of time.The bonds are almost always callable (i.e., the issuer has the right to buy them back at certaintimes at a predetermined prices). The holder always has the right to convert the bond once it hasbeen called. The call feature is therefore usually a way of forcing conversion earlier than the holderwould otherwise choose. Sometimes the holder’s call option is conditional on the price of thecompany’s stock being above a certain level.
Credit risk plays an important role in the valuation of convertibles. If we ignore credit risk, wewill get poor prices because the coupons and principal payments on the bond, assuming noconversion, will be overvalued. This is because they will be treated as riskless and discounted at therisk-free rate.
Ingersoll provides a way of valuing convertibles using a model similar to Mertan’s (1974) modeldiscussed in Section 26.5.!! He assumes geometric Brownian motion for the issuer’s total assetsand models the company’s equity, its convertible debt, and its other debt as claims contingent onthe value of the assets. Credit risk is taken into account because the debt holders get repaid in fullonly if the value of the assets exceeds the amount owing to them.
A simpler model that is more widely used in practice involves modeling the issuer’s stock price.A tree for the stock price is constructed in the usual way. The life of the tree is set equal to thelife of the convertible bond. The value of the convertible at the final nodes of the tree iscalculated based on any conversion options that the holder has at that time. We then roll backthrough the tree. At nodes where the terms of the instrument allow conversion, we test whetherconversion is optimal. We also test whether the position of the issuer can be improved by callingthe bonds. If so, we assume that the bonds are called and retest whether conversion is optimal.This is equivalent to setting the value at a node equal to
max[(min(Q,, Q2), Q3]
where Q, is the value given by the rollback (assuming that the bond is neither converted nor calledat the node), Q> is the call price, and Q; is the value if conversion takes place.
One complication is the choice of the discount rate used in conjunction with the tree. Supposefirst that the convertible is certain to remain a bond. It is then appropriate to use a “risky”discount rate that reflects the credit risk of the issuer. This leads to the value of the convertiblebeing correctly calculated as the market value of a regular nonconvertible bond. Suppose next thatthe bond is certain to be converted. It is then appropriate to use the risk-free interest rate as the
" See J. E. Ingersoll, “A Contingent Claims Valuation of Convertible Securities,” Journal of’ Financial Economics,4 (May 1977}, 289-322.Credit Derivatives 653
 
78.42156.840.00156.84
58.09116.180.00116.18
43.040.00100.00100.00
96.3231.880.00
J 190.00100.00
 
Figure 27.5 Tree for valuing convertible
discount rate. The value of the bond is then correctly calculated as the value of the equityunderlying the bond.
In practice, we are usually uncertain as to whether the bond will be converted. We thereforearrange the calculations so that the value of the bond at each node is divided into twocomponents: a component that arises from situations where the bond ultimately ends up asequity and a component that arises from situations where the bond ends up a debt. We apply arisk-free discount rate to the first component and a “risky” discount rate for the secondcomponent.!?
Example 27.6 As a simple example of the procedure for valuing convertibles, consider a nine-month zero-coupon bond issued by company XYZ with a face value of $100. Suppose that it can beexchanged for two shares of company XYZ’s stock at any time during the nine months. Assume alsothat it is callable for $115 at any time. The initial stock price is $50, its volatility is 30% per annum,and there are no dividends. The risk-free yield curve is flat at 10% per annum. The risky yield curvecorresponding to bonds issued by company XYZ is flat at 15% per annum. Figure 27.5 shows thestock price tree that can be used to value this convertible. The top number at each node is the stockPrice; the second number is the component of the bond’s value arising from situations where itultimately becomes equity: the third number is the component of the bond’s value arising from
? This approach is formalized in K. Tsiveriotis and C. Fernandes, “Valuing Convertible Bonds with Credit Risk,”Journal of Fixed Income, 8, no. 2 (September 1998), 95-102.654 CHAPTER 27
situations where it ultimately becomes debt; the fourth number is the total value of the bond. Thetree parameters are u = 1.1618, d = 0.8607, a = 1.0253, and p = 0.5467. At the final nodes, theconvertible is worth max(100, 2S,). At node G, the numbers show that the stock price is 78.42, thevalue of the bond is 156.84 and all of this value arises from situations where the bond ends up asequity. At node I, the stock price is 43.04, the value of the convertible is 100, and all of this arisesfrom situations where it ends up as debt.
As we roll back through the tree, we test whether conversion is optimal and whether the bondshould be called. At node D, rollback gives the value of the equity component of the convertible as
(0.5467 x 156.84 + 0.4533 x 116.18)e°!*05 — 134.98
The value of the debt component is zero. Calling or converting the bond does not change its valuesince it is already essentially equity. At node F, the equity component of the convertible is zero andthe debt component is worth 100e~°!5*9*5 or 96.32. Node E is more interesting. The equitycomponent of the convertible is worth
(0.5467 x 116.18 + 0.4533 x O)e79!*95 — 61.95The debt component is worth(0.5467 x 0 +.0.4533 x 100)e~°'5*5 — 43.66
The total value of the bond is 61.95 + 43.66 = 105.61. Clearly, the bond should be neither convertednor called.At node B, the equity component of the convertible is worth
(0.5467 x 134.98 + 0.4533 x 61.95)e0!*5 == 99.36
and the debt component is worth
(0.54 66je79 15025 _ 19.96
The total value of the bond is 99.36 + 19.06 = 118.42. It is optimal for the issuer to call the bond atnode B because this will cause immediate conversion and lead to the value of the bond being reducedto 2 x 58.09 = 116.18 at the node. The numbers at node B recognize that this will happen. The bondis worth 116.18, all of it equity, at node B. Continuing in this way, the value of the convertible at theinitial node, A, is 104.95. If the bond had no conversion or call option, its value would be
1002791597 =. 89,36.The value of the conversion option (net of the issuer’s call option) is therefore
104.95 ~ 89.36 = 15.59
When dividends on the equity or interest on the debt is paid, they must be taken into account. Ateach node, we first assume that the bond is debt. We include in the debt component of the value ofthe bond the present value of any interest payable on the bond in the next time step. Then, whentesting whether the bond should be converted, we take into account the present value of anydividends that will be received during the next time step. The calculations can be made more preciseby allowing the risk-free and “‘risky” rates to be time dependent and equal to the relevant forwardrates. The tree is then built as indicated in Section 18.4. Cash flows between time ¢ and f+ dt arediscounted at the appropriate forward rate applicable to the period between ¢ and t + dt.Credit Derivatives 655
SUMMARY
 
 
Credit derivatives enable banks and other financial institutions to actively manage their creditrisks. They can be used to transfer credit risk from one company to another and to diversify creditrisk by swapping one type of exposure for another.
The most conunon credit derivative ts a credit default swap. This is a contract where one companybuys insurance against another company defaulting on its obligations. The payoff is usually thedifference between the par value of a bond issued by the second company and its value immediatelyafter a default. Credit default swaps can be analyzed by calculating the present value of the expectedcost of the insurance in a risk-neutral world and the present value of the expected payoff.
A total return swap is an instrument where the total return on a portfolio of credit-sensitiveassets is exchanged for LIBOR plus a spread. A total return swap can be used to exchange acorporate bond for a bond providing a LIBOR-based return. It then eliminates market risk as wellas credit risk. Total return swaps are often used as financing vehicles. A company wanting topurchase a portfolio of bonds will approach a financial institution, who will buy the bonds on itsbehalf. The financial institution will then enter into a credit default swap where it pays the returnon the bonds to the company and receives LIBOR. The advantage of this type of arrangement isthat the financial institution reduces its exposure to defaults by the company.
A credit spread option is an option on a credit spread. Under one type of structure, the payoff iscalculated by comparing the credit spread in the market at a future time to a strike credit spread.Under another, it is calculated by comparing the price of a floating-rate bond to a strike price.
In collateralized debt obligations a number of different securities are created from a portfolio ofcorporate bonds or commercial loans. There are rules for determining how credit losses areallocated to the securities. The result of the rules is that securities with both very high and verylow credit ratings are created from the portfolio.
All over-the-counter derivative transactions involve some credit risk. The credit derivatives markethas made market participants more conscious of the need to adjust their pricing to allow forcounterparty credit risk. An option is an example of a derivative that is always an asset to one partyand a liability to the other. The party for which it is an asset can adjust for credit risk by judiciouslyincreasing the discount rate. Other derivative transactions (e.g., swaps) can be either assets orliabilities depending on movements in market variables. Usually a financial institution has severalderivative transactions outstanding with a counterparty. Estimating the expected loss from defaultsby the counterparty involves estimating the expected claim that will be made conditional on defaultat various future times.
Convertible bonds are bonds that can be converted to the issuer’s equity according to pre-specified terms. Credit risk has to be considered in the valuation of a convertible bond. This isbecause, if the bond is not converted, the promised payments on the bond are subject to credit risk.One procedure for valuing convertible bonds is to calculate the component of the value attrib-utable to the possibility of the bond ending up as equity separately from the component of thevalue attributable to the possibility of the bond ending up as debt.
SUGGESTIONS FOR FURTHER READING
 
 
Cooper, I., and A. Mello, “The Default Risk of Swaps,” Journal of Finance, 46 (1991), 597-620.Das, S., Credit Derivatives: Trading & Management of Credit & Default Risk, Wiley, Singapore, 1998.656 CHAPTER 27
Duffie, D., and M. Huang, “Swap Rates and Credit Quality,” Journal of Finance, 51, no. 3 (July 1996),921-49.
Goldman Sachs, Valuing Convertible Bonds as Derivatives, Quantitative Strategies Research Notes,Goldman Sachs, November 1994.
Hull, J. C, ‘Assessing Credit Risk in a Financial Institution’s Off-Balance-Sheet Commitments,”Journal of Financial and Quantitative Analysis, 24 (1989), 489-501.
Hull, J. C, and A. White, ‘The Impact of Default Risk on the Prices of Options and Other DerivativeSccurities,” Journal of Banking and Finance, 19, no. 2 (May 1995), 299-322.
Hull, J. C, and A. White, ‘“The Price of Default,” RISK, September 1992, pp. 101-3.
Hull, J. C., and A. White, “Valuing Credit Default Swaps I: No Counterparty Default Risk,” Journal ofDerivatives, 8, no. 1 (Fall 2000), 29-40.
Huil, J. C., and A. White, “Valuing Credit Default Swaps II: Modeling Default Correlations,” Journalof Derivatives, 8, no. 3 (Spring 2001), 12-22.
Jarrow, R. A., and 8. M. Turnbull, “Pricing Options on Derivative Securities Subject to Credit Risk,”Journal of Finance, 50 (1995), 53-85.
Johnson, H., and R. Stulz, “The Pricing of Options Under Default Risk,” Journal of Finance, 42 (1987),267-80.
Tavakoli, J. M., Credit Derivatives: A Guide to Instruments and Applications, Wiley, New York, 1998.
Tsiveriotis, K., and C. Fernandes, ‘Valuing Convertible Bonds with Credit Risk,” Journal of FixedIncome, 8, no. 2 (September 1998), 95-102.
~ om a mr mes eee
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
 
27.1. A credit default swap requires a premium of 60 basis points per year paid semiannually. Theprincipal is $300 million and the credit default swap is settled in cash. A default occurs after fouryears and two months, and the calculation agent estimates that the price of the reference bond is40% of its face value shortly after the default. List the cash flows and their timing for the sellerof the credit default swap.
27.2. Suppose that the risk-free zero curve is flat at 7% per annum with continuous compounding andthat defaults can occur at times | ycar, 2 years, and 3 years in a three-year credit default swap.Suppose that the recovery rate is 30% and the default probabilities are as in the final column ofTable 26.5. What is the credit default swap spread? Assume that payments are made semi-annually and that the accrued interest on the reference bond is always zero at the time of adefault.
27.3. Suppose that the five-year risk-free yield is 4% and the five-year yield on bonds issued by acorporation is 5.6% both with semiannual compounding. Estimate the CDS spread when thecorporation is the reference entity, the reference liability is an 8% coupon bond, and theexpected recovery rate is 40%. Assume that semiannual payments on the CDS. Useequation (27.3) *
27.4. What is the formula corresponding to equation (27.1) for valuing a binary credit default swap.
27.5. Prove the statement in the text that a binary swap spread when the expected recovery rate is 50%is typically about 80% higher than when the expected recovery rate is 10%. (Hint: Assume thatCredit Derivatives 657
27.6.
27,7,
27.8.
27.9.
27.10.
27.11.27.12.
27.13.
27.14.27.15.
27.16,
27.17.
27.18,
the payoff from a regular credit default swap is the payoff from a binary credit default swaptimes one minus the recovery rate.)
How does a first-to-default credit default swap work. Explain why the value of a first-to-defaultcredit default swap increases as the correlation between the reference entities in the basketdecreases.
Explain why a total return swap can be an efficient financing tool.Explain the difference between a total return swap and an asset swap.
Suppose that the spread between the yield on a three-year zero-coupon riskless bond and a three-year zero-coupon bond issued by a corporation is 1%. By how much does Black-Scholesoverstate the value of a three-year option sold by the corporation?
“A long forward contract subject to credit risk is a combination of a short position in ano-default put and a long position in a call subject to credit risk.” Explain this statement.
Explain why the credit exposure on a matched pair of forward contracts resembles a straddle.
Explain why the impact of credit risk on a matched pair of interest rate swaps tends to be lessthan that on a matched pair of currency swaps.
“When a bank is negotiating currency swaps, it should try to ensure that it is receiving the lowerinterest rate currency from a company with a low credit risk.” Explain.
Does put-call parity hold when there is default risk? Explain your answer.
The text discusses how the price of a European option can be adjusted for credit risk. Howwould you adjust the price of an American option for credit risk?
Suppose that a financial institution has entered into a swap dependent on the sterling interestrate with counterparty X and an exactly offsetting swap with counterparty Y. Which of thefollowing statements are true and which are false?a. The total present value of the cost of defaults is the sum of the present value of the cost ofdefaults on the contract with X plus the present value of the cost of defaults on the contractwith Y.b. The expected exposure in one year on both contracts is the sum of the expected exposure onthe contract with X and the expected exposure on the contract with Y.c. The 95% upper confidence limit for the exposure in one year on both contracts is the sumof the 95% upper confidence limit for the exposure in one year on the contract with X andthe 95% upper confidence limit for the exposure in one year on the contract with Y.Explain your answers.
A company enters into a one-year forward contract to sell 100 U.S. dollars for 150 Australiandollars. The contract is initially at the money. In other words, the forward exchange rate is 1.50.The one-year dollar risk-free rate of interest is 5% per annum. The one-year dollar rate ofinterest at which the counterparty can borrow is 6% per annum. The exchange rate volatility is12% per annum. Estimate the present value of the cost of defaults on the contract? Assume thatdefaults are recognized only at the end of the life of the contract.
Suppose that in Problem 27.20, the six-month forward rate is also 1.50 and the six-month dollarrisk-free interest rate is 5% per annum. Suppose further that the six-month dollar rate of interestat which the counterparty can borrow is 5.5% per annum. Estimate the present value of the costof defaults assuming that defaults can occur either at the six-month point or at the one-yearpoint? (If a default occurs at the six-month point, the company’s potential loss is the marketvalue of the contract.)658
27.19.
CHAPTER 27
Consider an 18-month zero-coupon bond with a face value of $100 that can be converted intofive shares of the company’s stock at any time during its life. Suppose that the current share priceis $20, no dividends are paid on the stock, the risk-free rate for all maturities is 6% per annumwith continuous compounding, and the share price volatility is 25% per annum. Assume that theyield on nonconvertible bonds issued by the company is 10% per annum for all maturities. Thebond is callable at $110. Use a three-time-step tree to calculate the value of the bond. What is thevalue of the conversion option (net of the issuer’s call option)?
ASSIGNMENT QUESTIONS
 
 
27.20.
27.21.
27.22.
27.23.
Suppose that the risk-free zero curve is flat at 6% per annum with continuous compounding andthat defaults can occur at timés 1 year, 2 years, 3 years, and 4 years in a four-year plain vanillacredit default swap with semiannual payments. Suppose that the recovery rate is 20% and theprobabilities of default at times 1 year, 2 years, 3 years and 4 years are 0.01, 0.015, 0.2 , and 0.25,respectively. The reference obligation is a bond paying a coupon semiannually of 8% per year.Defaults always take place immediately before coupon-payment dates on this bond. What is thecredit default swap spread? What would the credit default spread be if the the instrument were abinary credit default swap?
Suppose that the 1-, 2-, 3-, 4-, and 5-year interest rates are 4%, 4.8%, 5.3%, 5.5%, and 5.6%,respectively. The volatility of all swap rates are 20%. A bank has entered into an annual-pay swapwith a counterparty where the fixed rate exchanged for floating is 5.705%. The notional principalis $100 million. The spreads over the risk-free rates for 1-, 2-, 3-, 4-, and 5-year bonds issued by thecounterparty are 20, 40, 55, 65, and 75 basis points, respectively. Assume that defaults can occuronly on payment dates and ignore any losses arising from payments that are due to be exchangedon the date of default. Use the DerivaGem software to answer the following questions:a. What is the expected loss from defaults on a 5-year swap where the bank receives fixed?b. What is the expected loss from defaults on a 5-year swap where the bank receives floating?Explain why your answer here is higher than your answer to part (a).c. What is the spread required by the bank on a matched pair of interest rate swaps tocompensate for credit risk?d. What difficulties are there in modifying your analysis to allow for losses arising frompayments due to be made on the day of default?
Explain carefully the distinction between real-world and risk-neutral default probabilities. Whichis higher? A bank cnters into a credit derivative where it agrees to pay $100 at the cnd of onc ycarif'a certain company’s credit rating falls from A to BBB or lower during the year. The one-yearrisk-free rate is 5%. Using Table 26.8, estimate a value for the derivative. What assumptions areyou making? Do they tend to overstate or understate the value of the derivative?
A three-year convertible bond with a face value of $100 has been issued by company ABC. It paysa coupon of $5 at the end of each year. It can be converted into ABC’s equity at the end of the firstyear or at the end of the second year. At the end of the first year, it can be exchanged for 3.6 sharesimmediately after the coupon date. At the end of the second year it can be exchanged for 3.5 sharesimmediately after the coupon date. The current stock price is $25 and the stock price volatilityis 25%. No dividends are paid on the stock. The risk-free interest rate is 5% with continuouscompounding. The yield on bonds issued by ABC is 7% with continuous compounding.
a. Use a three-step tree to calculate the value of the bond.
b. How much is the conversion option worth?Credit Derivatives 659
a
. What difference does it make to the value of the bond and the value of the conversion optionif the bond is callable any time within the first two years for $115?
d. Explain how your analysis would’change if there were a dividend payment of $1 on the equity
at the 6-month, 18-month, and 30-month points. Detailed calculations are not required.
27.24. You have the option to enter into a five-year credit default swap at the end of one year for aswap spread of 100 basis points. The principal is $100 million. Payments are made on the swapsemiannually. The forward swap spread for the period between year | and year 6 is 90 basispoints, and the volatility of the forward swap spread is 15%, LIBOR is flat at 5% (continouslycompounded). The risk-neutral probability of a default by the reference entity during the firstyear is 0.015. What is the value of the option? Assume that the option ceases to exist if there is adefault during the first year. 
CHAPTER 2
Me ae om = OG
REAL OPTIONS
 
Up to now we have been almost entirely concerned with the valuation of derivatives dependent onfinancial assets. In this chapter we explore how the ideas we have developed can be extended to valueinvestments in real assets. Real assets include land, buildings, plant, and so on. Often there areembedded options. Valuation is difficult because market prices are not readily available.
We start by explaining the traditional approach to evaluating investments in real assets and theirshortcomings. We then move on to explain how the risk-neutral valuation principle developed inChapters 10, 12, and 21 can be extended to value investments in real assets. After that we describethe options that are embedded in real assets and explain how the risk-neutral valuation approachcan facilitate the valuation of these options.!
28.1 CAPITAL INVESTMENT APPRAISAL
 
The traditional approach to valuing a potential capital investment project is known as the “netpresent value”, or NPV, approach. The NPV of a project is the present value of its expected futureincremental cash inflows and outflows. The discount rate used to calculate the present value is a“risk-adjusted” discount rate, chosen to reflect the risk of the project. As the riskiness of theproject increases, the discount rate also increases.
As an example consider an investment that costs $100 million and will last five years. Theexpected cash inflow in each year is estimated to be $25 million. If the risk-adjusted discount rate is12%, the net present value of the investment is (in millions of dollars)
25 25 25 25 250+ 3+ tasthet he 9.88A negative NPV, such as the one we have just calculated, indicates that the project will reduce thevalue of the company to its shareholders and should not be undertaken. A positive NPV indicatesthat the project should be undertaken because it will increase shareholder wealth.
The risk-adjusted discount rate should be the return required by the company, or the company’sshareholders, on the investment. This can be calculated in a number of ways. One approach oftenrecommended involves the capital asset pricing model. The steps are as follows:
1. Take a sample of companies whose main line of business is the same as that of the projectbeing contemplated.
' Throughout this chapter the risk-neutral world we will consider js the traditional risk-neutral world we defined inChapter 21.
660Real Options 661
2. Calculate the betas of the companies and average them to obtain a proxy beta for theproject.
3. Set the required rate of return equal to the risk-free rate plus the proxy beta times the excessreturn of the market portfolio over the risk-free rate.
One problem with the traditional NPV approach is that many projects contain embeddedoptions. Consider, for example, a company that is considering building a plant to manufacturea new product. Often the company has the option to abandon the project if things do not work outwell. It may also have the option to expand the plant if demand for the output exceedsexpectations. These options usually have quite different risk characteristics from the base projectand require different discount rates.
To understand the problem here, return to the example at the beginning of Chapter 10. Thisinvolved a stock whose current price is $20. In three months’ time the price will be either $22or $18. Risk-neutral valuation shows that the value of a three-month call option on the stock witha strike price of 21 is 0.633. Footnote ! of Chapter 10 shows that, if the expected return required byinvestors on the stock in the real world is 16%, the expected return required on the call option is42.6%. A similar analysis shows that if the option is a put rather than a call then the expectedreturn required on the option is —52.5%. in practice it wouid be very difficuit to estimate theseexpected returns directly in order to value the options. (We know the returns only because we areable to value the options in another way.) The same point applies to options on real assets. There isno easy way of estimating the risk-adjusted discount rates appropriate for cash flows when theyarise from abandonment, expansion, and other options. This is the motivation for exploringwhether the risk-neutral valuation principle can be applied to options on real assets as well asto options on financial assets.
Another probiem with the traditional NPV approach lies in the estimation of the appropriaterisk-adjusted discount rate for the base project (i.e., the project without embedded options). Thecompanies that are used to estimate a proxy beta for the project in the three-step procedure abovehave expansion options and abandonment options of their own. Their betas reflect these optionsand may not therefore be appropriate for estimating a beta for the base project.
In the next section we explain how the risk-neutral valuation arguments in Chapters 10, 12,and 21 can be extended to value projects dependent on multiple sources of uncertainty. Themethod requires estimates of market prices of risk for each source of uncertainty. Approaches forestimating market prices of risk are outlined in Section 28.3.
28.2 EXTENSION OF THE RISK-NEUTRALVALUATION FRAMEWORK
 
Consider an asset whose price, f, depends on a variable @ and time t. (Later we will extend theanalysis to the situation where there is more than one variable.) Assume that the process followedby @ is
doBamadt+ sdz
where dz is a Wiener process. The parameters m and s are the expected growth rate in 6 and thevolatility of 6, respectively. We assume that the parameters depend only on @ and t. The variable 6662 CHAPTER 28
need not be a financial variable. It could be something as far removed from financial markets asthe temperature in the center of New Orieans.The asset price f follows a process of the formdf = ufdt+ofdz
The analysis in Section 21.1 shows that
aor
Co
 
=k (28.1)
where r is the risk-free rate and 4 is the market price of risk for 6.Because f is a function of 0 and t we can use It6’s lemma in Appendix 11A to express yz and a interms of m an s. The result isaf a afof 4 20? of
=moti 4SMpa monet ae tO ae
andaf= so—of=s 36Substituting into equation (28.1) leads to the following differential equation that must be satisfiedby f: ;af, af 220F—+0—(m ~ ds) + 4