Example: biology

Chapter 4 Structural Models of Credit Risk - Fields Institute

Chapter 4 Structural Models of Credit RiskBroadly speaking, Credit risk concerns the possibility of financial losses due to changes inthe Credit quality of market participants. The most radical change in Credit quality is adefault event. Operationally, for medium to large cap firms, default is normally triggeredby a failure of the firm to meet its debt servicing obligations, which usually quickly leadsto bankruptcy proceedings, such as Chapter 11 in the Thus default is considereda rare and singular event after which the firm ceases to operate as a viable concern,and which results inlargefinancial losses tosomesecurity holders. With some flexiblethinking, this view of Credit risk also extends to sovereign bonds issued by countries witha non-negligible risk of default, such as those of developing Structural Models , a default event is deemed to occur for a firm when its assetsreach a sufficiently low level compared to its liabilities.

Bond holders, on the other hand, receive min(K,A T)=A T −(A T −K)+ = K −(K −A T)+. Therefore the value D t for the debt at earlier times t < T can be obtained as the value of a zero-coupon bond minus a European put option. Of course the fundamental identity of accounting holds: A t = E t +D t, and all three assets are discounted risk ...

Tags:

  Bond

Information

Domain:

Source:

Link to this page:

Please notify us if you found a problem with this document:

Other abuse

Transcription of Chapter 4 Structural Models of Credit Risk - Fields Institute

1 Chapter 4 Structural Models of Credit RiskBroadly speaking, Credit risk concerns the possibility of financial losses due to changes inthe Credit quality of market participants. The most radical change in Credit quality is adefault event. Operationally, for medium to large cap firms, default is normally triggeredby a failure of the firm to meet its debt servicing obligations, which usually quickly leadsto bankruptcy proceedings, such as Chapter 11 in the Thus default is considereda rare and singular event after which the firm ceases to operate as a viable concern,and which results inlargefinancial losses tosomesecurity holders. With some flexiblethinking, this view of Credit risk also extends to sovereign bonds issued by countries witha non-negligible risk of default, such as those of developing Structural Models , a default event is deemed to occur for a firm when its assetsreach a sufficiently low level compared to its liabilities.

2 These Models require strongassumptions on the dynamics of the firm s asset, its debt and how its capital is main advantage of Structural Models is that they provide an intuitive picture, as wellas an endogenous explanation for default. We will discuss other advantages and some oftheir disadvantages in what The Merton Model (1974)The Merton model takes an overly simple debt structure, and assumes that the total valueAtof a firm s assets follows a geometric Brownian motion under the physical measuredAt= Atdt+ AtdWt,A0>0,( )where is the mean rate of return on the assets and is the asset volatility. We alsoneed further assumptions: there are no bankruptcy charges, meaning the liquidation valueequals the firm value; the debt and equity are frictionless tradeable and medium cap firms are funded by shares ( equity ) and bonds ( debt ).

3 TheMerton model assumes that debt consists of a single outstanding bond with face valueKand maturityT. At maturity, if the total value of the assets is greater than the debt, thelatter is paid in full and the remainder is distributed among shareholders. However, if4142 Chapter 4. Structural Models OF Credit RISKAT<Kthen default is deemed to occur: the bondholders exercise a debt covenant givingthem the right to liquidate the firm and receive the liquidation value (equal to the totalfirm value since there are no bankruptcy costs) in lieu of the debt. Shareholders receivenothing in this case, but by the principle of limited liability are not required to inject anyadditional funds to pay for the these simple observations, we see that shareholders have a cash flow atTequalto(AT K)+,and so equity can be viewed as a European call option on the firm s assets.

4 On the otherhand, the bondholder receives min(AT,K). Moreover, the physical probability of defaultat timeT, measured at timet, isPt[ =T]=Pt[AT K]=N[ dP2]wheredP2=( T t) 1(log(At/K)+( 2/2)(T t)).The valueEtat earlier timest < Tcan be derived using the classic martingale ar-gument (see exercise 24 for an alternative derivation). Assuming one can trade the firmvalueAt, we note thate rtAtis a martingale under the risk-neutral measureQwithmarket price of risk =( r)/ and Radon-Nikodym derivativedQdP= exp( WT 12 2T).( )Then we find the standard Black-Scholes call option formulaEt=EQ[e r(T t)(AT K)+] = BSCall(At, K, r, , T t)( )=AtN[d1] e r(T t)KN[d2]( )whered1=log(At/K)+(r+ 2/2)(T t) T t,d2=log(At/K)+(r 2/2)(T t) T t( ) bond holders, on the other hand, receivemin(K, AT)=AT (AT K)+=K (K AT)+.

5 Therefore the valueDtfor the debt at earlier timest < Tcan be obtained as the value ofa zero-coupon bond minus a European put option. Of course the fundamental identity ofaccounting holds:At=Et+Dt,and all three assets are discounted risk neutral martingales. A zero coupon defaultablebond with face value 1 and maturityTwill have the price Pt(T)=Dt/K, and has theyield spreadYSt(T)=1T tlogKe r(T t)Dt= 1T tlog(er(T t)AtK(1 N[d1]) +N[d2].)( ) THE MERTON MODEL (1974)43 Despite being derived from a debt structured with a single maturityT, this equationis often interpreted as giving a function ofT: it is imagined that if an additional bond ofsmall face value with a different maturity were issued by the firm, it would also be pricedaccording to ( ).

6 The qualitative behaviour of this term structure is that Credit spreadsstart at zero forT= 0, increase sharply to a maximum, and then decrease either to zeroat large times ifr 2/2 0 or a positive value ifr 2/2>0. This is in accordancewith the diffusive character of the model. For very short maturity times, the asset pricediffusion will almost surely never cross the default barrier. The probability density ofdefault then increases for longer maturities but starts to decrease again as the geometricBrownian motion drifts away from the behaviour is also observed in first passage and excursion Models , except thatspreads exhibit a faster decrease for longer maturities. It is at odds with empirical obser-vations in two respects: (i) observed spreads remain positive even for small time horizonsand (ii) tend to increase as the time horizon increases.

7 The first feature follows from thefact that there is always a small probability of immediate default. The second is a conse-quence of greater uncertainty for longer time horizons. One of the main reasons to studyreduced-form Models is that, as we will see, they can easily avoid such previously obtained formula for the physical default probability (that is under themeasureP) can be used to calculate risk neutral default probability provided we replace byr. Thus one finds thatQ[ > T]=N(N 1(P[ > T]) T).and as long as >0 we see that market implied ( risk neutral) survival probabilitiesare always less than historical the event of default, the bondholder receives only a fractionAT/K, called therecovery fraction, of the bond principalK: the fractional loss (K AT)/Kis called theloss given defaultor LGD.

8 As you will see in an exercise, the probability distribution ofLGD can be computed explicitly in the Merton that equity value increases with the firm s volatility (since its payoffis con-vex in the underlier), so shareholders are generally inclined to press for riskier positionsto be taken by their managers. The opposite is true for bondholders. So-called agencyproblems relate to the contradictory aims of shareholders, bondholders and other stake-holders . Structural Models like Merton s model depend on the unobserved variableAt. On theother hand, for publicly traded companies, the share price (and hence the total equity)is closely observed in the market. The usual ad-hoc approach to obtaining an estimatefor the firm s asset valuesAtand volatility in Merton s model uses the Black-Scholesformula for a call option, that is,Et= BSCall(At, K, r, , T t),( )whereKandTare determined by the firm s debt structure.

9 One combines this with asecond equation by equating the equity volatility to the coefficient of the Brownian term44 Chapter 4. Structural Models OF Credit RISK obtained by applying It o s formula to ( ), namely, At BSCall A= EEt.( )As we will see in Section , a consistent method is to use Duan s maximum likelihoodresult [5] to estimate and directly from the equity time seriesEi. Once an estimatefor is obtained in this way, it can be inserted back into the pricing formula ( ) inorder to produce estimates for the firm Merton model is only a starting point for studying Credit risk, and is obviouslyfar from realistic: The non-stationary structure of the debt that leads to the termination of operationson a fixed date, and default can only happen on that date.

10 Geske [10] extended theMerton model to the case of bonds of different maturities. It is incorrect to assume that the firm value is tradeable. In fact, the firm value andits parameters is not even directly observed. Interest rates should certainly be taken to be stochastic: this is not a serious draw-back, and its generalization was included in Merton s original paper. The short end of the yield spread curve in calibrated versions of the Merton modeltypically remains essentially zero for months, in strong contradiction with so-called first passage Models extend the Merton framework by allowing default tohappen at intermediate Black-Cox modelThe simplestfirst passage modelagain takes a firm with asset value given by ( ) andoutstanding debt with face valueKat maturityT.


Related search queries