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,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Options,Futures,and Other Derivatives,7th Edition,Copyright John C.Hull 2008,*,本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考，不能作为科学依据。本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考!,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Options,Futures,and Other Derivatives,7th Edition,Copyright John C.Hull 2008,*,本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考，不能作为科学依据。本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考!,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Options,Futures,and Other Derivatives,7th Edition,Copyright John C.Hull 2008,*,本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考，不能作为科学依据。本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考!,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Options,Futures,and Other Derivatives,7th Edition,Copyright John C.Hull 2008,*,本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考，不能作为科学依据。本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考!,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Options,Futures,and Other Derivatives,7th Edition,Copyright John C.Hull 2008,*,本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考，不能作为科学依据。本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考!,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Options,Futures,and Other Derivatives,7th Edition,Copyright John C.Hull 2008,*,本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考，不能作为科学依据。本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考!,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Options,Futures,and Other Derivatives,7th Edition,Copyright John C.Hull 2008,*,本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考，不能作为科学依据。本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考!,Chapter 13,Credit Risk,1/52,What is credit risk?,Credit risk arises from the possibility that borrowers and counterparties in derivatives transactions may default.,2,2,2/52,2,Contents,Approaches to estimating the probability that a company will default,The difference between risk-neutral and real-world probabilities of default,Credit risk of derivative,Default correlation,Gaussian copula models,3,3,3,3/52,Approaches to estimating default probabilities,Historical default probabilities of rating companies,From bonds prices,From equity prices,From derivatives prices,4/52,Historical cumulative average default rates(%),5/52,Interpretation,The table shows the probability of default for companies starting with a particular credit rating,The probability that a bond initially rated Baa will default during the second year is 0.506-0.181=0.325,Default probability change with time,6/52,Default Intensities vs Unconditional Default Probabilities,The unconditional default probability is the probability of default for a certain time period as seen at time zero,The conditional default probability is the probability of default for a certain time period conditional on no earlier default(say,default intensity or hazard rate),7/52,Define V(t)as cumulative probability of the company surviving to time t.,Taking limits,we get,Define Q(t)as the probability of default by time t.,Where is the average default intensity between 0 and t,8/52,Recovery rate,The recovery rate for a bond is usually defined as the price of the bond immediately after default as a percent of its face value,Recovery rates are significantly negatively correlated with default rates,9/52,Recovery rates,(Moodys:1982 to,Table 22.2,page 491),10/52,Using Bond Prices,Average default intensity over life of bond is approximately,Where s is the spread of the bonds yield over the risk-free rate and R is the recovery rate.,11/52,More Exact Calculation,Assume that a 5 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%(continuous compounding).,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),12/52,Calculations,13/52,Calculations(Cons.),We set 288.48Q=8.75 to get Q=3.03%,This analysis can be extended to allow defaults to take pace more frequently,Instead of assuming a constant unconditional probability of default we can assume a constant default intensity or a particular pattern for the variation of default probabilities with time.,With several bonds we can use more parameters to describe the term structure of default probability.,14/52,The Risk-Free Rate,The risk-free rate when default probabilities are estimated is usually assumed to be the LIBOR/swap zero rate(or sometimes 10 bps below them),To get direct estimates of the spread of bond yields over swap rates we can look at asset swaps,15/52,Asset Swaps,Asset swap spreads provide a direct estimate of the spread of bond yields over the LIBOR/swap curve.,If the asset swap spread is 150 bps and the LIBOR/swap zero curve is flat at 5%.The expected loss from default over the 5-year life of the bond is therefore$6.55.6.55=288.48*Q,Q=2.27%,16/52,Credit Default Swap Spreads(bps),17/52,Credit Default Swap Spreads(bps),18/52,Comparison historical vs bond,Calculation of default intensities using historical data are based on equation(22.1)and table(22.1);From equation(22.1),we have,The calculations using bond prices are based on equation(22.2)and bond yields published by Merrill Lynch.,19/52,Real World vs Risk Neutral Default Probabilities,7 year average,20/52,Risk Premiums Earned by Bond Traders,21/52,The default probability from historical data is significantly lower than that from bond prices,The ratio declines while the difference increases as a companys credit rating declines.,22/52,Real World vs.,Risk-Neutral Default Probabilities,The default probabilities 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 probabilities,23/52,Possible reasons for these results,Corporate bonds are relatively illiquid,The subjective default probabilities of bond traders may be much higher than 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 market.,24/52,Which 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 VaR and scenario analysis,25/52,Mertons model,Mertons model regards the equity as an option on the assets of the firm.,In a simple situation the equation value is,where is the value of the firm and is the debt repayment required.,26/52,Equity vs.Assets,An option pricing model enables the value of the firms equity today,to be related to the value of its assets today,and the volatility of its assets,The risk-neutral probability that the company will default on the debt is .,27/52,Volatilities,?,28/52,Example,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 equations yields,29/52,Example(Con.),The probability of default is,The market value of the debt is,The present value of the promised payment is 9.51,The expected loss is about(9.51-9.4)/9.51=1.2%,The recovery rate is(12.7-1.2)/12.7=91%,30/52,Implementation of Mertons model(e.g.Moodys KMV),Mertons model produces a good ranking of default probabilities(risk-neutral or real-world),Moody,企业把股票当于企业资产期权思想计算出风险中性世界违约距离，再利用拥有海量历史违约数据库，建立起风险中性违约距离与现实世界违约率之间对应关系，从而得到预期违约频率，作为违约概率预测指标。,31/52,贝尔斯登预期违约频率,32/52,从期权价格中引出风险中性违约概率,因为股票是企业资产期权，这么股票期权就是期权期权，其价格能够表示为：,利用最大熵方法（,Capuano,),就能够从企业同期限全部期权价格中预计出 和,D,33/52,从期权价格中能够推导出风险中性违约概率,利用上述方法，我们就可依据3月14日贝尔斯登将于3月22日到期期权价格，计算出贝尔斯登风险中性违约概率和企业价值概率分布。贝尔斯登于3月14日被摩根大通接管。,下列图显示，市场对贝尔斯登一周后命运产生巨大分歧，企业价值大涨大跌概率远远大于小幅变动概率，这么分布与正常情况分布有天壤之别。可见期权价格能够让我们清楚地看出市场在非常时期对未来特殊看法。,34/52,贝尔斯登风险中性违约概率和企业价值概率分布（3月14日）,35/52,风险中性违约概率,风险中性违约概率即使不一样于现实概率，但其改变能够反应现实世界违约概率改变。在金融危机时期，它可能比CDS价差能更敏感地反应出违约概率改变。,在贝尔斯登于3月14日被接管前后，依据上述方法计算出来风险中性概率天天改变比CDS价差更敏感。这是因为在金融危机期间，金融机构本身信用度大幅降低，造成在OTC市场交易CDS交易量急剧萎缩，价差大幅扩大，信号失真。,36/52,期权隐含中性违约概率与,CDS,价差,37/52,Credit Risk Mitigation,Netting:incremental effect,Collateralization,Downgrade triggers,38/52,Default correlation,The credit default correlation between two companies is a measure of their tendency to default at about the same time,Factors,(1)macroeconomic environment:good economy =low number of defaults,(2)Same industry or geographic area:companies can be similarly or inversely affected by an external event,(3)credit contagion:connections between companies can cause a ripple effect,39/52,Credit derivative,Credit derivatives are contracts where the payoff depends on the creditworthiness of one or more companies or countries,Buyers:banks or other financial institutions,Sellers:insurance company,Single name:credit default swap,CDS,40/52,How does CDS works?,This is a contract that provides insurance against the risk of a default by particular company.,The company is known as the reference entity and a default by the company is known as a credit event.,The buyer of the insurance obtains the right to sell bonds issued by the company for their face value when a credit event occurs.,The sellers of the insurance agrees to buy the bonds for their face value when a credit event occur.,41/52,Example,A 5-year credit default swap on March 1,.,The notional principal is$100 million.,The buyer agrees to pay 90 basis points annually for protection against default by the reference entity.,Default,protection,buyer,Default,protection,seller,90 basis points per year,Payment if default,by reference entity,42/52,Mechanism,If not default,reference entity pays$900,000 on March 1 of each-,If default,e.g.June 1,;(1)specifies physical settlement;(2)determine the mid-market value of the cheapest deliverable bond,or say,cash payment,In arrear payment,including a final accrual payment,CDS spread:the total amount paid per year,as a percent of the notional principal,to buy protection,43/52,CDS and Bond yields,A CDS can be used to hedge a position in a corporate bond.,The n-year CDS spread should be approximately equal to the excess of the par yield on an n-year corporate bond over the par yield on an n-year risk-free bond.,How to use it,44/52,CDS and Cheapest-to-deliver bond,Bonds typically have the same seniority,but they may not sell for the same percentage of face value immediately after a default.,Search a cheapest-to-deliver bond.,45/52,Valuation of credit default swaps,Mid-market CDS spreads,Example:,Suppose the probability during a year conditional on no earlier default is 2%.,Time(year),default probability,survival probability,1,0.02,0.98,2,0.0196,0.9604,3,0.0192,0.9412,4,0.0188,0.9224,5,0.0184,0.9039,46/52,Valuation of credit default swaps(cons.),(2)Default always happen halfway through a year and that payments on the credit default swap are made once a year at the end of each year.,(3)The risk-free interest rate is 5%per annum with continuous compounding and the recover rate is 40%.,47/52,1,Default 1,2,3,4,5,0,Default 2,Default 3,Default 4,Default 5,Payoff,Accrual payment,.,.,.,.,Payment 1,Payment 2,Payment 3,Payment 4,Payment 5,Survival probability,Default probability,48/52,PV of the expected payment,Assume notional principal is 1 and payment at rate of s per year.,Time(year),survival probability,expected payment,discount factor,pv of expected payment,1,0.98,0.98s,0.9512,0.9322s,2,0.9604,0.9604s,0.9048,0.8690s,3,0.9412,0.9412s,0.8607,0.8101s,4,0.9224,0.9224s,0.8187,0.7552s,5,0.9039,0.9039s,0.7788,0.7040s,total,4.0704s,49/52,PV of the expected payoff,Assume notional principal is 1,defaults always happen halfway of a year.,Time(year),default probability,recovery rate,expected payment,discount factor,pv of expected payment,1,0.02,0.4,0.012,0.9512,0.0117,2,0.0196,0.4,0.0118,0.9048,0.0109,3,0.0192,0.4,0.0115,0.8607,0.0102,4,0.0188,0.4,0.0113,0.8187,0.0095,5,0.0184,0.4,0.0111,0.7788,0.0088,total,0.0511,50/52,PV of the last accrued payment,Assume notional principal is 1,defaults always happen halfway of a year.,Time(year),default probability,accrual payment,discount factor,pv of expected payoff,1,0.02,0.5s,0.9753,0.0098s,2,0.0196,0.5s,0.9277,0.0091s,3,0.0192,0.5s,0.8825,0.0085s,4,0.0188,0.5s,0.8395,0.0079s,5,0.0184,0.5s,0.7985,0.0073s,total,0.0426s,51/52,Valuation at or after the negotiation,Marking to market a CDS,By product:,Estimating default probabilities and recover rate with CDS quoted spread.,52/52,52,
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