A. Useful Tools from Martingale Theory

This appendix covers a number of standard results from martingale theory and which are used in the main body of the text. The concepts are outlined in a very brief fashion. Proofs are often omitted or the reader is referred to the literature. The first section introduces the probabilis• tic setup of the material used in later sections. See, for example, Williams (1991) for a more detailed introduction to probability. The later sections cover some important results from martingale theory and stochastic calcu• lus. A more extensive treatment of this material can be found in any text on Brownian motion and stochastic calculus. Most of the material presented in this section is adapted from Karatzas and Shreve (1991) and Revuz and Yor (1994).

A.I Probabilistic Foundations

Definitions. A measurable space (il,9") consists of a sample space il and a collection of subsets of il called a l1-algebra. A count ably additive mapping J.L : 9" -+ JR + is called a measure on (il, 9") and the triple (il, 9", J.L) is called a measure space. A 9"-measurable function is a mapping h from (il,9") to a state space (S, S) such that for any A E S, h-1(A) E 9". A Borel l1-algebra is a l1-algebra containing all open sets of a topological space. For example, 'B(il) is the Borel l1-algebra of il. The mapping h is called a Borel function if it is 'B(il)• measurable. A random variable is a measurable function. For example, consider a collection of functions h from a topological space il with l1-algebra 9" to S = JRd and S = 'B(JRd). S = JRd is the Euclidean space and 'B(JRd) is the smallest l1-algebra of Borel sets of the topological space JRd such that each h is measurable. We say that the l1-algebra l1(h) is generated by h. Obviously, for a random variable X, I1(X) E 9". Otherwise the mappings would not be measurable. A process is an family of random variables and is written (Xt)O~t~T or simply X t in shorthand notation. The index t is often interpreted as time defined on the interval [0, T] for T < 00. 224 A. Useful Tools from Martingale Theory

For fixed sample points wEn, the function t ~ Xt(w) is called the sample path or trajectory of the process. However, a process can be viewed as a function of two variables (t,w) where t is the (time) index and wEn, i.e. we have the mapping (t, w) ~ X t (w) Vw E n. Therefore, a process is a joint mapping h from lR+ x n onto S. This mapping is measurable if we equip it with the corresponding a-algebrae of Borel sets, i.e., if we have h : (t,w) ~ Xt(w) : (lR+ x n, ~(lR+) ® ~ ~ (lRd, ~(lRd») such that for any A E ~(lRd), h-1(A) E ~(lR+) ®~. A is a measure space with /1(n) = 1 and /1(0) = o. In this case /1 is called a probability measure. It associates subsets of n (events) with a probability. For example, consider a probability space (n,~, P). If, for an event A E ~, peA) = 1, then we say that the event occurs , abbreviated as P-a.s. or a.s. An event A E ~ with peA) = 0 is called almost impossible. If A ¢ ~, then A is impossible. A probability measure Q is said to be equivalent to P if, for any event A E ~, peA) = 0 {:} Q(A) = O. This means that two probability measures P and Q are equivalent if and only if for any event which is almost impossible under P, it is also almost impossible under Q. If only peA) = 0 =} Q(A) = 0, then we say that Q is absolutely continuous with respect to P. A ~O~t~T = {~o, ... ,~T}, sometimes written (~dO~t~T or ~t in shorthand, is a family of sub-a-algebrae included in ~ which is non-decreasing in the sense that ~s s;;: ~t for s < t. Often, a filtration can be interpreted as representing the accumulation of information over time. A process generates a filtration. For example, consider the filtration ~t = a(XsE[O,tJ) generated by Xt. This is the smallest a-algebra with respect to which Xs is measurable for every s E [0, t]. For technical reasons, we usually assume that it also contains all P-null sets, i.e., if A E ~ and peA) = 0 , then A E ~t. \It E [0, T]. In this case it is called the natural filtration of X. Generally, we implicitly assume a filtration to be of the natural kind. A filtration to which the null sets are added is called augmented. A filtered probability space is a probability space equipped with a fil• tration and is often written (n,~, (~dO~t~T' P), (fl,~, (~tE[O,TJ)' P), or (fl,~, (~t)O~t~T' P). Consider a filtered probability space. If, for an event A E ~, peA) = 1 for all t E [0, T], then we say that A is given almost everywhere. In shorthand notation we write A P-a.e. or a.e. Let a filtered probability space (fl,~, (~dO

A.2 Process Classes

Definition A.2.1. Let £00 be the class of all adapted processes. 0 is a pro• cess. Define the following spaces:

£1 = {o E £00: loT 10ti dt < 00 a.s.},

£2 = { 0 E £00 : loT O;dt < 00 a.s.},

1{2 = { 0 E £2 : E (loT O;dt) < 00 } ,

£1,1 = {o E £00 : loTI loT2 10(s, t)ldsdt < 00 a.s.} ,

£2,1 = {o E £1,1 : loT O(s, u)2ds < 00 a.s.,

f (f lo(s.U)ldU), ds < = as}

We write 0 E £1(f.?,3',P) or simply 0 E £1 if it is clear from context on which probability space the process is defined. In most instances in this text, integrable (£1) or square-integrable (£2) processes are used.

A.3 Martingales

Definition A.3.1. A process Mt adapted to 3't and satisfying E [lMt I] < 00, 'it E [0, T] is called a P-submartingale if

and a P -supermartingale if

M t is a martingale if it is both a submartingale and a supermartingale, i.e., if Ep[MTI3'd = M t .

Remark A. 3.1. Informally, we call the process M t a local martingale if it does not satisfy the technical condition E(IMtl) < 00, Vt E [0, T]. 226 A. Useful Tools from Martingale Theory

Theorem A.3.1 (Dooh-Meyer). Let X be a right-continuous submartin• gale. Under some technical conditions, X has the decomposition

°:s t < 00, such that M t is right-continuous martingale and At an increasing process. Proof. Cf. Karatzas and Shreve (1991), p. 25. Definition A.3.2. For a martingale M t E £2, the pro• cess is (M, M)t = At, where At is the increasing process of the Doob-Meyer decomposition of X = M2. Quadratic variation is also known as cross• variation. Remark A.3.2. M is a unique process such that M2 - (M, M) is a martingale. (M, M) is often abbreviated (M)=(M, M). Definition A.3.3. Similar to the previous definition, for M and N two different continuous martingales, there exists a unique continuous increas• ing process (M, N) = ~((M + N) - (M - N) vanishing at zero such that M N - (M, N) is a martingale. Remark A. 3. 3. This definition immediately follows from the previous defini• tion. Consider two different martingales M and N. By the previous definition, the process (M +N)2_(M + N) is a martingale, but so is (M _N)2_(M - N) and their difference 4MN - (M + N) + (M - N). Therefore, the cross• variation process is (M, N) = ~((M + N) - (M - N). If M and N are independent martingales, then (M, N) = 0. Definition A.3A. A continuous is a continuous adapted process X which has the decomposition X = Xo + M + A where M is a con• tinuous local martingale and A a non-decreasing, continuous, adapted process of finite variation. Xo is a :Fo-measurable random variable. Theorem A.3.2 (Martingale Representation). Let M be a continuous local martingale. For any adapted local martingale X, there is a predictable process

X t = Xo + lot

Proof. Cf. Revuz and Yor (1994), Chapter IV.

Corollary A.3.1. Let W t be a JRd-valued Brownian motion. If X t is a con• tinuous martingale in (£2)n with X O = 0, then there exists a unique pre• dictable process

X t = Xo + lot

Proof. Cf. Karatzas and Shreve (1991), Section 3.D.

Remark A.3.4. For X t is a local martingale, it is sufficient that

A.4 Brownian Motion

Definition A.4.1. A probability space (il,:7, (:7t)09~T' P) is given. Let

(Wt)tE[O,Tj = (Wl, ... , Wtd) be a JRd-valued continuous process adapted to the filtration :7t . W t is called a or a Brownian motion if, "It, s E [0, T] s.t. s < t, the process increments W t - Ws are independent and identically distributed. The distribution is normal with mean zero and covariance matrix (t - s) Vd.

Remark A.4.1. We usually define W ti and wI, i f. j, to be independent. This implies that a d-dimensional Brownian motion W = (WI, ... , W d) has cross-variation (Wi, Wj)t = Jijt where J is the Kronecker delta. Therefore, (Wi)t = t. The covariance matrix is then given by (t-s)Vd = (t-s)Id, where Id is the d x d identity matrix. However, sometimes we define W: and wI, i f. j to be correlated Brow• nian motion. In this case, (Wi, Wj)t = J~ p!j ds, where p;j is called the correlation coefficient between Brownian motions W: and wI. For constant correlation coefficients, (Wi, Wj)t = pijt.

Definition A.4.2. Given a filtered probability space (il,:7, (:7dO

X t = Xo + lot o:s ds + lot AS dWs , (A.I) where 0: : JRn x JR+ -> JRn and 0' : JRnxd x JR+ -> JRnxd. 0' is called volatility matrix. Sometimes we write (A.I) with sums, i.e., Vi E {I, ... , n},

t d t X; = X~ + { o:! ds + L 1O'!j dW1 Jo j 0

= Xo + lot o:! ds + lot a! . dWs '

The cross variation process is given by (Xi, Xj)t = J~ a! . O'~ ds, where denotes the inner product (dot product) of vectors O'i and O'j. The correlation coefficient is defined as O'i . O'j Pij = IIO'illllO'jil and is also a process. II . II denotes the Euclidian norm. 228 A. Useful Tools from Martingale Theory

Definition A.4.3. The representation of Ito processes is slightly different if we use correlated Brownian motion Bt as a building block. In that case we write

X t = Xo + lot Q s ds + lot Vs dBs, where Q : jRn X jR+ -> jRn and v : jRn X jR+ -> jRn. The equivalent notation with sums is X t = L~ Xi with

This notation relates to the previous definition of Ito processes by the equation

v; dB; = 110";11 dB; = 0"; . dWt , 'Vi E {l..n}, where 0"; and Wt are defined as in Definition A.4.2. Let \It denote the instantaneous covariance matrix. Each element of Vt is given by d(Xi' Xj)t Vij = dt = PijViVj . We also call Vt a covariance matrix if its elements are given by d(Xi' Xjk To determine the symmetrical covariance matrix Vt from the triangular volatility matrix O"t, we compute O"tO"i = Vt. Conversely, O"t can be recovered from Vt by a Cholesky decomposition if \It is positive definite. Conveniently, a valid covariance matrix is always positive definite.

Example A.4.1. Consider the jRn-valued process Xl = Xo +O"iB; for i E [1,2] with B; a jR-valued Brownian motion and constant covariance matrix

V = ( O"f PO"l::2). pO" 1 0"2 0"2

The covariance matrix is given by the fact that (Bl, B2)t/t = p. Applying the Cholesky decomposition, we obtain

0" = (;:2 Jl ~ p20"J .

This result can be easily verified by checking V = 0"0" T. Therefore, X t (Xl, xl) is equivalent to yt = (~l, ~2) such that

~l = xci + 0"1 Wl , ~2 = X~ + p0"2wl + J!=P20"2W?, for a jR2-valued Brownian motion with (WI, W2) = o. A.5 Stochastic Integration 229

Remark A.4.2. Whether correlated or independent Brownian motion is used depends on the application at hand. Since Brownian motion is always explic• itly defined as independent or correlated whenever used in the main body of this text, we usually do not use a distinct symbol for correlated Brownian motion.

A.5 Stochastic Integration

Definition A.5.l. For any (Ot)O~t~T = (Of, ... , Of) in ('}{2)d, we write the stochastic integral with respect to a square-integrable martingale M as

It (0) is a unique, square-integrable martingale with quadratic variation pro• cess (I(O»)t = J~ 1108112d(M)8' If M = W a Brownian motion, then (I(O»)t = J~ 110811 2ds. Remark A. 5.1. It can be shown that the stochastic integral is also defined for the wider class 0 E £2 in which case It is a local martingale. If (M) is not absolutely continuous, the martingale M has to be pro• gressively measurable. If (M) is absolutely continuous, it is sufficient if the integrand is adapted since adaptivity and continuity imply progressive mea• surability. For a rigorous construction of the stochastic integral, see Karatzas and Shreve (1991) or Revuz and Yor (1994).

Theorem A.5.l (Ito). Let the function f: jR+ X lRd ----t lR be of class C1,2. For a continuous d-dimensional semimartingale X t = (Xl, ... ,xt) with de• composition Xt = Xo + A1t + Ail we have

or in differential notation

Proof. For example, by an application of a Taylor series. Cf. Karatzas and Shreve (1991), p. 150-153, or Revuz and Yor (1994), p. 139, 145. 230 A. Useful Tools from Martingale Theory

Example A.S.l. Consider the special case of X t a JRn-valued Ito process X; = i i ",d s: . {I } 'th B . Xo + Jort as d s + ~i=l Jort asii dwis' lor 2 = , ... , n WI W t a rowman motion in JRd. a i is an JRd-valued vector. Assuming that a and a satisfy the same boundedness condition as in Definition A.4.2, then Ito's formula has the following terms:

dA~ = a~ dt, d dM; = a; . dWt = L a;i dWl, i d d(X\ Xi)t = a; . a1 dt = L a;ka1k dt. k This is equivalent to

1 T dfx = fxdX + "2 trace{aa fxx )dt, where a is in JRnxd, fx in JRn, and fxx in JRnxn.

Example A.S.2. Consider the special case of Xt a JR2-valued Ito process X: = Xu + J~ a~ ds + J~ a! dW;, for i = {I,2} with Wl a correlated Brownian motion in JR. Assuming the same boundedness conditions as above, Ito's formula in differential notation is

df(XI,X2) = fx,dxI + fX2dx2 1 1 + "2fXIXld(XI) + "2fX2X2d(X2) + f XIX2 d(XI,X2), where

Vi E {I,2}, where PXIX2 denotes the correlation coefficient between Wl and W?

Theorem A.5.2 (Integration by parts). For X t and yt two continuous semi martingales,

XtYt = XoYo + lot Xs dYs + lot Ys dXs + (X, Y)t, or in differential form,

d(Xtyt) = X t dyt + yt dXt + d(X, Yk If X = Y, A.5 Stochastic Integration 231

Proof. Analogous to Ito's formula by setting f(Xt, Yi) = XtYi. Cf. Revuz and Yor (1994), p. 138, for a different proof. Example A.S.3. A simple example of a special case is that of a standard Brownian motion Wt. Clearly, we have wl = 2 J~ WS dWs + t. Example A.S.4. Consider the more general special case of two real-valued Ito processes xl and xl, i.e., Xl = X~ + J~ o:! ds + J~ O"! dWs for i E {I,2} and Wt = (Wl, ... , Wl)· 0: and 0" are defined as in Definition A.4.2 and satisfy the same boundedness conditions. In this case their product is

xl xl = XJX~ + lot x; dX; + lot x; dX; + lot O"!O"~ ds.

1 2 . Tnld· (Xl x2) ft ~d 1 k 2 k It o"s and o"s are vectors 10 lAo , I.e., , t = Jo ~k O"s' O"s' d s. Th·IS resu is obtained in a straightforward way by recognizing the multiplication rules dt dt = 0, dt dWt = 0, dWtkdWtk = dt, and dW! dWtk = 0, j -=I k. The analogy to the derivation of Ito's formula for Ito processes is evident.

Example A.S.S. Let X t and Zt be two . Compute the process F(X, Z) = i. For Y = f(X) = ~ we have F = XY and can apply the integration by parts theorem such that F = J X dY + J Y dX + (X, Y). Since Y is a function of Z, we apply Ito's formula. Since f'(X) = --b and f"(X) = -b, we have 1 1 dY = - Z2 dZ + Z3 d(Z) , (A.2) by Ito's formula. Substituting for dY, we have

By expression (A.2), the covariation can be expressed as

(X, Y) = (X, -Z)1 = (X, - JZ2 1 dZ + JZ3 1 d(X)).

By the multiplication rule, (X, (X)) = 0, and therefore (X, ~) = --b(X, Z). Thus,

Ft = _tX = It -dX1 - It -dZX + It -d(Z)X - -(X 1 Z). Zt 0 Z 0 Z2 0 Z3 Z2' Dividing by i and differentiating gives 232 A. Useful Tools from Martingale Theory

Theorem A.5.3 (Doleans-Dade). Let Lt be a continuous local martingale defined on (n,:7, (:rt)O

d£t = £t dLt , then £t (L) is given by the local martingale

£t(L) = exp (Lt - ~(L)t) .

£t is called the Doleans-Dade exponential or the stochastic exponential. Corollary A.5.1. A similar result applies to a driftless Ita process. For a process 1t in (£2)d, define Lt = J~ 1t . dWt . W t is an ~d-valued Brownian motion. A process (£t)O~t~T satisfying the stochastic differential equation d£t = £Ot . dWt is given by the local martingale

£t = exp (lot 1s . dWs - ~ lot 1I1s11 2 dS) . (A.3)

Proof. Consider the transformation £ = eX. Define

x = lot 1s . dWs - ~ lot \bs11 2 ds.

The stochastic differential equation for £ then follows as a straightforward application of Ito's formula. The proof for Theorem A.5.3 is analogous.

Corollary A.5.2 (Novikov). If E[exp(~(L))] < 00 then £ in A.S.3 is a martingale. Proof. Cf. Revuz and Yor (1994), p. 318. Remark A.S.2. The same result also applies to £ if defined as in expression (A.3), i.e., if Lt = J~ 1t . dWt. It can easily be seen that the condition in that case is E[exp(~ J[ \bsI1 2ds)] < 00.

Theorem A.5.4 (FUbini). For a process h(s, t) E £2,1 such that

is continuous a.e., an interchange of Lebesque and Ita integrals is permissible. Therefore, we have the equalities

t2 lotl lot2 h(s, t) dWs dt = lo lotl h(s, t) dt dWs

lotl lot2/\t h(s, t)l{s~t}dWs dt = lot21tl h(s, t)1{s9}dt dWs'

Proof. Cf. Karatzas and Shreve (1991), p. 233, Protter (1990), p. 159. A.6 Change of Measure 233

A.6 Change of Measure

Theorem A.6.l. A probability triple (il, 9'", P) is given. Let ( be the sto• chastic exponential from A.S.3. If a new probability measure Q defined on (il, 9'") is given by Q= i(TdP, VA E 9'",

then Q is equivalent to P.

Proof. Cf. Revuz and Yor (1994), p. 325. Remark A. 6.1. If ( is defined as the stochastic exponential in expression (A.3), then the above statement is valid if and only if'"Y is in £2. Definition A.6.l. (T = ~ is called the Radon-Nikodym derivative or Ra• don- Nikodym density of Q with respect to P.

Corollary A.6.l. For arbitrary t, such that 0 ~ t ~ T, (t is given by

Proof. By the (local) martingale property of ( (cf. Theorems A.5.3 and A.5.2.). Corollary A.6.2. Assume that ( is a martingale and X an adapted process from £ 1 such that EQ[Xtl = Ep[Xt(tl· Proof. Substituting the definition of Q into In XdP gives In X(dP.

Corollary A.6.3 (Bayes). Similarly, for 0 ~ s ~ t ~ T, we have

Proof. For every A E 9'"s,

by the definition of conditional expectation. By Theorem A.6.1, we obtain

i Ep [Xt(t 19'"sl dP. (A.4)

The law of iterated expectations (cf. Williams (1991), p.88), another appli• cation of Theorem A.6.1, and the definition of conditional expectation give the following equalities: 234 A. Useful Tools from Martingale Theory

Theorem A.6.2 (Girsanov). Given that a probability space is endowed with a filtration, {n,~, (~t)O

£1t = Mt -lot D-1 d(M, D)

is a continuous local martingale under {n,~, (~t)O~t~T' Q). For N, another local martingale, we have (£1, N) = (£1, N) = (M, N). If Q is an equivalent probability measure defined by Q = J e{L) dP (cf. Theorem A.5.3), then we have

£1 = M - JD-1d(M,D) = M - (M,L).

Again, £1 is defined on the same filtration as M.

Corollary A.6.4. Given a filtered probability space, {n,~, (~t)O

dQ ( [T 1 [T ) dP = exp 10 "fs' dWs - 2" 10 111811 2 ds (A.5)

and W t = (Wl, ... , Wl) a d-dimensional Brownian motion. If we assume that Novikov's condition holds such that (t = E[~] is a martingale. Then the process Wt = Wt -lot "fsds

is a d-dimensional Brownian motion defined on (n,~, (J"t)O$t~T' Q). Remark A.6.2. It is easy to see that the corollary is a special case of the theorem. Wt corresponds to M and Lt = J~ "f8 ·dWs. For D = e(L), we obtain the SDE de = e"fdW. We start with d(M, D) = d(W, e) = dW de. Then, substituting for de and applying the multiplication rules gives d(M, D) = e"fdt. Simplification and integration give the result of the corollary. Since D = e(L), we can also use the second part of the theorem, which yields the desired result immediately by the multiplication rules. We can show that the equivalent probability measure is invariant with respect to time. (t satisfies the integral equation (t = 1 + J~ (s"fs . dWs' With Novikov's condition satisfied, we have E[(t] = 1, 'It E [0, T]. Since integral and expectation are equivalent, the equivalent probability measure can also be defined as Qt = E[l{A} (t], VA E ~t and for any t E [0, T]. By the martingale property, Qt{A) = Qs{A), for any s, t E [0, T]. A.6 Change of Measure 235

Proof. For a proof of Girsanov's theorem for semimartingales, refer to Revuz and Yor (1994), Chapter 8. For the special case of Ito processes, see Karatzas and Shreve (1991), Section 3.5.B. References

ABKEN, P. (1993): "Valuation of Default-Risky Interest-Rate Swaps," Advances in Futures and Options Research, 6, 93-116. ABRAMOWITZ, M., AND I. A. STEGUN (1972): Handbook of Mathematical Functions. Dover Publications, New York, N.Y. ACHARYA, S., AND J.-F. DREYFUS (1989): "Optimal Bank Reorganization Policies and the Pricing of Federal Deposit Insurance," Journal of Finance, 44(5),1313- 1333. ALTMAN, E. I. (1984): "A Further Investigation of the Bankruptcy Cost Question," Journal of Finance, 39(4), 1067-1089. ALTMAN, E. I. (1989): "Measuring Corporate Bond Mortality and Performance," Journal of Finance, 44(4), 909-921. ALTMAN, E. I., AND V. M. KISHORE (1996): "Almost Everything You Wanted to Know about Recoveries on Defaulted Bonds," Financial Analysts Journal, 52(6), 57-64. AMIN, K. I. (1991): "On the Computation of Continuous-Time Option Prices Us• ing Discrete Approximations," Journal of Financial and Quantitative Analysis, 26(4),477-495. AMIN, K. I., AND J. N. BODURTHA (1995): "Discrete-Time Valuation of American Options with Stochastic Interest Rates," Review of Financial Studies, 8(1), 193- 234. AMIN, K. I., AND R. A. JARROW (1992): "Pricing Options on Risky Assets in a Stochastic Interest Rate Economy," , 2(4), 217-237. AMMANN, M. (1998): "Pricing Derivative Credit Risk," Ph.D. thesis, University of St.Gallen. AMMANN, M. (1999): Pricing Derivative Credit Risk, vol. 470 of Lecture Notes in Economics and Mathematical Systems. Springer Verlag, Berlin, Heidelberg, New York. ANDERSON, R. W., AND S. SUNDARESAN (1996): "Design and Valuation of Debt Contracts," Review of Financial Studies, 9(1), 37-68. ANDERSON, R., AND S. SUNDARESAN (2000): "A Comparative Study of Structural Models of Corporate Bond Yields: An Exploratory Investigation," Journal of Banking and Finance, 24, 255-269. ANDRADE, G., AND S. N. KAPLAN (1998): "How Costly is Financial (Not Eco• nomic) Distress? Evidence from Highly Leveraged Transactions That Became Distressed," Journal of Finance, 53(5), 1443-1493. ARTZNER, P., AND F. DELBAEN (1995): "Default Risk Insurance and Incomplete Markets," Mathematical Finance, 5(3), 187-195. ASQUITH, P., R. GERTNER, AND D. SCHARFSTEIN (1994): "Anatomy of Financial Distress: An Examination of Junk-Bond Issuers," Quarterly Journal of Eco• nomics, 109, 625-658. 238 References

BACK, K., AND S. R. PLISKA (1991): "On the Fundamental Theorem of Asset Pric• ing with an Infinite State Space," Journal of Mathematical Economics, 20(1), 1-18. BHASIN, V. (1996): "On the Credit Risk of OTC Derivative Users," Discussion paper, Board of Governors of the Federal Reserve System. BLACK, F., AND J. C. COX (1976): "Valuing Corporate Securities: Some Effects of Bond Indenture Provisions," Journal of Finance, 31(2), 351-367. BLACK, F., E. DERMAN, AND W. Toy (1990): "A One-Factor Model of Inter• est Rates and Its Application to Treasury Bond Options," Financial Analysts Journal, 46(1), 33-39. BLACK, F., AND M. SCHOLES (1973): "The Valuation of Options and Corporate Liabilities," Journal of Political Economy, 81(3), 637-654. BOYLE, P. P. (1988): "A Lattice Framework for Option Pricing with Two State Variables," Journal of Financial and Quantitative Analysis, 23( 1), 1-12. BOYLE, P. P., J. EVNINE, AND S. GIBBS (1989): "Numerical Evaluation of Multi• variate Contingent Claims," Review of Financial Studies, 2(2), 241-250. BREMAUD, P. (1981): Point Processes and Queues: Martingale Dynamics. Springer Verlag, Berlin, Heidelberg, New York. BRENNAN, M. J., AND E. S. SCHWARTZ (1980): "Analyzing Convertible Bonds," Journal of Financial and Quantitative Analysis, 15(4), 907-929. BRENNER, M. (1990): "Stock Index Options," in Financial Options, ed. by S. Figlewski, W. L. Silber, and M. G. Subrahmanyam, chap. 5. Irwin, New York, N.Y. BRITISH BANKERS' ASSOCIATION (1996): BBA Credit Derivatives Report 1996. BRIYS, E., AND F. DE VARENNE (1997): "Valuing Risky Fixed Rate Debt: An Extension," Journal of Financial and Quantitative Analysis, 32(2), 239--248. BRONSTEIN, I. N., AND K. A. SEMENDJAEV (1995): Taschenbuch der Mathematik. Verlag Harri Deutsch, Thun, Frankfurt am Main, 2nd edn. BUONCUORE, A., A. NOBILE, AND L. RICCIARDI (1987): "A New Integral Equation for the Evaluation of First-Passage Time Probability Densities," Advances in Applied Probability, 19, 784-800. CARVERHILL, A. (1994): "When is the Short Rate Markovian?," Mathematical Fi• nance, 4(4), 305-312. CHANCE, D. M. (1990): "Default Risk and the Duration of Zero-Coupon Bonds," Journal of Finance, 45(1), 265-274. CHEW, L. (1992): "A Bit of a Jam," RISK, 5(8), 86-91. CHICAGO MERCANTILE EXCHANGE (1998): Quarterly Bankruptcy Index: Futures and Options. CHRISTOPEIT, N., AND M. MUSIELA (1994): "On the Existence and Characteri• zation of Arbitrage-Free Measures in Contingent Claim Valuation," Stochastic Analysis and Applications, 12(1), 41-63. CLAESSENS, S., AND G. PENNACCHI (1996): "Estimating the Likelihood of Mexican Default from the Market Prices of Brady Bonds," Journal of Financial and Quantitative Analysis, 31 (1), 109-126. COOPER, I., AND M. MARTIN (1996): "Default Risk and Derivative Securities," Applied Mathematical Finance, 3(1), 53-74. COOPER, I. A., AND A. S. MELLO (1991): "The Default Risk of Swaps," Journal of Finance, 46(2), 597-620. CORNELL, B., AND K. GREEN (1991): "The Investment Performance of Low-Grade Bond Funds," Journal of Finance, 46(1), 29-48. COSSIN, D. (1997): "Credit Risk Pricing: A Literature Survey," Finanzmarkt und Portfolio Management, 11(4),398-412. References 239

COSSIN, D., AND H. PIROTTE (1997): "Swap Credit Risk: An Empirical Investiga• tion on Transaction Data," Journal of Banking and Finance, 21, 1351-1373. COSSIN, D., AND H. PIROTTE (1998): "How Well Do Classical Credit Risk Pricing Models Fit Swap Transaction Data," European Financial Management, 4(1), 65-77. Cox, J. C., AND C.-F. HUANG (1989): "Option Pricing and Its Applications," in Theory of Valuation, ed. by S. Bhattacharya, and G. M. Constantinides. Rowman & Littlefield, Savage, M.D. Cox, J. C., J. E. INGERSOLL, AND S. A. Ross (1980): "An Analysis of Variable Rate Loan Contracts," Journal of Finance, 35(2), 389-403. Cox, J. C., J. E. INGERSOLL, AND S. A. Ross (1985): "A Theory of the Term Structure of Interest Rates," Econometrica, 36(4), 385-407. Cox, J. C., S. A. Ross, AND M. RUBINSTEIN (1979): "Option Pricing: A Simplified Approach," Journal of Financial Economics, 7(3), 229-263. DALANG, R. C., A. MORTON, AND W. WILLINGER (1990): "Equivalent Martingale Measures and No-Arbitrage in Stochastic Securities Market Models," Stochas• tics and Stochastics Reports, 29, 185-201. DAS, S. R. (1995): "Credit Risk Derivatives," Journal of Derivatives, 2(3), 7-23. DAS, S. R., AND R. K. SUNDARAM (1998): "A Direct Approach to Arbitrage-Free Pricing of Credit Derivatives," Discussion paper, National Bureau of Economic Research. DAS, S. R., AND P. TUFANO (1996): "Pricing Credit-Sensitive Debt when Interest Rates, Credit Ratings and Credit Spreads are Stochastic," Journal of Financial Engineering, 5(2), 161-198. DELBAEN, F. (1992): "Representing Martingale Measures When Asset Prices Are Continuous and Bounded," Mathematical Finance, 2(2), 107-130. DELBAEN, F., AND W. SCHACHERMAYER (1994a): "Arbitrage and Free Lunch with Bounded Risk for Unbounded Continuous Processes," Mathematical Finance, 4( 4), 343-348. DELBAEN, F., AND W. SCHACHERMAYER (1994b): "A General Version of the Fun• damental Theorem of Asset Pricing," Mathematische Annalen, 300(3), 463-520. DELBAEN, F., AND W. SCHACHERMAYER (1995): "The No-Arbitrage Property un• der a Change of Numeraire," Stochastics and Stochastics Reports, 53, 213-226. DHARAN, V. G. (1997): "Pricing Path-Dependent Interest Rate Contingent Claims Using a Lattice," Journal of Fixed Income, 6(4), 40-49. DREZNER, Z. (1978): "Computation ofthe Bivariate Normal Integral," Mathematics of Computation, 32, 277-279. DUFFEE, G. R. (1998): "The Relation Between Treasury Yields and Corporate Bond Yield Spreads," Journal of Finance, 53(6), 2225-224l. DUFFEE, G. R. (1999): "Estimating the Price of Default Risk," Review of Financial Studies, 12(1), 197-226. DUFFEE, G. R., AND C. ZHOU (1996): "Credit Derivatives in Banking: Useful Tools for Loan Risk Management?," Discussion paper, Federal Reserve Board, Washington, D.C. 2055l. DUFFIE, D. (1996): Dynamic Asset Pricing Theory. Princeton University Press, Princeton, N.J., 2nd edn. DUFFIE, D., AND M. HUANG (1996): "Swap Rates and Credit Quality," Journal of Finance, 51(3), 921-949. DUFFIE, D., M. SCHRODER, AND C. SKIADAS (1996): "Recursive Valuation of De• faultable Securities and the Timing of Resolution of Uncertainty," Annals of Applied Probability, 6(4), 1075-1090. 240 References

DUFFIE, D., AND K. J. SINGLETON (1995): "Modeling Term Structures of De• taultable Bonds," Discussion paper, Graduate School of Business, Stanford Uni• versity. DUFFIE, D., AND K. J. SINGLETON (1997): "An Econometric Model of the Term Structure ofinterest-Rate Swap Yields," Journal of Finance, 52(4}, 1287-132l. DUFFIE, D., AND K. J. SINGLETON (1999): "Modeling Term Structures of Default• able Bonds," Review of Financial Studies, 12(4}, 687-720. DUTT, J. E. (1975): "On Computing the Probability Integral of a General Multi• variate t," Biometrika, 62, 201-205. DYBVIG, P. H., AND C.-F. HUANG (1988): "Nonnegative Wealth, Absence of Ar• bitrage, and Feasible Consumption Plans," Review of Financial Studies, 1(4}, 377-40l. EBERHART, A., W. MOORE, AND R. ROENFELDT (1990): "Security Pricing and De• viations from the Absolute Priority Rule in Bankruptcy Proceedings," Journal of Finance, 45(5}, 1457-1469. FALLOON, W. (1995): "Who's Missing from the Picture?," RISK, 8(4}, 19-22. FIGLEWSKI, S. C. (1994): "The Birth of the AAA Derivatives Subsidiary," Journal of Derivatives, 1( 4}, 80-84. FISCHER, E. 0., AND A. GRUNBICHLER (1991): "Riskoangepasste Pramien fUr die Einlagensicherung in Deutschland: Eine empirische Studie," Zeitschrift fur be• triebswirtschaftliche Forschung, 43(9}, 747-758. FLESAKER, B., L. HUGHSTON, L. SCHREIBER, AND L. SPRUNG (1994): "Taking all the Credit," RISK, 7(9}, 104-108. FONS, J. S. (1994): "Using Default Rates to Model the Term Structure of Credit Risk," Financial Analysts Journal, 50(5}, 25-32. FRANKS, J., AND W. TOROUS (1994): "A Comparison of Financial Recontracting in Distressed Exchanges and Chapter 11 Reorganizations," Journal of Financial Economics, 35(3}, 349-370. FRANKS, J. R., AND W. N. TOROUS (1989): "An Empirical Investigation of U.S. Firms in Reorganization," Journal of Finance, 44(3}, 747-769. FREY, R., AND D. SOMMER (1998): "The Generalization of the Geske-Formula for Compound Options to Stochastic Interest Rates Is Not Trivial - A Note," Journal of Applied Probability, 35(2}, 501-509. GEANAKOPLOS, J. (1990): "An Introduction to General Equilibrium with Incom• plete Asset Markets," Journal of Mathematical Economics, 19(1}, 1-38. GEMAN, H., N. E. KAROUI, AND J.-C. ROCHET (1995): "Changes of Numeraire, Changes of Probability Measure and Option Pricing," Journal of Applied Prob• ability, 32, 443-458. GESKE, R. (1977): "The Valuation of Corporate Liabilities as Compound Options," Journal of Financial and Quantitative Analysis, 12(4}, 541-552. GESKE, R., AND H. E. JOHNSON (1984): "The Valuation of Corporate Liabilities as Compound Options: A Correction," Journal of Financial and Quantitative Analysis, 19(2}, 231-232. GILSON, S. (1997): "Transactions Costs and Capital Structure Choice: Evidence from Financially Distressed Firms," Journal of Finance, 52(1}, 161-197. GRANDELL, J. (1976): Doubly Stochastic Poisson Processes, vol. 529 of Lecture Notes in Mathematics. Springer Verlag, Berlin, Heidelberg, New York. GRUNBICHLER, A. (1990): "Zur Ermittlung risikoangepasster Versicheruhgspramien fur die betriebliche Altersvorsorge," Zeitschrift fur Betriebswirtschaft, 60(3}, 319-34l. HAND, J., R. HOLTHAUSEN, AND R. LEFTWICH (1992): "The Effect of Bond Rating Announcements on Bond and Stock Prices," Journal of Finance, 47(2}, 733-750. References 241

HARRISON, J., AND S. R. PLISKA (1981): "Martingales and Stochastic Integrals in the Theory of Continuous Trading," Stochastic Processes and Their Applica• tions, 11, 215-260. HARRISON, J., AND S. R. PLISKA (1983): "A Stochastic Calculus Model of Contin• uous Trading: Complete Markets," Stochastic Processes and Their Applications, 15, 313-316. HARRISON, J. M., AND D. M. KREPS (1979): "Martingales and Arbitrage in Mul• tiperiod Securities Markets," Journal of Economic Theory, 20, 381-408. HART, D. (1995): "Managing Credit and Market Risk as a Buyer of Credit Deriva• tives," Journal of Commercial Lending, 77(6), 38-43. HART, O. D. (1975): "On the Optimality of Equilibrium When the Market Struc• ture Is Incomplete," Journal of Economic Theory, 11, 418-443. HEATH, D., AND R. JARROW (1987): "Arbitrage, Continuous Trading, and Margin Requirements," Journal of Finance, 42(5), 1129-1142. HEATH, D., R. JARROW, AND A. MORTON (1990): "Bond Pricing and the Term Structure of Interest Rates: A Discrete Time Approximation," Journal of Fi• nancial and Quantitative Analysis, 25(4),419-440. HEATH, D., R. JARROW, AND A. MORTON (1992): "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valua• tion," Econometrica, 60(1), 77-105. HELWEGE, J. (1999): "How Long Do Junk Bonds Spend in Default?," Journal of Finance, 54(1), 341-357. HENN, M. (1997): "Valuation of Credit Risky Contingent Claims," Unpublished Dissertation, Universitiit St. Gallen. Ho, T. S., AND R. F. SINGER (1982): "Bond Indenture Provisions and the Risk of Corporate Debt," Journal of Financial Economics, 10(4),375-406. Ho, T. S., AND R. F. SINGER (1984): "The Value of Corporate Debt with a Sinking• Fund Provision," Journal of Business, 57(3), 315-336. Ho, T.-S., R. C. STAPLETON, AND M. G. SUBRAHMANYAM (1993): "Notes on the Valuation of American Options with Stochastic Interest Rates," Discussion paper, Stern School of Business, New York University. Ho, T.-S., R. C. STAPLETON, AND M. G. SUBRAHMANYAM (1995): "Multivariate Binomial Approximations for Asset Prices with Nonstationary Variance and Covariance Characteristics," Review of Financial Studies, 8(4),1125-1152. Ho, T. S. Y., AND S.-B. LEE (1986): "Term Structure Movements and Pricing Interest Rate Contingent Claims," Journal of Finance, 41(5), 1011-1029. HOWARD, K. (1995): "An Introduction to Credit Derivatives," Derivatives Quar• terly, 2(2), 28-37. HSUEH, L., AND P. CHANDY (1989): "An Examination of the Yield Spread between Insured and Uninsured Debt," Journal of Financial Research, 12, 235-344. HUBNER, G. (2001): "The Analytic Pricing of Asymmetric Defaultable Swaps," Journal of Banking and Finance, 25, 295-316. HULL, J. (1997): Options, Futures, and Other Derivatives. Prentice-Hall, Upper Saddle River, N.J., 3rd edn. HULL, J., AND A. WHITE (1990): "Pricing Interest Rate Derivative Securities," Review of Financial Studies, 3(4), 573-592. HULL, J., AND A. WHITE (1992): "The Price of Default," RISK, 5(8), 101-103. HULL, J., AND A. WHITE (1993a): "Efficient Procedures for Valuing European and American Path-Dependent Options," Journal of Derivatives, 1(1), 21-31. HULL, J., AND A. WHITE (1993b): "One-Factor Interest Rate Models and the Val• uation of Interest Rate Derivative Securities," Journal of Financial and Quan• titative Analysis, 28(2), 235-254. 242 References

HULL, J., AND A. WHITE (1995): "The Impact of Default Risk on the Prices of Options and Other Derivative Securities," Journal of Banking and Finance, 19(2), 299-322. HURLEY, W. J., AND L. D. JOHNSON (1996): "On the Pricing of Bond Default Risk," Journal of Portfolio Management, 22(2), 66-70. INGERSOLL, J. E. (1987): Theory of Financial Decision Making. Rowman & Little• field, Savage, M.D. INTERNATIONAL SWAPS AND DERIVATIVES ASSOCIATION (1988-1997): ISDA Market Survey. IRVING, R. (1996): "Credit Derivatives Come Good," RISK, 9(7), 22-26. JAMSHIDIAN, F. (1989): "An Exact Option Formula," Journal of Finance, 44(1), 205-209. JAMSHIDIAN, F. (1991a): "Bond and Options Evaluation in the Gaussian Interest Rate Model," Research in Finance, 9, 131-170. JAMSHIDIAN, F. (1991b): "Forward Induction and Construction of Yield Curve Diffusion Models," Journal of Fixed Income, 1(1), 62-74. JAMSHIDIAN, F. (1993): "Options and Futures Evaluation with Deterministic Volatilities," Mathematical Finance, 3(2), 149-159. JARROW, R. A., D. LANDO, AND S. M. TURNBULL (1997): "A Markov Model for the Term Structure of Credit Risk Spreads," Review of Financial Studies, 1O(2), 481-523. JARROW, R. A., AND D. B. MADAN (1991): "A Characterization of Complete Security Markets on a Brownian Filtration," Mathematical Finance, 1(3), 31- 43. JARROW, R. A., AND D. B. MADAN (1995): "Option Pricing Using the Term Struc• ture of Interest Rates to Hedge Systematic Discontinuitites in Asset Returns," Mathematical Finance, 5(4), 311-336. JARROW, R. A., AND S. M. TURNBULL (1992a): "Drawing the Analogy," RISK, 5(9), 63-70. JARROW, R. A., AND S. M. TURNBULL (1992b): "A Unified Approach for Pricing Contingent Claims on Multiple Term Structures," Discussion paper, Johnson Graduate School of Management, Cornell University. JARROW, R. A., AND S. M. TURNBULL (1995): "Pricing Derivatives on Financial Securities Subject to Credit Risk," Journal of Finance, 50(1), 53-85. JARROW, R. A., AND S. M. TURNBULL (1996a): "The Impact of Default Risk on Swap Rates and Swap Values," Discussion paper, Queen's University. JARROW, R. A., AND S. M. TURNBULL (1996b): "An Integrated Approach to the Hedging and Pricing of Eurodollar Derivatives," Discussion paper, John• son Graduate School of Management, Cornell University. JEFFREY, A. (1995): "Single Factor Heath-Jarrow-Morton Term Structure Mod• els Based on Markov Spot Interest Rate Dynamics," Journal of Financial and Quantitative Analysis, 30(4), 619-642. JENSEN, M. (1991): "Corporate Control and the Politics of Finance," Journal of Applied Corporate Finance, 4(2), 13-33. JENSEN, M., AND W. MECKLING (1976): "Theory of the Firm: Managerial Behav• ior, Agency Costs, and Ownership Structure," Journal of Financial Economics, 3(4), 305-360. JOHNSON, H., AND R. STULZ (1987): "The Pricing of Options with Default Risk," Journal of Finance, 42(2), 267-280. JOHNSON, R. (1967): "Term Structures of Corporate Bond Yields as a Function of Risk of Default," Journal of Finance, 22, 313-345. References 243

JONES, E. P., S. P. MASON, AND E. ROSENFELD (1984): "Contingent Claim Anal• ysis of Corporate Capital Structures: An Empirical Investigation," Journal of Finance, 39(3). KABANOV, Y., AND D. KRAMKOV (1994): "No-Arbitrage and Equivalent Martin• gale Measures: An Elementary Proof of the Harrison-Pliska Theorem," Theory of Probability and Its Applications, 39(3), 523-527. KABANOV, Y. M., AND D. O. KRAMKOV (1998): "Asymptotic Arbitrage in Large Financial Markets," Finance and Stochastics, 2(2), 143-172. KARATZAS, I., AND S. E. SHREVE (1991): Brownian Motion and Stochastic Calcu• lus. Springer Verlag, Berlin, Heidelberg, New York, 2nd edn. KARATZAS, I., AND S. E. SHREVE (1998): Methods of Mathematical Finance. Springer Verlag, Berlin, Heidelberg, New York. KAU, J., AND D. KEENAN (1995): "An Overview of the Option-Theoretic Pricing of Mortgages," Journal of Housing Research, 6, 217-244. KIJIMA, M., AND K. KOMORIBAYASHI (1998): "A Model for Valuing Credit Risk Derivatives," Journal of Derivatives, 6(1), 97-108. KIM, J., K. RAMASWAMY, AND S. SUNDARESAN (1993): "Does Default Risk in Coupons Affect the Valuation of Corporate Bonds?: A Contingent Claim Model," Financial Management, 22(3), 117-131. KLEIN, P. (1996): "Pricing Black-Scholes Options with Correlated Credit Risk," Journal of Banking and Finance, 20(7), 1211-1129. KLEIN, P., AND M. INGLIS (1999): "Valuation of European Options Subject to Financial Distress and Interest Rate Risk," Journal of Derivatives, 6(3), 44-56. KREPS, D. M. (1981): "Arbitrage and Equilibrium in Economies with Infinitely Many Commodities," Journal of Mathematical Economics, 8(1), 15-35. LANDO, D. (1997): "Modelling Bonds and Derivatives with Default Risk," in Math• ematics of Financial Derivatives, ed. by M. Dempster, and S. Pliska. Cambridge University Press, Cambridge, U.K. LANDO, D. (1998): "On Cox Processes and Credit Risky Securities," Review of Derivatives Research, 2(2/3), 99-120. LELAND, H. E. (1994a): "Bond Prices, Yield Spreads, and Optimal Capital Struc• ture with Default Risk," Discussion paper, University of California at Berkeley. LELAND, H. E. (1994b): "Corporate Debt Value, Bond Covenants, and Optimal Capital Structure," Journal of Finance, 49(4), 1213-1252. LELAND, H. E., AND K. B. TOFT (1996): "Optimal Capital Structure, Endogenous Bankruptcy, and the Term Structure of Credit Spreads," Journal of Finance, 51(3),987-1019. LI, A., P. RITCHKEN, AND L. SANKARASUBRAMANIAN (1995): "Lattice Models for Pricing American Interest Rate Claims," Journal of Finance, 50(2), 719-737. LI, H. (1998): "Pricing of Swaps with Default Risk," Review of Derivatives Re• search, 2(2/3), 231-250. LITTERMAN, R., AND T. IBEN (1991): "Corporate Bond Valuation and the Term Structure of Credit Spreads," Journal of Portfolio Management, 17(3), 52-64. LONGSTAFF, F. A., AND E. S. SCHWARTZ (1994): "A Simple Approach to Valuing Risky Fixed and Floating Rate Debt and Determining Swap Spreads," Discus• sion paper, Anderson Graduate School of Management, University of California at Los Angeles. LONGSTAFF, F. A., AND E. S. SCHWARTZ (1995a): "A Simple Approach to Valuing Risky Fixed and Floating Rate Debt," Journal of Finance, 50(3), 789-819. LONGSTAFF, F. A., AND E. S. SCHWARTZ (1995b): "Valuing Credit Derivatives," Journal of Fixed Income, 5(1), 6-12. MADAN, D. B., AND H. UNAL (1998): "Pricing the Risks of Default," Review of Derivatives Research, 2(2/3), 121-160. 244 References

MARGRABE, W. (1978): "The Value of an Option to Exchange One Asset for An• other," Journal of Finance, 33(1}, 177-186. MASON, S. P., AND S. BHATTACHARYA (1981): "Risky Debt, Jump Processes, and Safety Covenants," Journal of Financial Economics, 9(3}, 281-307. MELLA-BARRAL, P., AND W. PERRAUDIN (1997): "Strategic Debt Service," Journal of Finance, 52(2}, 531-556. MERTON, R. C. (1973): "Theory of Rational Option Pricing," Bell Journal of Eco• nomics an Management Science, 4, 141-183. MERTON, R. C. (1974): "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates," Journal of Finance, 2(2}, 449-470. MERTON, R. C. (1977): "An Analytic Derivation of the Cost of Deposit Insur• ance and Loan Guarantees: An Application of Modern Option Pricing Theory," Journal of Banking and Finance, 1(1}, 3-1l. MERTON, R. C. (1978): "On the Cost of Deposit Insurance When There are Surveil• lance Costs," Journal of Business, 51, 439-452. MUSIELA, M., AND M. RUTKOWSKI (1997): Arbitrage Pricing of Derivative Securi• ties: Theory and Applications. Springer Verlag, Berlin, Heidelberg, New York. MYERS, S. C. (1977): "Determinants of Corporate Borrowing," Journal of Finan• cial Economics, 5(2}, 147-175. NELSON, D. B., AND K. RAMASWAMY (1990): "Simple Binomial Processes as Dif• fusion Approximations in Financial Models," Review of Financial Studies, 3(3}, 393-430. NIELSEN, L. T., J. SAA-REQUEJO, AND P. SANTA-CLARA (1993): "Default Risk and Interest-Rate Risk: The Term Structure of Default Spreads," Discussion paper,INSEAD. NIELSEN, S. S., AND E. I. RONN (1997): "The Valuation of Default Risk in Cor• porate Bonds and Interest Rate Swaps," Advances in Futures and Options Re• search, 9,175-196. OFFICE OF THE COMPTROLLER OF THE CURRENCY (1997-2000): Quarterly Deriva• tives Fact Sheets. PEARSON, N., AND T.-S. SUN (1994): "Exploiting the Conditional Density in Esti• mating the Term Structure: An Application to the Cox, Ingersoll, Ross Model," Journal of Finance, 49, 1279-1304. PIERIDES, Y. A. (1997): "The Pricing of Credit Risk Derivatives," Journal of Eco• nomic Dynamics and Control, 21(1O}, 1479-161l. PITTS, C., AND M. SELBY (1983): "The Pricing of Corporate Debt: A Further Note," Journal of Finance, 38(4}, 1311-1313. PLISKA, S. R. (1997): Introduction to Mathematical Finance: Discrete Time Models. Blackwell Publishers, Malden, M.A. PRESS, W. H., W. T. VETTERLING, S. A. TEUKOLSKY, AND B. P. FLANNERY (1992): Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge, U.K., 2nd edn. PROTTER, P. (1990): Stochastic Integration and Differential Equations. Springer Verlag, Berlin, Heidelberg, New York. RAMASWAMY, K., AND S. SUNDARESAN (1986): "The Valuation of Floating-Rate Instruments: Theory and Evidence," Journal of Financial Economics, 17(2}, 251-272. REVUZ, D., AND M. YOR (1994): Continuous Martingales and Brownian Motion. Springer Verlag, Berlin, Heidelberg, New York, 2nd edn. RITCHKEN, P., AND L. SANKARASUBRAMANIAN (1995): "Volatility Structures of Forward Rates and the Dynamics of the Term Structure," Mathematical Fi• nance, 5(1}, 55-72. References 245

RODRIGUEZ, R. J. (1988): "Default Risk, Yield Spreads, and Time to Maturity," Journal of Financial and Quantitative Analysis, 23(1), 111-117. ROGERS, L. (1994): "Equivalent Martingale Measures and No-Arbitrage," Stochas• tics and Stochastics Reports, 51, 41-49. RONN, E. I., AND A. K. VERMA (1986): "Pricing Risk-Adjusted Deposit Insurance: An Option-Based Model," Journal of Finance, 41(4), 871-895. RUBINSTEIN, M. (1991): "Somewhere over the Rainbow," RISK, 4(10), 63-66. RUBINSTEIN, M. (1994): "Return to Oz," RISK, 7(11),67-71. SARIG, 0., AND A. WARGA (1989): "Some Empirical Estimates of the Risk Struc• ture of Interest Rates," Journal of Finance, 44(5), 1351-1360. SCHACHERMAYER, W. (1994): "Martingale Measures for Discrete-Time Processes with Infinite Horizon," Mathematical Finance, 4(1), 25-55. SCHICH, S. T. (1997): "An Option-Pricing Approach to the Costs of Export Credit Insurance," Geneva Papers on Risk and Insurance Theory, 22(1), 43-58. SCHONBUCHER, P. J. (1998): "Term Structure Modelling of Defaultable Bonds," Review of Derivatives Research, 2(2/3), 161-192. SELBY, M., AND S. HODGES (1987): "On the Evaluation of Compound Options," Management Science, 33(3), 347-355. SHIMKO, D. C., N. TEJIMA, AND D. R. V. DEVENTER (1993): "The Pricing of Risky Debt When Interest Rates Are Stochastic," Journal of Fixed Income, 3(2), 58-65. SHLEIFER, A., AND R. VISHNY (1992): "Liquidation Values and Debt Capacity: A Market Equilibrium Approach," Journal of Finance, 47(4), 1343-1366. SMITHSON, C. (1995): "Credit Derivatives," RISK, 8(12), 38-39. SOLNIK, B. (1990): "Swap Pricing and Default Risk: A Note," Journal of Interna• tional Financial Management and Accounting, 2(1), 79-91. SORENSEN, E. H., AND T. F. BOLLIER (1994): "Pricing Swap Default Risk," Fi• nancial Analysts Journal, 50(3), 23-33. SOSIN, H. B. (1980): "On the Valuation of Federal Loan Guarantees to Corpora• tions," Journal of Finance, 35(5), 1209-1221. SUN, T. S., S. SURESH, AND W. CHING (1993): "Interest Rate Swaps: An Empirical Investigation," Journal of Financial Economics, 34(1), 77-99. SUNDARESAN, S. (1991): "Valuation of Swaps," in Recent Developments in Inter• national Banking and Finance, ed. by S. J. Khoury, chap. 12. Elsevier (North• Holland). TAQQU, M. S., AND W. WILLINGER (1987): "The Analysis of Finite Security Mar• kets Using Martingales," Advances in Applied Probability, 19, 1-25. TIAN, Y. (1992): "A Simplified Binomial Approach to the Pricing of Interest Rate Contingent Claims," Journal of Financial Engineering, 1(1), 14-37. TITMAN, S., AND W. TORous (1989): "Valuing Commercial Mortgages: An Empir• ical Investigation of the Contingent Claims Approach to Pricing Risky Debt," Journal of Finance, 44(2), 345-373. VASICEK, O. (1977): "An Equilibrium Characterization of the Term Structure," Journal of Financial Economics, 5(2), 177-188. WEINSTEIN, M. (1983): "Bond Systematic Risk and the Option Pricing Model," Journal of Finance, 38(5), 1415-1429. WEISS, L. (1990): "Bankruptcy Resolution: Direct Costs and Violation of Priority of Claims," Journal of Financial Economics, 27(2), 285-314. WILLIAMS, D. (1991): Probability with Martingales. Cambridge University Press, Cambridge, U.K. WILMOTT, P., S. HOWISON, AND J. DEWYNNE (1995): The Mathematics of Finan• cial Derivatives. Cambridge University Press, Cambridge, U.K. 246 References

ZHENG, C. (2000): "Understanding the Default-Implied Volatility for Credit Spreads," Journal of Derivatives, 7(4), 67-77. ZHOU, C. (1997): "A Jump-Diffusion Approach to Modeling Credit Risk and Valu• ing Defaultable Securities," Discussion paper, Federal Reserve Board, Washing• ton, D.C. ZIMMERMANN, H. (1998): State-Preference Theorie und Asset Pricing: Eine Einfuhrung. Physica-Springer Verlag, Heidelberg. List of Figures

1.1 Outstanding OTC interest rate options...... 3 1.2 Average cumulated default rates for U.S. investment-grade bonds. 5 1.3 Average cumulated default rates for U.S. speculative-grade bonds. 5 1.4 Classification of credit risk models...... 10

3.1 Lattice with bankruptcy process...... 59

4.1 Implied interest rate and bond volatility ...... 105 4.2 Implied bond prices ...... 107 4.3 Implied term structures ...... 107 4.4 Implied yield spreads ...... 108 4.5 Price reductions for different maturities ...... 109

5.1 Bivariate lattice ...... 150 5.2 Bankruptcy process with stochastic recovery rate ...... 154 5.3 Multidimensional lattice with bankruptcy process ...... 155

6.1 Implied credit-risky term structure ...... 190 6.2 Forward yield spread ...... 191 6.3 Absolute forward bond spread ...... 191 6.4 Implied risky term structures for different correlations ...... 193 List of Tables

1.1 U.S. corporate bond yield spreads 1985-1995 ...... 2 1.2 Credit derivatives use of U.S. commercial banks ...... 7

3.1 First passage time models ...... 55

4.1 Price reductions for different psv ...... 102 4.2 Price reductions for different Pv D ...... 102 4.3 Price reductions for various correlations ...... 102 4.4 Price reductions for different Pv p ...... 108 4.5 Price reductions for different Pvp, Psv, Psp ...... 110 4.6 Deviations from forward bond price ...... 112 4.7 Deviations from forward stock price ...... 112

5.1 Price reductions for different correlations ...... 165 5.2 Price reductions for different correlations ...... 165 5.3 Price reductions for different correlations ...... 166 5.4 Price reductions with default cost ...... 167 5.5 Term structure data ...... 168 5.6 Prices of vulnerable bond options ...... 169 5.7 Implied default probabilities ...... 170 5.8 Prices of options on credit-risky bond ...... 171 5.9 Price reductions for various Pv p under HJM ...... 172

6.1 Prices of credit spread options on yield spread ...... 189 6.2 Yield spread to bond spread conversion ...... 192 6.3 Prices of credit spread options on bond spread ...... 192 6.4 Prices of credit spread options with Vasicek interest rates ...... 194 Index

a-algebra, 223 - corporate, 4, 144 - coupon, 51 a.e., 15, 224 - credit-risky, 6 a.s., 15, 224 - Treasury, 6 arbitrage, 13, 15, 17, 19, 20 - zero-coupon, 33, 145 - approximate, 25 bond spread, 190 asymptotic, 25 Borel, 223 - first type, 15 Borel algebra, 223 - opportunity, 19 Brownian motion, 227-229 - second type, 15 - correlated, 227, 228 arrival rate, 141 - geometric, 78, 85 Arrow-Debreu price, 159 - independent, 227 asset, 48, 78 - deflating, 78 call-put parity, see put-call parity - riskless, 14 Cholesky decomposition, 228 - traded, 86 claim, 19 asset pricing - asset, 89 - fundamental theorem, 17, 18, 25 - attainable, 15 assets-to-debt ratio, 50 - contingent, 15, 19 assets-to-deposits ratio, 71 - credit-risky, 78 - debt, 89 bankcruptcy - deterministic, 84 - cost, 50 - European, 77 bankruptcy, 11, 142 - junior, 69 - boundary, 53 - payoff,78 - cost, 52 - replicate, 17 - endogenous, 52 - senior, 69 - index, 175 - seniority, 78 - premature, 53 clearing house, 4 process, 58, 153 collateral, 3, 6 - strict priority, 52 computational cost, 171 - time, 145 contingent claim, see claim bankruptcy cost, 148 convenience yield, 6 basket, 89 correlation, 101, 227 Bayes rule, 233 correlation coefficient, 227 benchmark, 23 counterparty default risk, see credit binomial tree, 149 risk Black-Cox, 53 counterparty risk, 66 Black-Scholes, 26-30 covariance, 227 bond, 4 covariance matrix, 228 - Brady, 51 credit derivative, 7, 70, 175-216 - convertible, 54 - binary, 177 252 Index

~ commercial bank, 7 diffusion, 227 compound approach, 183 distance to default, 50 counter party risk, 205~215 distribution ~ credit spread, 178, 194 ~ beta, 65 ~ digital, 177 ~ exponential, 59, 61 ~ exchange-traded, 175 ~ lognormal, 42, 151 ~ first type, 176 ~ risk-neutral, 42 ~ knock-out, 178 Doleans-Dade exponential, 232 ~ pure, 176, 177 Doob-Meyer decomposition, 226 rating-based, 178 dot product, see inner product second type, 178 drift, 151 credit quality, see credit risk duality, 16 credit rating, 1 Duffie-Singleton, 65, 68 credit risk, 1, 205 duration, 51 ~ change, 71 forward contract, 110 equilibrium, 17, 69 swap, 176 equivalent, see measure, equivalent ~ systematic, 48 Euclidian norm, 85, 227 ~ two-sided, 69, 100 Eurocurrency, 68 credit risk model Eurodollar, 175 exchange option, see option, exchange ~ calibration, 109 ~ firm value, 48~53 exposure ~ credit, 4 ~ first passage time, 53~58, 83 ~ hybrid, 141 Farka's lemma, 16 ~ intensity, 58~66 Feynman-Kac, 30 ~ traditional, 48 filtered probability space, see probabil- credit spread, 73 ity space, filtered credit-linked notes, 178 filtration, 14, 224 cross-variation, see variation, quadratic ~ augmented, 224 ~ natural, 224 debt forward, 4, 70, 99, 110 ~ junior, 52, 54 ~ credit spread, 178 ~ senior, 54 forward contract ~ strategic service, 52 ~ counterparty risk, 110 ~ variable-rate, 51 ~ vulnerable, 99 debt insurance, 70 forward measure, 38~41, 91 debt-to-asset ratio, 146 forward price, 94, 100 default, 11 ~ credit-adjusted, 111 ~ intensity, 59 forward rate, 33, 90 ~ option, 176 ~ evolution, 43 ~ process, 58 Fourier transforms, 82 ~ rates, 4, 5 free lunch, 25 ~ swap, 176 friction, 14 ~ time, 58 Fubini,232 default threshold, 55 function default boundary, 83 ~ Borel, 22:3 default risk, see credit risk ~ measurable, 223 default-free, 11 deflator, 78 gains process, 14 deposit insurance, 71, 175, 179 generator matrix, 62 derivative geometric Brownian motion, 41 ~ default-free, 6 Geske, 51 ~ vulnerable, 66, 77 Girsanov's theorem, 234 Index 253

Green's function, 159 local, 225 - right-continuous, 226 hazard rate, 59, 141 - square-integrable, 229 Heath-Jarrow-Morton, 33-38, 43-45 martingale representation, 20, 226 measure, 223 index units, 89 - absolutely continuous, 224 indicator function, 141, 142 - change, 233 induction - empirical, 48 - backward, 160 - equivalent, 15, 20, 224, 233 - forward, 159 - existence, 22, 25 inner product, 14, 227 - forward-neutral, 38, 91 integrable, 225 - invariant, 22 - square, 225 - martingale, 13, 15, 20, 78 integration by parts, 230 - probability, 14 intensity, 141 risk-neutral, 25 interest rate - unique, 13 - Gaussian, 90 measure space, 223 - stochastic, 157, 168, 187, 193 Merton, 47, 48, 67, 77 Ito's formula, 229 money market account, 14, 19, 33 moral hazard, 52 Jarrow-Lando-Turnbull, 62 multiplication rules, 231 Jarrow-Turnbull, 58, 68 Johnson-Stulz, 66 netting, 69 non-linear pricing, 146 Klein, 80 normal distribution, 28 Kronecker delta, 227 - multi-variate, 51 Novikov,232 Lando, 65 numeraire, 13, 14, 20, 23, 88 lattice, 58, 149 - change, 38 - multi-variate, 149 - node, 58 OCC,7 - recombine, 158 optimal exercise, 66 liability, 4, 48, 78 option - deterministic, 79 - American, 66, 153, 160 - stochastic, 85, 96 - bond, 95, 99 loan guarantee, 71, 175 compound, 51, 72, 163, 183 Longstaff-Schwartz, 54 - credit spread, 178 credit-risky, 80 Madan-Unal, 65 - credit-risky bond, 162, 178 mapping default put, 176 - additive, 223 - European, 170 margin, 20 - exchange, 30, 90, 198-215 Margrabe, 30-33, 77, 99 - put, 71 market, 15 - vulnerable, 66 - arbitrage-free, 15 OTC, 2, 175 - complete, 13, 15 - finite, 13 PDE, 29, 82 frictionless, 20 price - incomplete, 13 - deflated, 14, 22 -- viable, 17 - relative, 14, 22 market price of risk, 22 price system, 15, 17 Markov, 58, 152, 157, 172 principal, 146 -- chain, 62 probability martingale, 225 - survival, 63, 144 254 Index

probability measure, see measure, 224 SDE,22 probability space, 224 SEC, 6 - filtered, 224 self-financing, 19 process, 223, 224 semimartingale, 141, 226, 229, 231, 234 - Ito, 227 separating hyperplanes, 16 - adapted, 14, 224 set - bankruptc~ 58, 142 - Borel,223 - binomial, 41 - open, 223 - bond price, 37 short rate, 34 - compensated, 141 - adjusted, 65 - convergence, 41 - time-homogenous, 104 - Cox, 142 - Vasicek, 103 - decomposition, 226 space - discrete, 224 - Euclidian, 223 - discrete approximation, 41 - filtered, 14 - doubly stochastic Poisson, 142 - infinite, 18 - driftless, 40 - measurable, 223 - jump, 141 - probability, 14 - jump-diffusion, 57 - sample, 13, 14, 223 - point, 142 - state, 13 - Poisson, 141 spread derivative, 70, 71 - predictable, 14, 224 state - sample path, 224 - absorbing, 63 - square-root, 56 state claim, 159 - trajectory, 224 state variable, 59, 141, 144 process class, 225 stochastic exponential, see Doh~ans- put-call parity, 84, 163 Dade exponential stochastic integral, 229 Radon-Nikodym derivative, 233 , 17, 55, 141 random variable, 223 structural models, see credit risk - binomial, 151 models, firm value - bivariate, 81 submartingale, 225 - normal, 81, 151 - right-continuous, 226 supermartingale, 225 , 151 swap,4,68 rating, 62 - credit risk, 176 - agency, 64 - currency, 69 recovery rate, 53, 58, 78, 96, 177 - interest rate, 2 - exogenous, 62 - settlement, 69 fixed, 65, 83 - total return, 177 - stochastic, 153 - zero, 67 Taylor, 229 reduced-form models, see credit risk TED spread, 176 models, intensity term structure, 105, 106 reference asset, 176 - credit-risky, 106, 161 replication - defaultable, 161 - unique, 15 - hump-shaped, 106 risk premia, 63 - risk-free, 160 risk-neutral, 48 - volatility, 162 riskless, 11, 19 time horizon - infinite, 18 safety covenant, 53 trading strategy, 13, 14, 19 savings account, see money market - replicating, 15 account - self-financing, 13-15 Index 255

- tame, 20 variation transition matrix, 62 - quadratic, 226 Treasury, 6 Vasicek, 55, 103 tree volatility function, 91 - binary, 158, 172 volatility matrix, 227 - path-dependent, 172 vulnerable option, see option, - recombine, 152 vulnerable under-investment, 52 warrant, 166 wealth process, 14 value Wiener process, see Brownian motion - negative, 4 - notional, 3, 7 yield spread, 2, 4, 69, 189, 190