. Treasury Risk Management 205 05_200306_CH05/Beaumont 8/15/03 12:52 PM Page 205 securities), Treasury bill yields would be expected to edge lower relative to Eurodollar yields and the TED spread would widen. Examples of events that might contribute to perceptions of market uncertainty would include a weak stock market, banking sector weakness as reflected in savings and loan or bank failures, and a national or international calamity. Accordingly, one way for investors to create a strategy that benefits from an expectation that equity market volatility will increase or decrease by more than generally expected is via a purchase or sale of a fixed income spread trade. Investors could view this as a viable alternative to delta-hedging an equity option to isolate the value of volatility (V) within the option. Finally, here is an example of an interrelationship between products and credit risk. Studies have been done to demonstrate how S&P 500 futures con- tracts can be effective as a hedge against widening credit spreads in bonds. That is, it has been shown that over medium- to longer-run periods of time, bond credit spreads tend to narrow when the S&P 500 is rallying, and vice versa. Further, bond credit spreads tend to narrow when yield levels are declining. In sum, and in general, when the equity market is in a rallying mode, so too is the bond market. This is not altogether surprising since the respective equity and bonds of a given company generally would be expected to trade in line with one another; stronger when the company is doing well and weaker when the company is not doing as well. CASH FLOW INTERRELATIONSHIPS Chapter 2 described the three primary cash flows: spot, forwards and futures, and options. These three primary cash flows are interrelated by shared variables, and one or two rather simple assumptions may be all that’s required to change one cash flow type into another. Let us now use the tri- angle approach to highlight these interrelationships by cash flows and their respective payoff profiles. A payoff profile is a simple illustration of how the return of a particu- lar cash flow type increases or decreases as its prices rises or falls. Consider Figure 5.16, an illustration for spot. As shown, when the price of spot rises above its purchase price, a pos- itive return is enjoyed. When the price of spot falls below its purchase price, there is a loss. Figure 5.17 shows the payoff profile for a forward or future. As read- ers will notice, the profile looks very much like the profile for spot. It should. Since cost-of-carry is what separates spot from forwards and futures, the distance between the spot profile (replicated from Figure 5.16 and shown as a dashed line) and the forward/future profile is SRT (for a non — cash-flow paying security). As time passes and T approaches a value 206 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 05_200306_CH05/Beaumont 8/15/03 12:52 PM Page 206 of zero, the forward/future profile gradually converges toward the spot pro- file and actually becomes the spot profile. As drawn it is assumed that R remains constant. However, if R should grow larger, the forward/future pro- file may edge slightly to the right, and vice versa if R should grow smaller (at least up until the forward/future expires and completely converges to spot). Risk Management 207 0 O Price Return Positive returns Negative returns Price at time of purchase FIGURE 5.16 Payoff profile. 0O O Price Return Positive returns Negative returns Profile for forward/future Forward price at time of initial trade Spot price at time of initial trade Profile for spot Equal to SRT. Convergence between forward/future profile and spot profile will occur as time passes. FIGURE 5.17 Payoff profile for a forward or future. 05_200306_CH05/Beaumont 8/15/03 12:52 PM Page 207 Figure 5.18 shows the payoff profile for a call option. The earlier pro- file for spot is shown in a light dashed line and the same previous profile for a forward/future is shown in a dark dashed line. Observe how the label of “Price” on the x-axis has been changed to “Difference between forward price and strike price” (or F Ϫ K). An increasingly positive difference between F and K represents a larger in-the-money value for the option and the return grows larger. Conversely, if the difference between F and K remains constant or falls below zero (meaning that the price of the under- lying security has fallen), then there is a negative return that at worst is lim- ited to the price paid for the option. As drawn, it is assumed that R and V remain constant. However, if R or V should grow larger, the option profile may edge slightly to the right and vice versa if R or V should grow smaller (at least up until the option expires and completely converges to spot). A put payoff profile is shown in Figure 5.19. The lines are consistent with the particular cash flows identified above. With the benefit of these payoff profiles, let us now consider how com- bining cash flows can create new cash flow profiles. For example, let’s cre- ate a forward agreement payoff profile using options. As shown in Figure 5.20, when we combine a short at-the-money put and a long at-the-money call option, we generate the same return profile as a forward or future. Parenthetically, a putable bond has a payoff profile of a long call option, as it is a combination of being long a bullet (noncallable) bond and 208 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 0 O Return Positive returns Negative returns Profile for forward/future Inflection point where F = K Profile for spot Difference between forward price and strike price Distance is equal to SRT Distance is equal to value of volatility Price of option at time of initial trade FIGURE 5.18 Call payoff profile. 05_200306_CH05/Beaumont 8/15/03 12:52 PM Page 208 a long put option. A callable bond has a payoff profile of a short put option as it is a combination of being long a bullet bond and a short call option. Since a putable and a callable are both ways for an investor to benefit from steady or rising interest rates, it is unusual for investors to have both puta- bles and callables in a single portfolio. Accordingly, it is important to rec- ognize that certain pairings of callables and putables can result in a new cash flow profile that is comparable to a long forward/future. Let us now look at a combination of a long spot position and a short for- ward/future position. This cash flow combination ought to sound familiar because it was first presented in Chapter 4 as a basis trade (see Figure 5.21). Next let us consider how an active delta-hedging strategy with cash and forwards and/or futures can be used to replicate an option’s payoff profile. Specifically, let us consider creating a synthetic option. Risk Management 209 0 K – F Return Positive returns Negative returns FIGURE 5.19 Put payoff profile. + = Long call option Short put option Long forward/future FIGURE 5.20 Combining cash flows. 05_200306_CH05/Beaumont 8/15/03 12:52 PM Page 209 . risk adjustment is performed by reduc- 2 18 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 05_200306_CH05/Beaumont 8/ 15/03 12:52 PM Page 2 18 ing the risky return at the project or. and 2 08 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 0 O Return Positive returns Negative returns Profile for forward/future Inflection point where F = K Profile for spot Difference. value 206 FINANCIAL ENGINEERING, RISK MANAGEMENT, AND MARKET ENVIRONMENT 05_200306_CH05/Beaumont 8/ 15/03 12:52 PM Page 206 of zero, the forward/future profile gradually converges toward the spot pro- file