期貨期權(quán)及其衍生品配套課件全34章Ch22

1、credit riskchapter 22options, futures, and other derivatives, 7th international edition, copyright john c. hull 20081credit ratings in the s&p rating system, aaa is the best rating. after that comes aa, a, bbb, bb, b, ccc, cc, and c the corresponding moodys ratings are aaa, aa, a, baa, ba, b,caa, ca

2、, and c bonds with ratings of bbb (or baa) and above are considered to be “investment grade”options, futures, and other derivatives, 7th international edition, copyright john c. hull 20082historical data historical data provided by rating agencies are also used to estimate the probability of default

3、options, futures, and other derivatives, 7th international edition, copyright john c. hull 20083cumulative ave default rates (%) (1970-2006, moodys, table 22.1, page 490)options, futures, and other derivatives, 7th international edition, copyright john c. hull 20084interpretation the table shows the

4、 probability of default for companies starting with a particular credit rating a company with an initial credit rating of baa has a probability of 0.181% of defaulting by the end of the first year, 0.506% by the end of the second year, and so onoptions, futures, and other derivatives, 7th internatio

5、nal edition, copyright john c. hull 20085do default probabilities increase with time? for a company that starts with a good credit rating default probabilities tend to increase with time for a company that starts with a poor credit rating default probabilities tend to decrease with time options, fut

6、ures, and other derivatives, 7th international edition, copyright john c. hull 20086default intensities vs unconditional default probabilities (page 490-91) the default intensity (also called hazard rate) is the probability of default for a certain time period conditional on no earlier default the u

7、nconditional default probability is the probability of default for a certain time period as seen at time zero what are the default intensities and unconditional default probabilities for a caa rate company in the third year?options, futures, and other derivatives, 7th international edition, copyrigh

8、t john c. hull 20087default intensity (hazard rate) the default intensity (hazard rate) that is usually quoted is an instantaneous if v(t) is the probability of a company surviving to time toptions, futures, and other derivatives, 7th international edition, copyright john c. hull 20088ttdttetqtetvtv

9、ttvttvt)()(1)()()()()()(0is timeby default ofy probabilit cumulative theto leads thisrecovery ratethe recovery rate for a bond is usually defined as the price of the bond immediately after default as a percent of its face valueoptions, futures, and other derivatives, 7th international edition, copyr

10、ight john c. hull 20089recovery rates(moodys: 1982 to 2006, table 22.2, page 491)options, futures, and other derivatives, 7th international edition, copyright john c. hull 200810estimating default probabilities alternatives:use bond pricesuse cds spreadsuse historical datause mertons modeloptions, f

11、utures, and other derivatives, 7th international edition, copyright john c. hull 200811using bond prices (equation 22.2, page 492)average default intensity over life of bond is approximatelywhere s is the spread of the bonds yield over the risk-free rate and r is the recovery raters1options, futures

12、, and other derivatives, 7th international edition, copyright john c. hull 200812more exact calculation assume that a five year corporate bond pays a coupon of 6% per annum (semiannually). the yield is 7% with continuous compounding and the yield on a similar risk-free bond is 5% (with continuous co

13、mpounding) price of risk-free bond is 104.09; price of corporate bond is 95.34; expected loss from defaults is 8.75 suppose that the probability of default is q per year and that defaults always happen half way through a year (immediately before a coupon payment. options, futures, and other derivati

14、ves, 7th international edition, copyright john c. hull 200813calculations (table 22.3, page 493)time(yrs)defprobrecovery amountrisk-free valuelgddiscount factorpv of exp loss0.5q40106.7366.730.9753 65.08q1.5q40105.9765.970.9277 61.20q2.5q40105.1765.170.8825 57.52q3.5q40104.3464.340.8395 54.01q4.5q40

15、103.4663.460.7985 50.67qtotal288.48qoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200814calculations continued we set 288.48q = 8.75 to get q = 3.03% this analysis can be extended to allow defaults to take place more frequently with several bonds we can u

16、se more parameters to describe the default probability distributionoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200815the risk-free rate the risk-free rate when default probabilities are estimated is usually assumed to be the libor/swap zero rate (or som

17、etimes 10 bps below the libor/swap rate) to get direct estimates of the spread of bond yields over swap rates we can look at asset swapsoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200816real world vs risk-neutral default probabilities the default probab

18、ilities backed out of bond prices or credit default swap spreads are risk-neutral default probabilities the default probabilities backed out of historical data are real-world default probabilitiesoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200817a compa

19、risoncalculate 7-year default intensities from the moodys data (these are real world default probabilities)use merrill lynch data to estimate average 7-year default intensities from bond prices (these are risk-neutral default intensities)assume a risk-free rate equal to the 7-year swap rate minus 10

20、 basis pointoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200818real world vs risk neutral default probabilities, 7 year averages (table 22.4, page 495)options, futures, and other derivatives, 7th international edition, copyright john c. hull 200819risk p

21、remiums earned by bond traders (table 22.5, page 496)options, futures, and other derivatives, 7th international edition, copyright john c. hull 200820possible reasons for these results corporate bonds are relatively illiquid the subjective default probabilities of bond traders may be much higher tha

22、n the estimates from moodys historical data bonds do not default independently of each other. this leads to systematic risk that cannot be diversified away. bond returns are highly skewed with limited upside. the non-systematic risk is difficult to diversify away and may be priced by the marketoptio

23、ns, futures, and other derivatives, 7th international edition, copyright john c. hull 200821which world should we use? we should use risk-neutral estimates for valuing credit derivatives and estimating the present value of the cost of default we should use real world estimates for calculating credit

24、 var and scenario analysisoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200822mertons model (page 498-499) mertons model regards the equity as an option on the assets of the firm in a simple situation the equity value ismax(vt d, 0)where vt is the value o

25、f the firm and d is the debt repayment requiredoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200823equity vs. assets an option pricing model enables the value of the firms equity today, e0, to be related to the value of its assets today, v0, and the volat

26、ility of its assets, svoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200824ev n dden ddvdrttddtrtvvv0012102212()()ln ()();wheresssvolatilities options, futures, and other derivatives, 7th international edition, copyright john c. hull 200825sssevveevvn dv0

27、010()this equation together with the option pricing relationship enables v0 and sv to be determined from e0 and seexample a companys equity is $3 million and the volatility of the equity is 80% the risk-free rate is 5%, the debt is $10 million and time to debt maturity is 1 year solving the two equa

28、tions yields v0=12.40 and sv=21.23%options, futures, and other derivatives, 7th international edition, copyright john c. hull 200826example continued the probability of default is n(-d2) or 12.7% the market value of the debt is 9.40 the present value of the promised payment is 9.51 the expected loss

29、 is about 1.2% the recovery rate is 91%options, futures, and other derivatives, 7th international edition, copyright john c. hull 200827the implementation of mertons model choose time horizon calculate cumulative obligations to time horizon. this is termed by kmv the “default point”. we denote it by

30、 d use mertons model to calculate a theoretical probability of default use historical data or bond data to develop a one-to-one mapping of theoretical probability into either real-world or risk-neutral probability of default.options, futures, and other derivatives, 7th international edition, copyrig

31、ht john c. hull 200828credit risk in derivatives transactions (page 502-504)three cases contract always an asset contract always a liability contract can be an asset or a liabilityoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200829general result assume t

32、hat default probability is independent of the value of the derivative consider times t1, t2,tn and default probability is qi at time ti. the value of the contract at time ti is fi and the recovery rate is r the loss from defaults at time ti is qi(1-r)emax(fi,0). defining ui=qi(1-r) and vi as the val

33、ue of a derivative that provides a payoff of max(fi, 0) at time ti, the cost of defaults isoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200830niiivu1if contract is always a liability (equation 22.9)maturity this for yieldfree-risk the is and derivative t

34、he of seller theby issued bonds coupon zero on yieldthe is value. free-default the is and derivative the of value actual the is time at payoff a provides derivative whereyyfftefftyy*0*0)(0*0.*options, futures, and other derivatives, 7th international edition, copyright john c. hull 200831credit risk

35、 mitigation netting collateralization downgrade triggersoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200832default correlation the credit default correlation between two companies is a measure of their tendency to default at about the same time default c

36、orrelation is important in risk management when analyzing the benefits of credit risk diversification it is also important in the valuation of some credit derivatives, eg a first-to-default cds and cdo tranches. options, futures, and other derivatives, 7th international edition, copyright john c. hu

37、ll 200833measurement there is no generally accepted measure of default correlation default correlation is a more complex phenomenon than the correlation between two random variablesoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200834binomial correlation m

38、easure (page 516)one common default correlation measure, between companies i and j is the correlation betweena variable that equals 1 if company i defaults between time 0 and time t and zero otherwisea variable that equals 1 if company j defaults between time 0 and time t and zero otherwisethe value

39、 of this measure depends on t. usually it increases at t increases.options, futures, and other derivatives, 7th international edition, copyright john c. hull 200835binomial correlation continueddenote qi(t) as the probability that company a will default between time zero and time t, and pij(t) as th

40、e probability that both i and j will default. the default correlation measure isoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200836)()()()()()()()(22tqtqtqtqtqtqtptjjiijiijijsurvival time correlation define ti as the time to default for company i and qi(

41、ti) as the probability distribution for ti the default correlation between companies i and j can be defined as the correlation between ti and tj but this does not uniquely define the joint probability distribution of default timesoptions, futures, and other derivatives, 7th international edition, co

42、pyright john c. hull 200837gaussian copula model (page 514-515) define a one-to-one correspondence between the time to default, ti, of company i and a variable xi byqi(ti ) = n(xi ) or xi = n-1q(ti)where n is the cumulative normal distribution function. this is a “percentile to percentile” transform

43、ation. the p percentile point of the qi distribution is transformed to the p percentile point of the xi distribution. xi has a standard normal distribution we assume that the xi are multivariate normal. the default correlation measure, rij between companies i and j is the correlation between xi and

44、xjoptions, futures, and other derivatives, 7th international edition, copyright john c. hull 200838binomial vs gaussian copula measures (equation 22.14, page 516)the measures can be calculated from each otheroptions, futures, and other derivatives, 7th international edition, copyright john c. hull 2

45、00839function ondistributiy probabilitnormal bivariate cumulative the is wherethat somtqtqtqtqtqtqxxmtxxmtpjjiijiijjiijijjiij)()()()()()(;,)(;,)(22rrcomparison (example 22.4, page 516) the correlation number depends on the correlation metric used suppose t = 1, qi(t) = qj(t) = 0.01, a value of rij e

46、qual to 0.2 corresponds to a value of ij(t) equal to 0.024. in general ij(t) rij and ij(t) is an increasing function of t options, futures, and other derivatives, 7th international edition, copyright john c. hull 200840example of use of gaussian copula (example 22.3, page 515)suppose that we wish to

47、 simulate the defaults for n companies . for each company the cumulative probabilities of default during the next 1, 2, 3, 4, and 5 years are 1%, 3%, 6%, 10%, and 15%, respectively options, futures, and other derivatives, 7th international edition, copyright john c. hull 200841use of gaussian copula

48、 continued we sample from a multivariate normal distribution to get the xi critical values of xi aren -1(0.01) = -2.33, n -1(0.03) = -1.88, n -1(0.06) = -1.55, n -1(0.10) = -1.28,n -1(0.15) = -1.04options, futures, and other derivatives, 7th international edition, copyright john c. hull 200842use of

49、 gaussian copula continued when sample for a company is less than -2.33, the company defaults in the first year when sample is between -2.33 and -1.88, the company defaults in the second year when sample is between -1.88 and -1.55, the company defaults in the third year when sample is between -1,55

50、and -1.28, the company defaults in the fourth year when sample is between -1.28 and -1.04, the company defaults during the fifth year when sample is greater than -1.04, there is no default during the first five years options, futures, and other derivatives, 7th international edition, copyright john c. hull 200843a one-factor model for the correlation structure (equation 22.10 , page 515) the correlation between xi and xj is