Pricing and Portfolio Optimization Analysis

in Defaultable Regime-Switching Markets

Agostino Capponi∗ Jose E. Figueroa-Lopez† Jeffrey Nisen‡

Abstract

We analyze pricing and portfolio optimization problems in defaultable regime switching markets driven by a

underlying continuous-time Markov process. We contribute to both of these problems by obtaining new representa-

tions of option prices and optimal portfolio strategies under regime-switching. Using our option price representation,

we develop a novel efficient method to price claims which may depend on the full path of the underlying Markov

chain. This is done via a change of probability measure and a short-time asymptotic expansion of the claim’ s price

in terms of the Laplace transforms of the symmetric Dirichlet distribution. The proposed approach is applied to

price not only simple European claims such as defaultable bonds, but also a new type of path-dependent claims

that we term self-decomposable as well as the important class of vulnerable call and put options on a stock. In

the portfolio optimization context, we obtain explicit constructions of value functions and investment strategies

for investors with Constant Relative Risk Aversion (CRRA) utilities, built on the Hamilton-Jacobi-Bellman (HJB)

framework developed in Capponi and Figueroa-Lopez (2011). We give a precise characterization of the investment

strategies in terms of corporate bond returns, forward rates, and expected recovery at default, and illustrate the

dependence of the optimal strategies on time, losses given default, and risk aversion level of the investor through a

detailed economical analysis.

AMS 2000 subject classifications: 93E20, 60J20.

Keywords and phrases: Credit Risk, Regime Switching Models, Option Pricing, Portfolio Optimization, Hamilton-

Jacobi-Bellman framework.

1 Introduction

Regime switching models are aimed at capturing the idea that the macro-economy is subject to regular and unpre-

dictable changes, which in turn affect the price of financial securities. For example, structural changes of macro-

economic conditions, such as inflation and recession may induce changes in the stock returns, or in the term structure

of interest rates, and similarly, periods of high market turbulence and liquidity crunches may increase the default risk

of financial institutions. This has been empirically verified in the stock market, as stated by Ang and Bekaert (2002-b),

who found the existence of two regimes characterized by different levels of volatility. Similar findings have also been

documented in the bond market (see Ang and Bekaert (2002-a), Ang and Bekaert (2002-c), and Dai et al. (2007)).

Most recently, in the credit market, Giesecke et al. (2011) found the existence of three regimes, associated with high,

middle, and low default risk, via an empirical analysis of the corporate bond market over the course of the last 150

years.

These considerations have led many researchers to use regime switching models for asset pricing and, more recently,

for portfolio optimization problems. In the pricing framework, most of the literature has focused on payoffs of European

type with the exception of few works (see, e.g., Guo and Zhang (2004) and references therein). In the context of stock

∗School of Industrial Engineering, Purdue University, West Lafayette, IN, 47907, USA (capponi@purdue.edu).†Department of Statistics, Purdue University, West Lafayette, IN, 47907, USA (figueroa@purdue.edu).‡Department of Statistics, Purdue University, West Lafayette, IN, 47907, USA (jnisen@purdue.edu).

1

options, Guo (2001) considered a market consisting of two regimes, and provided a semi-analytical formula for the

option price, based on occupation time densities. Buffington and Elliott (2002) generalized the method to the case

of multiple regimes, under the assumption that the generator of the Markov chain is time homogenous, and derived

expressions for the price of European claims. Yao et al. (2006) developed a fixed point iteration scheme to recover

prices of European options in a regime switching model, assuming time homogenous generators. Regime switching

models for default-free interest rates derivatives have been studied by Elliott and Wilson (2007) and Elliott and Siu

(2008), and by Kuen Siu (2010) in the case where the counterparty is defaultable. In the context of credit risk, regime

switching models have been successfully employed by Bielecki et al. (2008-a) and Bielecki et al. (2008-b), who analyzed

pricing and hedging of a defaultable game option under a Markov modulated default intensity framework. Norberg

(2000) studied no-arbitrage pricing of derivatives on the regime parameters with an eye towards computation.

A most recent branch of literature has considered regime switching models for dynamic portfolio optimization.

Sotomayor and Cadenillas (2009) considered expected utility maximization from consumption and terminal wealth

over an infinite horizon in a market consisting of stocks and a money market account, where the Markov chain is

observable. Taksar and Zeng (2010) considered a similar model, but assumed that the Markov chain is hidden. Zhang

et al. (2010) solved the portfolio selection problem after completing the continuous-time Markovian regime switching

model with jump securities. Nagai and Runngaldier (2008) studied a finite horizon portfolio optimization problem for a

risk averse investor with power utility, assuming that the coefficients of the risky assets in the economy are nonlinearly

dependent on hidden Markov-chain modulated economic factors.

In this paper, we consider a macro-economy with finitely many observable economical regimes, containing state

information regarding the equity (drift and volatility), credit (hazard intensity and loss given default), and interest rate

market (short rate). We assume three liquidly traded securities, namely, a money market account, a risky (default-free)

stock, and a defaultable bond. The dynamics of the securities are assumed to depend on the macro-economic regimes,

which are modeled using a finite state continuous time Markov process. We follow the reduced form approach to

credit risk, and model the default event using a doubly stochastic framework. Classical (yet fundamental) problems of

mathematical finance, such as contingent claim pricing and portfolio optimization, are still problematic in the context

of regime-switching models, in part, due to the lack of easily computable expression for option prices and optimal

portfolios. More explicit characterizations of the latter two quantities that allow efficient computational method are

of great need in the field. Our paper contributes to both of the previously mentioned fundamental problems, which,

as we will see, are nevertheless linked by the necessity of developing an efficient computational method for pricing

defaultable bonds. The latter are needed for the numerical computation of portfolio optimization strategies. We now

proceed to explicitly describe our contributions to both problems.

In the pricing space, we develop a novel algorithm for valuing claims in a regime switching model consisting of an

arbitrary finite number of regimes, whose dynamics is governed by a possibly time varying generator. Our algorithm

allows pricing a class of path dependent financial instruments, which we refer to as self-decomposable claims, i.e. claims

whose payoff only depends on the regime of the underlying Markov chain, and can be decomposed in terms of payoffs of

shorter maturity claims, of possibly different type (see Section 3 for the precise statement). Such a class encompasses

the most basic instruments, such as bonds, whose price may also be recovered from no-arbitrage arguments via the

solution of a coupled system of ordinary differential equations (ODE’s), but it also includes more exotic instruments,

where prices can only be recovered via Monte-Carlo methods. To this purpose, we demonstrate our algorithm on

barrier options on volatility, which turn out to be self-decomposable in terms of shorter maturity barrier options,

and bond prices. Notice that in our macro-economic model, regime parameters, such as short rate, volatility, and

default intensity may be seen as proxies for interest rate, volatility, and credit spread indices, and therefore options on

these underlying financial measurements provide means for the investor to hedge against interest rate risk, market and

default risk, in different economic regimes.

Our methodology first exploits a change of measure technique, transforming the risk-neutral probability measure

into an equivalent measure under which the Markov process becomes “homogenous” in that the transition intensities

and probabilities are constant. We then express the price of the claim in terms of a series expansion of Laplace

2

transforms of the symmetric Dirichlet distribution. As the methodology performs a decomposition of the claim price

in terms of shorter dated claims, it introduces an approximation error, which, however, can be controlled through an

explicit upper bound which we provide. Our algorithm is computationally fast and, even for simple claims such as

bonds, is able to achieve a high level of accuracy in the price, within a time complexity which compares favorably

with standard ODE methods. As a far reaching application of our method, we also propose a new method to price

vulnerable call/put options on the stock (where there is an additional risk factor represented by the Brownian motion),

a subject which has received significant attention in the literature, as documented above.

In the portfolio optimization space, we employ the novel HJB framework developed in Capponi and Figueroa-Lopez

(2011), and develop a detailed numerical and economic analysis of value functions and investment strategies in a

defaultable regime switching market, populated by a representative CRRA investor facing both default and regime

switching risk. This extends the current literature on regime switching portfolio optimization in that our framework is

able to capture not only the modes experienced by the stock and default-free bond market, but also those experienced

by the credit market. This is highly relevant in today’s financial markets, where defaultable instruments have become

increasingly attractive to investors, as they are able to provide higher leverage and risk-return profiles. Few exceptions,

including Bielecki and Jang (2006), Callegaro et al. (2010), Lakner and Liang (2008), and Jiao and Pham (2010),

include defaultable instruments in the portfolio optimization framework, but they assume that the uncertainty in the

asset price dynamics is governed by a continuous process, where the unique jump leads to default.

We show that the optimal bond investment strategy and pre-default value function can be uniquely recovered as

the solution of a coupled system composed by ordinary differential equations and nonlinear equations. Under mild

assumptions, we provide conditions guaranteeing local existence and uniqueness of the solution of the coupled system

and show numerically, via a fixed point algorithm, that global convergence is typically achieved. Interestingly, in a

different context of liquidity risk, where investors can only trade in stocks at Poisson random times, Pham and Tankov

(2009) also find that the optimal control problem leads to solving a coupled system of integro-partial differential

equations. We also provide necessary and sufficient conditions under which a CRRA investor goes long or short in

the defaultable security, and show that these depend on the interplay between corporate bond returns, instantaneous

forward rate of the defaultable bond, and expected recovery (the precise statement is given in Section 4.2.3). We

numerically illustrate how the strategy of the investor behaves as a function of time, risk aversion level of the investor,

and loss experienced at default, under a meaningful “realistic” economic scenario.

The rest of the paper is organized as follows. Section 2 sets up the defaultable regime switching model. Section

3 presents a novel efficient method for pricing claims in the regime switching model. Section 4 studies the portfolio

optimization problem for a CRRA investor and characterizes the “directionality” of the bond investment strategy.

Section 5 performs comparative statics on the defaultable bond investment strategy, showing its dependence on time,

losses and risk aversion level of the investor. Section 6 concludes the paper. Proofs and numerical details are relegated

to the Appendix.

2 The defaultable regime-switching model

We consider a market consisting of a risk-free asset, a risky (default-free) asset, and a defaultable bond with respective

price processes Btt≥0, Stt≥0, and p(t, T )0≤t≤T defined on a complete probability space (Ω,G,G,P). Here, Pdenotes the real world or historical probability measure and G := (Gt) is an enlarged filtration given by Gt := Ft ∨Ht,where F := Ftt≥0 represents the reference filtration and Ht = σ(H(u) : u ≤ t) is the filtration generated by an

exogenous default process H(t) := 1τ≤t, after completion and regularization on the right (see Belanger et al. (2004)

for details). We assume the canonical construction of the default time τ in terms of a given hazard process htt≥0, so

that

τ := inft ∈ R+ :

∫ t

0

hudu ≥ χ, (1)

3

where χ is an exponential random variable defined on the probability space (Ω,G,P) and independent of F. In that

case, it follows that

ξPt := H(t)−∫ t

0

(1−H(u−))hudu (2)

is a G-martingale under P (see Bielecki and Rutkowski (2001), Section 6.5).

We place ourselves in an regime-switching market model. More specifically, we define an F-adapted continuous-time

Markov process Xtt≥0 with finite state space e1, e2, . . . , eN, where hereafter ei = (0, ..., 1, ...0)′ ∈ RN and ′ denotes

the transpose. Throughout, pi,j(t, s) := P(Xs = j|Xt = i) for t ≤ s represents the transition probabilities of X and

A(t) := [ai,j(t)]i,j=1,2,...,N denotes the generator, defined by

ai,j(t) = limh→0

pi,j(t, t+ h)

h, (i 6= j), ai,i(t) := −

∑j 6=i

ai,j(t). (3)

The following semi-martingale representation is well known (cf. Elliott et al. (1994)):

Xt = X0 +

∫ t

0

A′(s)Xsds+MP(t), (4)

where MP(t) = (MP1 (t), . . . ,MP

N (t))′ is a RN -valued P-martingale process. The following terminology will also be

needed:

Ct :=

N∑i=1

i1Xt=ei. (5)

We consider three market instruments, whose dynamics are driven by Xt. We have a risk-free asset Bt with

dynamics

dBt = rtBtdt, (6)

where rt takes a constant value ri if the economy regime variable Xt is at the ith state ei. Equivalently, r(t,Xt) :=

〈r,Xt〉, where 〈·, ·〉 denotes the standard inner product in RN and r = (r1, r2, . . . , rN )′ are positive constants. The

risky (default-free) asset St follows the dynamics

dSt = µtStdt+ σtStdWt, S0 = s, (7)

where Wt is an F-adapted Wiener process independent of Xt and

µt := µ(t,Xt) := 〈µ,Xt〉 , σt := σ(t,Xt) := 〈σ,Xt〉 (8)

for constant vectors µ = (µ1, µ2, . . . , µN )′ and σ = (σ1, σ2, . . . , σN )′, representing the respective appreciation rates and

volatilities that the risky asset can take depending on the different economic regimes. Further, we assume that the

dynamics of the default intensity h and loss rates L are also driven by the underlying Markov process X, i.e.

ht := 〈h,Xt〉 , and Lt := 〈L,Xt〉 , (9)

for constant vectors h = (h1, . . . , hN )′ and L = (L1, . . . , LN )′. We emphasize that the distribution of the hazard rate

process ht = 〈h,Xt〉 under the risk-neutral measure is different from that under the historical measure. Therefore,

our framework allows the incorporation of a default risk premium, defined as the ratio between risk-neutral and

historical intensity, through the change of measure of the underlying Markov chain. We adopt the recovery-of-market

value assumption, under which we obtain (see Duffie and Singleton (1999), Theorem 1) that the price process of the

defaultable security is given by its risk-neutral conditional expectation

p(t, T ) = 1τ>tEQ[e−

∫ Tt

(rs+hsLs)ds

∣∣∣∣Ft]. (10)

4

Hereafter, we assume that Q is an equivalent risk-neutral pricing measure such that, under Q, W is still a standard

Wiener process and X is an independent continuous-time Markov process with generator AQ (see Section 11.2 in

Bielecki and Rutkowski (2001)).

Under mild differentiability and boundedness conditions on the risk-neutral generator AQ of X, the pre-default

dynamics of the bond price p(t, T ) under the historical measure P was obtained in Capponi and Figueroa-Lopez (2011)

(Proposition 3.2) as

dp(t, T ) = p(t−, T )

[rt + ht(Lt − 1) +D(t)] dt+

⟨ψ(t), dMP(t)

⟩〈ψ(t), Xt−〉

− dξPt, (11)

where (MP(t))t is the N -dimensional (F,P)-martingale defined in (4), (ξPt )t is the (G,P)-martingale defined in (2), and

D(t) := 〈(D1(t), . . . , DN (t))′, Xt〉 with

Di(t) :=

N∑j=1

(ai,j(t)− aQi,j(t))ψj(t)

ψi(t)=∑j 6=i

(ai,j(t)− aQi,j(t))(ψj(t)

ψi(t)− 1

). (12)

Here, ψ(t) := (ψ1(t), . . . , ψN (t))′, where ψi(t) is the pre-default price of the defaultable bond given that the macro-

economy is in the ith regime:

ψi(t;T ) := EQ[e−

∫ Tt

(rs+hsLs)ds∣∣∣Xt = ei

]. (13)

To lighten the notation, we will sometimes use ψi(t) instead of ψi(t;T ) when the dependence on T is clear from the

context.

3 Novel algorithm for pricing claims in regime switching models

In this section, we develop a novel efficient algorithm for pricing a claim whose payoff depends on the underlying

economic regime in place (Xt). We start by considering a simple European claim of the form Ξ(XT ) for a deter-

ministic function Ξ (see Section 3.1 below), and proceed to consider more general path-dependent claims, termed

self-decomposable claims, whose payoffs can be decomposed into shorter maturity payoffs (see Section 3.2). We illus-

trate this method for a type of barrier option on the Markov process (Xt). In Section 3.3, we apply our approach

to develop a new method to price vulnerable call/put options on the stock. The latter type of options have received

growing interest in the literature.

3.1 The basic algorithm

Let us consider a simple vulnerable European claim with expiration T whose payoff is of the form Ξ(XT ) for a

deterministic function Ξ : e1, e2, . . . , eN → R. As it is well known (see, e.g., Theorems 9.23 and 9.24 in McNeil et al.

(2006)), under the recovery-of-market value assumption, the price of this vulnerable claim is given by

Ψi[Ξ](t;T ) := EQ[

Ξ(XT )e−∫ Tt

(rs+hsLs)ds∣∣∣Xt = ei

]. (14)

The key idea of our approach to compute (14) lies in changing the risk-neutral probability measure Q into an equivalent

measure Q such that Xtt≤T is a homogeneous Markov process under Q. Such a probability measure Q exists whenever

aQi,j is strictly positive for all t > 0 and i 6= j (see Lemma A.2 of Capponi and Figueroa-Lopez (2011) for details).

Using this change of probability measure, it follows (see again Capponi and Figueroa-Lopez (2011)) that

Ψi[Ξ](t;T ) = EQ[

Ξ(XT )e−∫ Ttr(s)′Xsds−

∑s∈(t,T ]:∆Xs 6=0X

′s−K(s)Xs

∣∣∣Xt = ei

], (15)

where K(t) := [Ki,j(t)]i,j and r(t) := (r1(t), . . . , rN (t))′ are defined as

Ki,j(t) := − log(

(N − 1)aQi,j(t))

1i 6=j , ri(t) := ri + hiLi − 1− aQi,i(t).

5

As explained before, Q has the virtue that the generator of (Xt) under Q is given by ai,j = 1/(N − 1) (i 6= j) and

ai,i = −1. Our first step is to condition on the number of transitions of X. Since the number of transitions is Poisson

distributed with unit intensity, we get

Ψi[Ξ](t;T ) =

∞∑m=0

e−(T−t) (T − t)m

m!Φi,m[Ξ](T − t), (16)

where

Φi,m[Ξ](ζ) := EQi

[Ξ(Xm) exp

−

m∑n=0

∫ ζU(n+1)

ζU(n)

r(T − ζ + s)′Xnds −m∑n=1

X ′n−1K(T − ζ + ζU(n))Xn

]. (17)

Here, EQi [·] = EQ

[·| X0 = i

], Xi is the embedded Markov chain of X, and U(1) < U(2) < · · · < U(m) are the ordered

statistics of m i.i.d. uniform [0, 1] variables independent of X, fixing U(0) := 0 and U(m+1) := 1.

We now derive a formula for Φi,m[Ξ](ζ) when the risk-neutral generator AQ is time-invariant. The formula will be

expressed in terms of the Laplace transform of the “symmetric” Dirichlet distribution, defined by

Lm(λ1, . . . , λm) := m!

∫Tm

e−∑mj=1 λjxjdx, (18)

where Tm := (λ1, . . . , λm) ∈ Rm : λi ≥ 0,∑mi=1 λi ≤ 1. The proof of the following result is given in the Appendix A.

Lemma 3.1. Suppose that aQi,j(t) ≡ aQi,j (hence, K and r are also time-invariant). Then, for m ≥ 1, we have that

Φi,m[Ξ](ζ) =1

(N − 1)m

∑(e1,...,em)

Ξ(em)e−ζr′em−

∑mn=1 e

′n−1KenLm(ζr′(e0 − em), . . . , ζr′(em−1 − em)), (19)

where e0 = ei and the above summation is over all “paths” (e1, . . . , em) such that ej ∈ e1, . . . , eN and ej 6= ej−1, for

j = 1, . . . ,m.

Remark 3.2. Note that

Φi,0[Ξ](ζ) = Ξ(i)e−ζri , Φi,1[Ξ](ζ) =1

(N − 1)

∑j∈1,...,N\i

Ξ(j)e−ζrj−Ki,j1

ζ(ri − rj)

(1− e−ζ(ri−rj)

).

For a general m, the following Taylor approximation around the origin will turn out to be quite useful to compute the

Dirichlet Laplace transform (18):

Lm(λ1, . . . , λm) = 1−∑mi=1 λim+ 1

+

∑mi=1 λ

2i + 1

2

∑i 6=j λiλj

(m+ 1)(m+ 2)+O

(‖(λ1, . . . , λm)‖3/2

). (20)

In order to evaluate the option price, the infinite series (16) will be truncated and the Laplace transform in (19)

may also be approximated by (20). Both of these approximations are valid when time-to-maturity ζ = T − t is small.

Hence, it useful to express the option price (15) in terms of the price of options with short-expiration. Concretely, fix

a small mesh δ := T/k for a positive integer k and let

Iu,v := −∫ v

u

r′Xsds−∑

s∈(u,v]:∆Xs 6=0

X ′s−KXs. (21)

Then, using the tower and Markov properties, the time 0 price of the vulnerable claim may be computed as follows

Ψi[Ξ](0;T ) = EQ [eI0,T Ξ(XT )∣∣X0 = ei

]= EQ

[eI0,δEQ [Ξ(XT )eIδ,T |Fδ

]∣∣∣X0 = ei

]=: EQ

[eI0,δ Ξ(Xδ)

∣∣∣X0 = ei

]= Ψi[Ξ](0; δ), (22)

6

where the new payoff function Ξ is defined as Ξ(ej) = Ψj [Ξ](0;T − δ). Note that (22) is the price of an option with

short maturity δ and, hence, it can be accurately computed by taking M terms in (16) and (possibly) using (20). In

order to evaluate the option’s payoff Ξ, we apply again the procedure (22), replacing T by T − δ. Computationally, we

can create a recursive or iterative implementation in order to evaluate (22). The pseudo-code of the proposed iterative

algorithm is given in Appendix C (Algorithm 1 therein).

Remark 3.3. The previous method can be adapted to deal with smooth time dependent functions r(t) and K(t). Indeed,

as in (22) and using notation (5), we will have

Ψi[Ξ](t;T ) = EQ [eIt,T Ξ(XT )∣∣Xt = ei

]= EQ [eIt,t+δΨCt+δ [Ξ](t+ δ;T )

∣∣Xt = ei]. (23)

Under sufficient smoothness on r(t) and K(t), It,t+δ can be accurately computed as if r and K were time-invariant

during the period [t, t+ δ].

Remark 3.4. The complexity of the proposed algorithm can be evaluated as follows. Let

ΨMi [Ξ](t;T ) =

M−1∑m=0

e−(T−t) (T − t)m

m!Φi,m[Ξ](T − t), (24)

The computational complexity to evaluate ΨMi [Ξ](t;T ) is O(NM−1). Indeed, for each m, it is required to evaluate

Φi,m[Ξ](T − t). This, in turn, requires at most NM−1 evaluations of the Laplace transform of the symmetric Dirichlet

distribution (one for each path (e0, . . . , em) that e0 = ei and ej 6= ej−1 for j = 1, . . . ,m as seen in (19)). Therefore,

the total computational complexity of ΨMi is O(NM−1) and, thus, the algorithm has complexity O(NM ) to compute the

prices conditional on all starting regimes. We also remark that, for a fixed M , the computation of Ψi[Ξ](0;T ) can be

sped up greatly by saving the paths (e1, . . . , em) needed for (19) at the beginning and reusing them for each evaluation

of Φi,m[Ξ](ζ) with ζ ∈ δ, . . . , kδ.

Our next result gives a precise error bound for our algorithm under certain mild conditions. Its proof is given in

the Appendix A.

Proposition 3.5. Let M and k be fixed positive integers and let δ = T/k. Then, the proposed Algorithm (see Algorithm

1 in Appendix C) will result in a price approximation Ψi,k[Ξ](0;T ) such that

maxi∈1,...,N

∣∣∣Ψi[Ξ](0;T )− Ψi,k[Ξ](0;T )∣∣∣ ≤ TM

kM−1.

Remark 3.6. As indicated in the Remark 3.2, Φi,0(Ξ) and Φi,1(Ξ) can be computed easily. Hence, using only these

two values, the bond prices can be computed up to an error of order T 2/k using k iterations with a maximal polynomial

complexity of O(N3).

3.1.1 Pricing of defaultable bond prices

In order to assess the accuracy and computational speed of the novel method described above, we compute the pre-

default bond prices (13) using our method and a standard numerical solution of the Feynman-Kac representation for

the bond price (13). For completeness, this is given in the next lemma (see Appendix A for its proof).

Lemma 3.7. The bond price processes ψ(t) = (ψ1(t), . . . , ψN (t)) satisfy the coupled system of ordinary differential

equations (ODE)

dψi(t) = (ri + hiLi − aQi,i(t))ψi(t)dt−∑j 6=i

aQi,j(t)ψj(t),

ψi(T ) = 1, i = 1, . . . , N. (25)

7

As a direct consequence of the previous lemma, in the time-invariant case (i.e., aQi,j(t) ≡ aQi,j), the solution of (25)

can be expressed in closed-form as

ψ(t;T ) = e−(T−t)Fψ1′, (26)

where the components of the matrix Fψ are [Fψ]i,i = ri+hiLi−aQi,i and [Fψ]i,j = −aQi,j . Table 1 shows the time-0 bond

prices for different maturities computed using our method and a Runge-Kutta type numerical solution1 of the system

(25) under the “realistic” parameter setup of Tables 3 and 4 given in the Section 5.1 below. Figure 1 shows the bond

prices for different times t corresponding to the maturities of T = 1 year and T = 20 years. It is evident that our method

is highly accurate even for maturities as long as 50 years. Furthermore, according to our computational experiments,

our method is in most cases more efficient than either solving the ODE system by the Runge-Kutta algorithm or

computing the exponential (26) using a Pade type approximation. For the sake of completeness, Appendix D compares

the processor time for these three methods.

X0 = e1 X0 = e2 X0 = e3

T (yrs.) NM ODE NM ODE NM ODE

0.25 0.9921 0.9921 0.9884 0.9884 0.9686 0.9686

0.50 0.9837 0.9837 0.9772 0.9772 0.9393 0.9393

1.00 0.9659 0.9659 0.9555 0.9555 0.8864 0.8864

2.00 0.9282 0.9281 0.9146 0.9146 0.7991 0.7990

5.00 0.8140 0.8136 0.8029 0.8031 0.6274 0.6273

10.0 0.6488 0.6484 0.6430 0.6431 0.4701 0.4701

15.0 0.5170 0.5166 0.5131 0.5131 0.3691 0.3690

20.0 0.4119 0.4116 0.4090 0.4090 0.2931 0.2930

25.0 0.3282 0.3280 0.3259 0.3259 0.2334 0.2333

30.0 0.2616 0.2613 0.2597 0.2597 0.1859 0.1858

50.0 0.1055 0.1053 0.1047 0.1047 0.0750 0.0749

Table 1: Time t = 0 bond prices for different time-to-maturities T using the ODE method and the new method (NM)

with parameters M = 2 and δ = 2.5 years (see Algorithm 1 in Appendix C).

3.2 Pricing of decomposable claims on the underlying Markov process

We now demonstrate the applicability of our approach to price other European claims with possibly path-dependent

payoffs. As it is evident from (22), our approach heavily relies on being able to decompose the payoff of the claim into

payoffs of shorter maturity. The following broad definition attempts to give a more precise meaning to this concept:

Definition 3.1. Consider a family of payoffs Σt,T 0≤t≤T , where for each 0 ≤ t ≤ T , Σt,T represents a payoff

depending on the path of X on [t, T ]. We write Σt,T := Σ(Xst≤s≤T ). We say that the family Σt,T 0≤t≤T is

self-decomposable if, for any 0 < t < t′ < T , the following decomposition holds true:

Σt,T = f(Σt,t′) + g(Σt,t′)Σt′,T ,

for some measurable functions f, g : R 7→ R.

We now proceed to describe our approach. Following our strategy for simple claims, it is natural that a feasible

method to price a self-decomposable claim Σt,T (X) ≡ Σ(Xst≤s≤T ) will consist of the following two general steps:

1This solution was obtained using the MATLAB function ode45.

8

0 0.2 0.4 0.6 0.8 1

0.9

0.92

0.94

0.96

0.98

1

Time (t)

ψ i(t)

NM(1)ODE(1)NM(2)ODE(2)NM(3)ODE(3)

10 12 14 16 18 20

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Time (t)ψ i(t

)

NM(1)ODE(1)NM(2)ODE(2)NM(3)ODE(3)

Figure 1: Bond price comparison for a one-year and a 20-year bond using the ODE method and the new method (NM)

parameters M = 2 and δ = 2.5 years (see Algorithm 1 in Appendix C).

(Decomposition) Fix a δ = (T − t)/k for a positive integer k and apply the following decomposition with t′ := t+ δ:

EQ [eIt,T Σt,T∣∣Xt = ei

]= EQ

[eIt,t′ f(Σt,t′)EQ[eIt′,T |Ft′ ]

∣∣∣X0 = ei

]+ EQ

[eIt,t′ g(Σt,t′)EQ [eIt′,T Σt′,T

∣∣Ft′]∣∣∣X0 = ei

]=: EQ [eIt,t′ f(Σt,t′)Ξ(Xt′)

∣∣X0 = ei]

+ EQ[eIt,t′ g(Σt,t′)Ξ(Xt′)

∣∣∣X0 = ei

],

where Ξ(el) := EQ[eIt′,T |Xt′ = el] and Ξ(el) := EQ [eIt′,T Σt′,T∣∣Xt′ = el

]. We then repeat the above decomposi-

tion to evaluate the payoffs Ξ(·) until T − t′ is small enough.

(Near-expiration approximation) We proceed to apply an efficient approximation to evaluate claims of the form

EQ [eIt,t′h(Σt,t′)Ξ(Xt′)∣∣Xt = ei

], when t is close to t′.

Remark 3.8. The above method can also be extended to deal with claims whose payoffs can be decomposed in terms

of the payoffs of other type of claims. For instance, we can say that two families of payoffs, say Σ(0)t,T 0≤t≤T and

Σ(1)t,T 0≤t≤T , are mutually self-decomposable if, for any t < t′ < T ,

Σ(k)t,T = fk(Σ

(0)t,t′) + gk(Σ

(0)t,t′)Σ

(0)t′,T + hk(Σ

(1)t,t′) + `k(Σ

(1)t,t′)Σ

(1)t′,T ,

for each k = 0, 1, and some measurable functions fk, gk, hk, `k.

As an illustration, we now consider the risk-neutral pricing of European barrier and digital contracts written on

the volatility process (σt)t. One may view the process σt := σ′Xs as a proxy to the volatility of a market index.

Instruments written on this process may be used to hedge volatility risk associated with periods of macro-economic

bust or boom akin to that experienced by the U.S. economy leading into the 2008 crisis. Let us define the following

family of path-dependent payoffs:

Σt,T := 1maxt≤s≤T

σ′Xs ≥ B, and Σt,T := 1

maxt≤s≤T

σ′Xs < B.

9

The following simple relationships show that the family of payoffs Σt,T 0≤t≤T and Σt,T 0≤t≤T are self-decomposable:

(1) Σt,T = Σt,t′Σt′,T , (2) Σt,T = Σt,t′ + Σt′,T − Σt,t′Σt′,T = Σt,t′ + (1− Σt,t′)Σt′,T , (t < t′ < T ). (27)

Now, let us consider the following European knock-out style barrier option:

ΨOUTi [Ξ](t;T ) := EQ

[e−

∫ TtrsdsΣt,TΞ(XT )

∣∣∣Xt = ei

]= EQ

[eIt,T Σt,TΞ(XT )

∣∣∣Xt = ei

],

where we had used the same change of probability measure Q as in (15) and the following process analog to (21):

Iu,v := −∫ v

u

r′Xsds−∑

s∈(u,v]:∆Xs 6=0

X ′s−KXs.

Then, the following decomposition follows from (27):

ΨOUTi [Ξ](t;T ) = EQ

[eIt,t′ Σt,t′ EQ

[Σt′,TΞ(XT )eIt′,T

∣∣∣Ft′]∣∣∣Xt = ei

]= EQ

[eIt,t′ Σt,t′Ψ

OUTCt′

[Ξ](t′;T )∣∣∣Xt = ei

]= ΨOUT

i [ΨOUTCt′

[Ξ](t′;T )](t; t′). (28)

For a near-expiration approximation method for ΨOUTi [Ξ](t; t′) (i.e. when ζ := t′ − t ≈ 0), we use again (16) and (17):

ΨOUTi [Ξ](t; t′) = e−ζ

M−1∑m=0

ζm

m!ΦOUTi,m (ζ),

with

ΦOUTi,m (ζ) =1

(N − 1)m

∑(e1,...,em)

Ξ(em)1maxj(σ′ej)<Be−ζr′em−

∑mn=1 e

′n−1KenLm(ζr′(e0 − em), . . . , ζr′(em−1 − em)).

Note that the knock-in style barrier option ΨINi [Ξ](t;T ) := EQ[exp−

∫ TtrsdsΣt,TΞ(XT )|Xt = ei] can be easily

computed using the relation Σt,T = 1 − Σt,T . In general one only needs to price either a knock-in or a knock-out

contract, as the value of the other follows immediately from the knock-in/knock-out parity. Table 2 presents the prices

of both knock-out digital and call options written on the volatility process obtained by applying our methodology.

Comparison with Monte Carlo prices is also presented. These results show the high accuracy of our method even for

long maturity options.

3.3 Pricing of vulnerable call/put options

As a final (yet very important) application of our approach, we now consider the pricing of vulnerable call or put

options. Consider the following vulnerable call option price at time t:

Π(t, T ; s, ei) := EQ[

(ST −K)+e−

∫ Tt

(rs+hsLs)ds∣∣∣Xt = ei, St = s

].

As before, we will change the probability measure into Q so that, in terms of the process It,T defined in (21),

Π(t, T ; s, ei) = EQ [ (ST −K)+eIt,T∣∣Xt = ei, St = s

].

Next, we have that the risk-neutral dynamics of the stock price is given by

ST = St exp

∫ T

t

b′Xsds+

∫ T

t

σ′XsdWs

,

10

Knock-Out Digital Contracts Knock-Out Barrier Call Options

Ξ(XT ) ≡ 1 Ξ(XT ) := (σXT −K)+

B = σ3 B = σ3, K = 0.075, Units = 10−3

X0 = e1 X0 = e2 X0 = e1 X0 = e2

T NM MC NM MC NM MC NM MC

0.50 0.9719 0.9693 0.9525 0.9523 3.5081 3.5574 21.1632 21.0587

1.0 0.9419 0.9455 0.9093 0.9100 5.8869 5.7910 18.2870 18.2697

2.5 0.8482 0.8466 0.7980 0.7964 9.0631 9.0351 13.1068 13.0927

5.0 0.7013 0.6986 0.6506 0.6481 9.1533 9.2635 9.3528 9.5317

10.0 0.4732 0.4721 0.4373 0.4336 6.4919 6.3608 6.0298 5.9426

15.0 0.3187 0.3201 0.2944 0.2955 4.3833 4.3335 4.0508 4.1597

20.0 0.2146 0.2141 0.1983 0.1974 2.9521 2.9365 2.7274 2.8137

25.0 0.1445 0.1445 0.1335 0.1341 1.9879 1.9577 1.8366 1.7792

30.0 0.0973 0.0985 0.0899 0.0912 1.3386 1.3136 1.2367 1.2111

35.0 0.0655 0.0676 0.0605 0.0608 0.9014 0.9038 0.8328 0.8228

Table 2: Knock-out digital and call options on the volatility process using the new method (NM) and the Monte Carlo

method (MC). The call option prices are expressed on the 10−3 scale and the results for X0 = e3 have been omitted

since they are knocked out at contract initiation. Here, r = (0.01, 0.1, 0.3)′ and σ = (0.05, 0.1, 0.2)′.

where the evolution of Xt is determined by the risk-neutral generator AQ under Q (see, e.g., Elliott et al. (2005)).

Above, b := (b1, . . . , bN )′ is given by bi := ri − σ2i /2. In particular, given σ(Xu : t ≤ u ≤ T ) and St = s, we can

see Sut≤u≤T as a geometric Brownian motion with a deterministic time-varying volatility and initial value s. As

it is well-known, one can express the call price for such a model in terms of the Black-Scholes formula with constant

volatility σ, short-rate r, spot price s0, maturity ζ, and strike K:

BS(ζ; s0, σ

2, r,K)

:= e−rζEQ(s0e

σWζ+(r−σ2/2)ζ −K)

+.

Concretely, denoting

ζ := T − t, σ2 :=1

T − t

∫ T

t

(σ′Xu)2du, b :=1

T − t

∫ T

t

b′Xudu, s0 := sebζ+ζσ2/2 = se

∫ Ttr′Xudu,

we have

EQ [ (ST −K)+|σ(Xu : t ≤ u ≤ T ), St = s] = EQ[(sebζ+ζσ

2/2 × eσWζ−ζσ2/2 −K)+

]= BS(ζ; s0, σ

2, 0,K).

Then, we obtain

Π(t, T ; s, ei) = EQ[EQ [eIt,T (ST −K)+|σ(Xu : t ≤ u ≤ T )

]∣∣∣Xt = ei, St = s]

= EQ [eIt,T BS(ζ; s0, σ2, 0,K)

∣∣Xt = ei, St = s].

Let us now focus on the time-invariant case, where AQ is time-invariant and, hence, Π(t, T ; s, ei) = Π(0, T − t; s, ei).We set

F (ζ; s, ei) := EQ [ (Sζ −K)+|S0 = s,X0 = ei] = EQ[eI0,ζBS

(ζ; se

∫ ζ0r′Xudu, ζ−1

∫ ζ

0

(σ′Xu)2du, 0,K)∣∣∣∣∣S0 = s,X0 = ei

].

Our approach is based on two ideas. Firstly, if ζ is small, note that

F (ζ; s, ei) ≈ F (ζ; s, ei) := EQ[eI0,ζBS

(ζ; seζr

′Xζ , (σ′Xζ)2, 0,K

)∣∣∣S0 = s,X0 = ei

], (29)

11

up to an error O(ζ). This is because∫ ζ

0r′Xudu = ζr′Xζ and

∫ ζ0

(σ′Xu)2du = ζ(σ′Xζ)2 if there are no transitions of the

process (Xt) during [0, ζ]. Since the expression in (29) can be seen as a European claim of the form (14) with maturity

ζ and payoff Ξ(ej) := BS(ζ; seζrj , σ2

j , 0,K), one can evaluate this “first order approximation” using our Algorithm 1

in Appendix C.

For a general maturity ζ, we proceed as in (22). Concretely, for δ < ζ, we have the recursive relationship

F (ζ; s, ei) = EQ [eI0,δF (ζ − δ;Sδ, Xδ)∣∣S0 = s,X0 = ei

]= EQ

[eI0,δF

(ζ − δ; sebδ+σ

√δW1 , Xδ

)∣∣∣X0 = ei

],

where now σ2 := (ζ − δ)−1∫ ζδ

(σ′Xu)2du and b := (ζ − δ)−1∫ δζb′Xudu. As before, the last expression is approximated

by

F (ζ; s, ei) := EQ[eI0,δF

(ζ − δ; seδb

′Xδ+σ′Xδ√δW1 , Xδ

)∣∣∣X0 = ei

]= EQ

[eI0,δ

∫ ∞−∞

F(ζ − δ; seδb

′Xδ+√δσ′Xδz, Xδ

) 1√2πe−z

2/2dz

∣∣∣∣X0 = ei

]In principle, for a fixed s, one can see the right-hand side above as the price of European claim of the form (14) with

maturity δ and payoff Ξ(ej) := EQ[F(ζ − δ; seδbj+σj

√δW1 , ej

) ]. But, since this approach would require to compute

F (ζ − δ; p, ej) for all p and ej , this would be computationally inefficient. To resolve this issue, we restrict all possible

initial prices s to be in the lattice L∆,B := sei∆ : i ∈ −B,−B+1, . . . , B−1, B for a small ∆ and a positive integer

B. Then, we can approximate F (ζ; sei∆, ej) as follows:

F (ζ; sei∆, ei) ≈ EQ[eI0,δ Ξi(Xδ)

∣∣∣X0 = ei

],

with

Ξi(ej) :=

B∑k=−B

F (ζ − δ; sek∆, ej)

∫ zi,jk

zi,jk−1

1√2πe−z

2/2dz,

where

zi,jk :=(k − i+ 1/2)∆− δbj

σj√δ

, (k = −B,−B + 1, . . . , B − 1), zi,j−B−1 := −∞, zi,jB :=∞.

Note that the points zi,jk ’s are chosen so that the midpoint zi,jk of the interval [zi,jk−1, zi,jk ] is such that s expk∆ =

s expi∆ + δbj +√δσj z

i,jk and, hence, Ξi above is a Riemann-Stieltjes sum approximation of the payoff

Ξi(ej) :=

∫ ∞−∞

F(ζ − δ; sei∆+δbj+

√δσjz, ej

) 1√2πe−z

2/2dz.

Figure 2 shows the comparison of a plain Monte Carlo method to our novel method above when the initial regime

is 1 and 3 (regime 2 is quite similar to regime 3). As seen there, the new method significantly improves the quality

of the approximation compared to the first order approximation (29), especially for longer maturities. For the sake of

completeness, we have included the precise algorithm in Appendix C (see Algorithm 2 therein). Error analysis and

further extensions of the previous method will be postponed for a future publication given space limitations.

4 Continuous time portfolio optimization

In this section, we develop a numerical and economical analysis of portfolio optimization problems in defaultable

regime switching markets populated by a CRRA investor. Section 4.1 recalls the HJB optimization framework derived

12

70 80 90 100 110 120 130 1400

10

20

30

40

50

60

Initial Stock Price, S0

Cal

l Opt

ion

Pric

es

Vulnerable Call Option Prices given X0 = 1

Monte Carlo Estimate (T=3, # of Rept.=5000)

New Method (T=3, δ=T, ∆=0.05, B=250, M=2, k=2)

New Method (T=3, δ=T/2, ∆=0.05, B=250, M=2, k=2)

Monte Carlo Estimate (T=6, # of Rept.=5000)

New Method (T=6, δ=T, ∆=0.05, B=250, M=2, k=2)

New Method (T=6, δ=T/2, ∆=0.05, B=250, M=2, k=2)

70 80 90 100 110 120 130 1400

5

10

15

20

25

30

35

40

45

Initial Stock Price, S0

Cal

l Opt

ion

Pric

es

Vulnerable Call Option Prices given X0 = 3

Monte Carlo Estimate (T=3, # of Rept.=5000)New Method (T=3, δ=T, ∆=0.05, B=250, M=2, k=2)New Method (T=3, δ=T/2, ∆=0.05, B=250, M=2, k=2)Monte Carlo Estimate (T=6, # of Rept.=5000)New Method (T=6, δ=T, ∆=0.05, B=250, M=2, k=2)New Method (T=6, δ=T/2, ∆=0.05, B=250, M=2, k=2)

Figure 2: Comparison of vulnerable call option prices using Monte Carlo and the new method of the Algorithm 2 in

Appendix C for maturities of T=3 years and T=6 years. Above, k=2 refers to the number of discretization points on

the PriceClaim Method (i.e. δ = T/k therein).

in Capponi and Figueroa-Lopez (2011). The new results are presented in Section 4.2, where explicit solutions for the

post-default value functions are obtained and conditions for the existence of solutions to the coupled system of ODEs

and nonlinear equations characterizing the pre-default value function and the bond investment strategy are given.

Moreover, we also provide a precise characterization of the “directionality” of the bond investment strategy in terms

of corporate returns, instantaneous forward rate, and expected recovery at default.

4.1 The portfolio optimization framework

This section reviews the main results given in Capponi and Figueroa-Lopez (2011), which are needed for the following

analysis. We first recall the HJB setup and then proceed to give the corresponding verification theorems.

4.1.1 Hamiltonian-Jacobi Bellman setup

We consider the classical Merton’s optimal portfolio problem for the defaultable regime-switching market introduced in

Section 2. Concretely, for a fixed time horizon R ≤ T , an initial value (x, z, v) ∈ E := e1, e2, . . . , eN×0, 1× (0,∞),

and a suitable trading strategy π = (πB , πS , πp), let us define the objective functional

JR(x, z, v;π) := EP[U(V πR )

∣∣∣∣X0 = x,H0 = z, V π0 = v

], (30)

where U : [0,∞)→ R∪∞ is a strictly increasing and concave utility function and Vu0≤u≤T is the investor’s wealth

process

dV πu = V πu−

πBu

dBuBu

+ πSudSuSu

+ πPudp(u, T )

p(u−, T )

. (31)

Here, πu = (πBu , πSu , π

pu) represents the percentage of wealth invested in the money-market account, the risky (default-

free) stock, and the defaultable bond, respectively. Our goal is to maximize the objective functional J(x, z, v;π) for a

13

suitable class of feedback or Markov admissible strategies πu := (πBu , πSu , π

Pu ) defined as

πu = (πBCu−

(u, Vu− , H(u−), πSCu−

(u, Vu− , H(u−), πPCu−

(u, Vu− , H(u−)),

for some functions πBi , πPi , π

Si : [0,∞) × [0,∞) × 0, 1 → R such that πBi (u, v, z) + πSi (u, v, z) + πPi (u, v, z) = 1. As

usual, we consider instead the following dynamical optimization problem:

ϕR(t, v, i, z) := supπ∈At(v,i,z)

EP[U(V π,t,vR )

∣∣∣∣Vt = v,Xt = ei, H(t) = z

], (32)

for each (v, i, z) ∈ (0,∞)× 1, 2, . . . , N × 0, 1, where

dV π,t,vu = V π,t,vu−

[ ru + πSu (µu − ru) + πPu (1−H(u−))[hu(Lu − 1) +D(u)]

du

+ πSuσudWu + πPu (1−H(u−))

⟨ψ(u), dMP(u)

⟩〈ψ(u), Xu−〉

− πPu dξPu], u ∈ [t, R],

V π,t,vt = v. (33)

The dynamics (33) follows from plugging the dynamics (6), (7), and (11) into (31) and using the condition πBi (u, v, z)+

πSi (u, v, z) + πPi (u, v, z) = 1. The class of processes At(v, i, z) denotes a suitable class of F-predictable locally bounded

feedback trading strategies

πu := (πSu , πPu ) := (πSCu− (u, V π,t,vu− , H(u−)), πPCu− (u, V π,t,vu− , H(u−))), u ∈ [t, R],

such that (33) admits a unique strong solution V π,t,vu u∈[t,R] and the solvency condition V π,t,vu > 0 for any u ∈ [t, R]

is satisfied when Xt = ei and H(t) = z.

Remark 4.1. As discussed in Capponi and Figueroa-Lopez (2011), in order for the solvency condition to hold true, it

is necessary that πP satisfies

Mi(s) := maxj 6=i:ψi(s)<ψj(s)

(− ψi(s)

ψj(s)− ψi(s)

)< πPi (s, v, z) < 1, (34)

for any v > 0, 0 < s ≤ τ , z ∈ 0, 1, and i = 1, . . . , N . We fix Mi := −∞ if ψi(s) ≥ ψj(s) for all j 6= i.

4.1.2 Verification theorems

We now consider the HJB equations associated to the optimization problem (32). We divide the problem into two cases:

one corresponding to the post-default scenario and the other corresponding to the pre-default scenario. Concretely, we

set

ϕR(t, v, i) = ϕi,0(t, v) = ϕR(t, v, i, 0), (pre-default case) (35)

and

ϕR(t, v, i) = ϕi,1(t, v) = ϕR(t, v, i, 1), (post-default case). (36)

Note that, in the post-default case, we have that p(t, T ) = 0, for any τ < t ≤ T . Consequently, πPt = 0 for τ < t ≤ T

and we can take π = πS as our control.

Below, ηi := (µi−ri)/σi denotes the Sharpe ratio of the risky asset under the ith state of economy and C1,20 denotes

the class of functions $ : [0, R]× R+ × 1, . . . , N → R+ such that

$(·, ·, i) ∈ C1,2((0, R)× R+) ∩ C([0, R]× R+), $v(s, v, i) ≥ 0, $vv(s, v, i) ≤ 0,

for each i = 1, . . . , N . We have the following verification result for the post-default value function, proven in Capponi

and Figueroa-Lopez (2011).

14

Theorem 4.2. Suppose that there exists a function w ∈ C1,20 that solves the nonlinear Dirichlet problem

wt(s, v, i)−η2i

2

w2v(s, v, i)

wvv(s, v, i)+ rivwv(s, v, i) +

∑j 6=i

ai,j(s) (w(s, v, j)− w(s, v, i)) = 0, (37)

for any s ∈ (0, R) and i = 1, . . . , N , with terminal condition w(R, v, i) = U(v). We assume additionally that w satisfies

(i) |w(s, v, i)| ≤ D(s) + E(s)v, (ii)

∣∣∣∣ wv(s, v, i)wvv(s, v, i)

∣∣∣∣ ≤ G(s)(1 + v), (38)

for some locally bounded functions D,E,G : R+ → R+. Then, the following statements hold true:

(1) w(t, v, i) coincides with the optimal value function ϕR(t, v, i) = ϕR(t, v, i, 1) in (32), when At(v, i, 1) is constrained

to the class of t-admissible feedback controls πSs = πCs

(s, Vs) such that πi(·, ·) ∈ C([0, R] × R+) for each i =

1, . . . , N and

|vπi(s, v)| ≤ G(s)(1 + v), (39)

for a locally bounded function G. If the solution w is non-negative, then condition (39) is not needed.

(2) The optimal feedback control πSs s∈[t,R), denoted by πSs , can be written as πSs = πCs

(s, Vs) with

πi(s, v) = − ηiσi

wv(s, v, i)

vwvv(s, v, i). (40)

Let us now define

θi(t) := hiLi −∑j 6=i

aQi,j(t)

(ψj(t)

ψi(t)− 1

). (41)

The following verification result for the pre-default optimal value function was proved in Capponi and Figueroa-Lopez

(2011).

Theorem 4.3. Suppose that the conditions of Theorem 4.2 are satisfied and, in particular, let w ∈ C1,20 be the solution

of (37). Assume that w ∈ C1,20 and pi = pi(s, v), i = 1, . . . , N , solve simultaneously the following system of equations:

θi(s)wv(s, v, i)− hi wv(s, v(1− pi), i) +∑j 6=i

ai,j(s)

(ψj(s)

ψi(s)− 1

)wv

(s, v

[1 + pi

(ψj(s)

ψi(s)− 1

)], j

)= 0, (42)

wt(s, v, i)−η2i

2

w2v(s, v, i)

wvv(s, v, i)+ rivwv(s, v, i) +

piθi(s)vwv(s, v, i) + hi [w(s, v(1− pi), i)− w(s, v, i)]

+∑j 6=i

ai,j(s)

[w

(s, v

(1 + pi

(ψj(s)

ψi(s)− 1

)), j

)− w(s, v, i)

]= 0, (43)

for t < s < R, with terminal condition w(R, v, i) = U(v). We also assume that pi(s, v) satisfies (34) and (39)

(uniformly in v and i) and w satisfies (38). Then, the following statements hold true:

(1) w(t, v, i) coincides with the optimal value function ϕR(t, v, i) = ϕR(t, v, i, 0) in (32), when At(v, i, 0) is constrained

to the class of t-admissible feedback controls (πSs , πPs ) = (πS

Cs−

(s, Vs− , H(s−)), πPCs−

(s, Vs− , H(s−))) such that

πSi (·, ·, z), πPi (·, ·, z) ∈ C([0, R]× R+),

for each i = 1, . . . , N , πS satisfies (39) for a locally bounded function G, and πP satisfies (34) and (39) (uniformly

in v, i, z). If the solution w is non-negative, then these bound conditions are not needed.

(2) The optimal feedback controls are given by πSs := πSCs−

(s, Vs, H(s)) and πPs := πPCs−

(t, Vt, H(s)) with

πSi (s, v, z) = − ηiσi

wv(s, v, i)

vwvv(s, v, i)(1− z)− ηi

σi

wv(s, v, i)

vwvv(s, v, i)z, (44)

πPi (s, v, z) = pi(s, v)(1− z). (45)

15

4.2 Power utility

This section analyzes the optimal value functions and investment strategies of the power utility investor. Section

4.2.1 gives notation and terminology. Section 4.2.2 specializes Theorem 4.2 and 4.3 to the case where the terminal

wealth of the investor is given by U(v) = vγ

γ , with 0 < γ < 1, and construct solutions for value functions and

investment strategies. Section 4.2.3 provides conditions under which a power utility investor would go long or short in

the defaultable bond security.

4.2.1 Notation and terminology

Throughout, Rn×m (respectively, Rn×m+ ) denotes the set of n×m (resp., positive) real matrices A. Given A ∈ Rn×m,

[A]i,j denotes its (i, j) entry. Next, we give some definitions, which will be used to characterize the optimal strategies.

Let us recall that Ct is given by (5) and ψ(t, T ) := (ψ1(t, T ), . . . , ψN (t, T ))′ denotes the pre-default regime conditioned

bond prices defined in (13). In all definitions to follow, we assume the macro-economy to be in the ith regime at

t. Let AΥ(t) = [aΥi,j(t)]i,j=1,...,N be the infinitesimal generator of the Markov process (Xt), under a given equivalent

probability measure Υ.

Definition 4.1. For any s < t, we have the following terminology:

• The expected corporate bond return per unit time under the measure Υ, during the interval [t, s], is defined as

EΥi (t, s) :=

1

s− tEΥ

[ψCs,T (s, T )− ψi(t, T )

ψi(t, T )

∣∣∣∣Xt = ei

]. (46)

• The expected instantaneous corporate bond return, under the measure Υ, is defined as

EΥi (t) := lim

s→t+EΥi (t, s). (47)

• The instantaneous forward rate of the defaultable bond at time t is defined as

gi(t) := −∂ logψi(t, T )

∂T

∣∣∣∣T=t

. (48)

Note that the above definitions are meaningful because the function ψi(t) is differentiable in time, as it has been

shown in Capponi and Figueroa-Lopez (2011) (Lemma A.2). We have the following useful results, which will be used

later (their proofs are reported in Appendix B.1).

Lemma 4.4. The instantaneous forward rate is given by

gi(t) = −[∑j 6=i

aQi,jψj(t, T )− ψi(t, T )

ψi(t, T )

]. (49)

Lemma 4.5. The instantaneous corporate bond return, under the equivalent measure Υ, is given by

EΥi (t) =

∑j 6=i

aΥi,j(t)

ψj(t, T )− ψi(t, T )

ψi(t, T ). (50)

4.2.2 Construction of solutions

In this part, we develop a numerical analysis of value functions and investment strategy. Let us recall that the investor’s

horizon R is assumed to be less than the maturity T of the defaultable bond. We start giving the expression for the

post-default value function and post-default stock investment strategy. The proof is reported in Appendix B.2.

16

Proposition 4.6. Assume that the aQi,j’s and ai,j’s are continuous in [0, T ]. Then,

(i) The optimal post-default value function is given by

ϕR(t, v, i) = vγK(t, i), (0 ≤ t ≤ R),

where K(t) = [K(t, 1),K(t, 2), . . .K(t,N)]′ is the unique positive solution of the linear system of first order

differential equations

Kt(t) = F (t)K(t), K(R) =1

γ1, (51)

with 1 = [1, . . . , 1]′ ∈ RN and

[F (t)]i,j =

−(γri − η2

i

2γγ−1 + ai,i(t)

), if i = j,

ai,j(t). if j 6= i.(52)

(ii) The optimal percentage of wealth invested in stock at time t in a post default scenario is given by

πSj (t) =µj − rjσ2j

1

1− γ. (53)

We also have the following corollary (see Appendix B for its proof).

Corollary 4.7. Assume the rate matrix F defined in (52) to be time invariant. Then, we have that

(1) The post-default value function is given by

K(t) = e(t−R)F 1

γ1′. (54)

(2) For each i ∈ 1, . . . , N, K(t, i) is a decreasing function of t.

Therefore, similarly to the logarithmic case analyzed in Capponi and Figueroa-Lopez (2011), we find that post-

default value functions and stock investment strategies may be computed explicitly. We now consider the pre-default

case. The following result gives sufficient conditions for the existence of the pre-default value function provided that a

certain non-linear system of ODE is well-posed.

Proposition 4.8. Assume that the aQi,j’s and ai,j’s are continuous in [0, T ] and let K(t) = [K(t, 1),K(t, 2), . . .K(t,N)]′

be the unique positive solution of the linear system of first order differential equations (51). Suppose

J(t) = [J(t, 1), J(t, 2), . . . J(t,N)]′, and p(t) = [p(t, 1), . . . , p(t,N)]′,

solve simultaneously the system of equations:

Jt(t) = G(t, p(t))J(t) + d(t, p(t)), J(R) =1

γ1, (55)

θi(t)J(t, i)− hiK(t, i)(1− pi(t))γ−1 +∑j 6=i

ai,j(t)J(t, j)

(ψj(t)

ψi(t)− 1

)(1 + pi(t)

(ψj(t)

ψi(t)− 1

))γ−1

= 0, (56)

where G : R+ ×RN → RN×N and d : R+ ×RN → RN×1 are given by

[G(t, p)]i,j = −ai,j(t)(

1 + pi

(ψj(t)

ψi(t)− 1

))γ, (i 6= j),

[G(t, p)]i,i = −(−η

2i

2

γ

γ − 1+ riγ + piγθi(t)− hi + ai,i(t)

),

[d(t, p)]i = −hi(1− pi)γK(t, i), p = [p1, . . . , pN ]′. (57)

17

Then, the optimal pre-default value function is given by

ϕRt (t, v, i) = vγJ(t, i). (58)

The optimal percentage of wealth invested in stock in the pre-default scenario is given by

πSj (t) =µj − rjσ2j

1

1− γ,

while the optimal percentage of wealth invested in bond is πPj (t) = 1τ>tpj(t).

The proof of the previous proposition follows immediately by plugging the function ϕRt (t, v, i) in Eq. (58) inside the

coupled system given by Eq. (42) and Eq. (43). The optimal stock strategy follows immediately from Theorem 4.3,

item (2), using Eq. (58). Therefore, the optimal demand in stock is myopic and independent from the value functions

and from the default event, while the optimal defaultable bond strategy is non-myopic and dependent on the relation

between historical and risk neutral regime switching probabilities. Note that the system (55)-(56) can be formulated

as a non-linear system of differential equations on R+ ×RN+ of the form:

Jt(t) = G(t, J(t))J(t) + d(t, J(t)), J(R) =1

γ1, (59)

where G : R+ ×RN+ → RN×N and d : R+ ×RN+ → RN×N are defined for J = [J1, . . . , JN ]′ ∈ RN+ and t ≥ 0 as

G(t, J) = G(t, p(t, J)), and d(t, J) = d(t, p(t, J)),

with G and d given as in Proposition 4.8, and p(t, J) := [p1(t, J), . . . , pN (t, J)]′ defined implicitly by the system of

equations

θi(t)Ji − hiK(t, i)(1− pi(t, J))γ−1 +∑j 6=i

ai,j(t)Jj

(ψj(t)

ψi(t)− 1

)(1 + pi(t, J)

(ψj(t)

ψi(t)− 1

))γ−1

= 0. (60)

The following Lemma shows that indeed p(t, J) is well-defined for (t, J) ∈ R+×RN+ (its proof is given in the Appendix

B.2).

Lemma 4.9. Assume J ∈ RN+ . The system (60) admits a unique real solution pi(t, J) in the interval (Mi(t), 1), where

Mi(t) is defined as in (34). Moreover, if for each i, j = 1, . . . , N , ai,j and aQi,j are differentiable functions of t, then

(t, J)→ p(t, J) is differentiable at each (t, J) ∈ R+ × (0,∞)N .

We can prove that the non-linear system (59) has a unique solution in a local neighborhood (t, J) ∈ R+ × RN+ :

|t− R| < a, |Ji − 1/γ| < b, i = 1, . . . , N for some a > 0 and b > 0. For illustration purposes, let us consider in detail

the case N = 2. In that case, the system (55-56) takes the form:

Jt(t, 1) = −a1,2(t)J(t, 2)

(1 + p1(t, J(t))

(ψ2(t)

ψ1(t)− 1

))γ(61)

− (ξ1(t) + γθ1(t)p1(t, J(t))) J(t, 1)− h1K(t, 1)(1− p1(t, J(t)))γ ,

Jt(t, 2) = −a2,1(t)J(t, 1)

(1 + p2(t, J(t))

(ψ1(t)

ψ2(t)− 1

))γ(62)

− (ξ2(t) + γθ2(t)p2(t, J(t))) J(t, 2)− h2K(t, 2)(1− p2(t, J(t)))γ ,

J(R, 1) = J(R, 2) =1

γ, (63)

18

where ξi(t) := −η2i

2γγ−1 + riγ − hi + ai,i(t) and the functions p1(t, J), p2(t, J) : R+ ×R2

+ → R are defined implicitly by

the following equations for any J := [J1, J2]:

0 = θ1(t)J1 − h1K(t, 1)(1− p1(t, J))γ−1 + a1,2(t)J2

(ψ2(t)

ψ1(t)− 1

)(1 + p1(t, J)

(ψ2(t)

ψ1(t)− 1

))γ−1

, (64)

0 = θ2(t)J2 − h2K(t, 2)(1− p2(t, J))γ−1 + a2,1(t)J1

(ψ1(t)

ψ2(t)− 1

)(1 + p2(t, J)

(ψ1(t)

ψ2(t)− 1

))γ−1

. (65)

Note that while the range of one of the functions pi’s is bounded, the other function will be unbounded. For instance,

if ψ2(t)/ψ1(t) > 1, then p1(t, J) will take values on the bounded domain (−(ψ2(t)/ψ1(t)− 1)−1, 1), while p2(t, J) will

take values on (−∞, 1). In turn, this fact makes the right hand-side of equation (62) potentially unbounded and also

is the main reason why it is not possible to obtain global existence without further restrictions (see Remark B.2 in

Appendix B.2 for more information). The following result shows the local existence and uniqueness of the solution

(the proof of Proposition 4.10 is reported in Appendix B.2).

Proposition 4.10. Suppose that ai,j(t) and aQi,j(t) are differentiable functions. Then, for any b > γ, there exists

an α := α(b) > 0 and a unique function J : (R − α,R] → [b−1, b]N satisfying (61-62) with terminal condition

J(R, 1) = J(R, 2) = γ−1.

Remark 4.11. Under the conditions of Proposition 4.10, it is known (see, e.g., Theorem 1.263 in Chicone (2006))

that if (R − α,R + α) (with α, α ∈ (0,∞]) is the maximal interval of existence of the solution of (61-63) and α <∞,

then either |J(t)| → ∞, J(t, 1) → 0, or J(t, 2) → 0 as t → R − α. Moreover, the solution of (61-63) can be found by

the standard Picard’s fixed-point algorithm. Hence, for instance, one can show numerically whether the solution is well

defined in the whole interval [0, R] by analyzing whether the numerical solution blows up or converges to 0.

We conclude the section by noticing that the pre-default scenario for the power investor is different from the one

faced by the logarithmic investor. Indeed, in the logarithmic case, the two systems decouple, see Capponi and Figueroa-

Lopez (2011) for details, and the pre-default value function may be obtained explicitly in terms of a matrix exponential

for time invariant generators.

4.2.3 Analysis of the bond investment strategy

In this section, we provide conditions under which a power utility investor would go long or short in the defaultable

bond. Let us define a measure P, equivalent to the historical measure P, via the generator AP = [aPi,j ] of the Markov

process given by

aPi,j(t) := ai,j(t)J(t, j)

J(t, i), (j 6= i), and aPi,i(t) := −

N∑k=1,k 6=i

aPi,j(t), (66)

where J(t, j) > 0 is the time component of the optimal pre-default value function defined by Eq. (55), (56), and (57).

Intuitively, the measure P is redistributing the mass of the historical distribution P towards those regimes j which

have higher values of the pre-default value function with respect to regime i. We have the following result (its proof is

reported in Appendix B.3).

Lemma 4.12. Under the assumptions of Proposition 4.8, we have that pi(t) > 0 if and only if

EPi (t) + gi(t) > hi

(K(t, i)

J(t, i)− Li

). (67)

In analogy with hi(1 − Li), representing the expected recovery rate in the i-th regime, we refer to the quantity

hi

(K(t,i)J(t,i) − Li

)as the adjusted expected recovery rate. We say that the long condition of the power utility investor is

19

satisfied when the relationship (67) holds. In financial terms, Lemma 4.12 says that a power investor computes the

expected return of the corporate bond under a probability measure equivalent to the historical measure, but adjusted

for default risk through the ratio of pre-default value functions. Then, he decides to go long in the security only if

such return plus the instantaneous forward rate is higher than the adjusted expected recovery rate. High values of

the instantaneous forward rate indicate high levels of default risk perceived by the market, and consequently lower

the price of the bond. Therefore, if the macro-economy is in regimes with high default risk premium, the logarithmic

investor would bear the default risk incurred by going long in the bond security. This result is in agreement with

Bielecki and Jang (2006), who also found that the pre-default optimal investment strategy in the defaultable bond is

an increasing function of the default event risk premium.

We next characterize the directionality of the strategy for the limiting case of a CRRA investor, namely an investor

with utility U(v) = log(v). Before doing so, we recall a lemma proven in Capponi and Figueroa-Lopez (2011) showing

that, under the ith economy regime, the optimal percentage of wealth invested in the defaultable bond by a logarithmic

investor is given by πPi (t) = 1τ>tpi(t), where pi(t) is identified in the following Lemma.

Lemma 4.13 (Capponi and Figueroa-Lopez (2011)). The system of equations

θi(s)−hi

1− pi+∑j 6=i

ai,j(s)ψj(s)− ψi(s)

ψi(s) + pi(ψj(s)− ψi(s))= 0, (68)

for i = 1, . . . , N , admits a unique real solution pi(s) in the interval (Mi, 1), where Mi ∈ [−∞, 0) is defined as in (34).

Moreover, if for each i, j = 1, . . . , N , ai,j and aQi,j are continuous functions, then p(s, i) is a continuous function of s.

We use the above lemma to provide necessary and sufficient conditions under which the logarithmic investor goes

long on the defaultable bond. The proof of the next lemma is reported in B.3.

Lemma 4.14. We have that pi(t) > 0 if and only if

EPi (t) + gi(t) > hi(1− Li). (69)

Similarly to the case of the power investor, if the relation in Eq. (69) holds, we say that the long condition of the

logarithmic investor holds. The long condition of the logarithmic investor is similar to the one of the power investor,

except that the former computes the expected return of the corporate bond under the historical measure, and decides

to go long in the defaultable security only if such return plus the instantaneous forward rate exceed the expected

recovery rate (as opposed to the adjusted one).

The following corollary provides sufficient conditions for the logarithmic investor to always go short in the defaultable

security. The proof is reported in Appendix B.3.

Corollary 4.15. For a logarithmic investor, the following statements hold:

(i) If N = 1, then for each fixed t, we have p(t) = 1− 1L1

< 0.

(ii) For fixed t, i, if aQi,j(t) = ai,j(t) for any j 6= i, then pi(t) < 0.

Corollary 4.15 show that in the mono-regime scenario, or in the case when aQi,j(t) = ai,j(t), the corporate bond

return gets reduced by the instantaneous forward credit spreads gi(t, T ) by an amount which makes it smaller than

the expected recovery at default. This leads the investor to go always short in the security, because the compensation

offered by the market is not enough to compensate him for the credit risk incurred. Moreover, item (i) of Corollary

4.15 shows that (1) in case of zero recovery on the defaultable bond (Li = 1), the logarithmic investor would not trade

at all in the defaultable security, and (2) the amount of bond units shorted is a decreasing function of the loss incurred

at default.

Although it is generally impossible to obtain explicit formulas for the bond investment strategy of a power investor,

it is possible to do so in special cases. The remainder of this section shows that this is indeed the case, if we consider

a square root utility investor, i.e. γ = 12 , and assume that N = 1. Below, we use the notation Lγ1 = L1γ − 1.

20

ai,j 1 2 3

1 -0.10474 0.08865 0.01609

2 0.84799 -0.848 0.00001

3 0.69561 0.00001 -0.69562

h L

1 0.741% 10%

2 4.261% 40%

3 11.137% 90%

Table 3: Left panel shows the historical generator of the Markov chain obtained in Giesecke et al. (2011). The rows

indicate the starting state, while the columns indicate the ending state of the chain. The right panel shows the default

intensities as reported in Giesecke et al. (2011) as well as our loss rates given default associated to three regimes.

Lemma 4.16. The optimal investment in the defaultable bond for a square root utility investor is given by

p1(t) =(L1 − 1)

(−e2h1tL

γ

+ e2h1RLγ1L1(Lγ1 +γ)

)e2h1tL

γ1L1(γ − 1) + e2h1RL

γ1 (L1 − 1)L1 (Lγ1 + γ)

(70)

Remark 4.17. It can be easily checked that p1(t) < 0. The numerator of Eq. (70) is positive because γ = 12 and

0 ≤ L1 ≤ 1. The denominator is negative because e2h1tLγ

> e2h1RLγ

and |γ− 1| > (L1− 1) (Lγ + γ), thus yielding that

p1(t) < 0.

Comparing Eq. (70) with item (i) of Corollary 4.15, we can immediately see that in mono-regime scenarios, for the

logarithmic investor the strategy only depends on loss given default, whereas for the power investor it also depends

on the default intensity and on time. Similarly to the logarithmic investor, we find that limL1→0 p1(t) = −∞ and

limL1→1 p1(t) = 0. Therefore, as in the case of the logarithmic investor, the investor does not allocate any wealth to

the defaultable bond if the loss L1 = 1. In case of a very low intensity h1, this may be explained by the fact that,

although with high probability default will not occur, in case when it does there is zero recovery, and thus a risk averse

investors will tend to avoid the exposure to default risk. In case of very high intensities, this happens because although

the investor will likely realize a profit by shorting the bond due to the high default probability, the bond selling price

will be close to zero if L1 → 1 and h1 → ∞ (see Eq. (10)). This, in turn, will make the realized profit equal to zero.

Moreover, we find the asymptotics limh1→0 p1(t) = 1− 1/L21 and limh1→∞ p1(t) = 2(1− 1/L1).

5 Comparative statics and economic interpretation

We present a comparative statics analysis of the corporate bond strategies and value functions as illustrated in Section

4. The objective is to investigate how the interplay between the historical and risk-neutral generator of the Markov

chain, time to maturity, default intensity and loss parameters, affect the directionality of the strategy. Moreover, we

illustrate how the risk aversion level γ of the power utility investor affects the bond investment strategy, including the

limiting case (γ = 0) of the logarithmic investor. We describe the simulation scenario in Section 5.1 and present the

comparative statics results in Section 5.2.

5.1 The simulation scenario

In order to present a realistic simulation setting, we take the estimates of the historical generator of the Markov chain

obtained by Giesecke et al. (2011), who employed a three-state homogenous regime switching model to examine the

effects of an array of financial and macro-economic variables in explaining variations in the realized default rates of

the U.S. corporate bond market over the course of 150 years. For completeness, we report their value estimates in

Table 3. Giesecke et al. (2011) also estimate the annual default rates for each regime. We report them in Table 3,

along with the corresponding losses, which we choose to be increasing in the credit riskiness of the regime. Table 3

shows three distinct regimes, hereon referred to as “low”, “middle”, and “high” default regime. It also indicates that

21

aQi,j 1 2 3

1 -0.380313 0.33687 0.043443

2 0.254397 -0.254397 0

3 0.208683 0.000006 -0.208689

Table 4: The generator of the Markov chain under the risk-neutral measure. The rows indicate the starting state,

while the columns indicate the ending state of the chain.

the probability of remaining in a low-default regime is very large, while the other two regimes are much less persistent.

Since our objective is to measure the impact of the default event on the optimal strategies, we assume that the annual

interest rate is the same across all regimes and equal to 3%. We also assume that the the annual stock volatility is equal

to 5% in all regimes. We set the drift of the stock equal to 7%, 5%, and 3%, respectively in the low, middle and high

default regime. The risk-neutral generator of the Markov chain is given in Table 4, and chosen so that the risk-neutral

probability of moving to riskier (safer) regimes is higher (lower) than the corresponding historical probabilities. This

is consistent with empirical findings showing the existence of a positive default risk premium. We take the investment

horizon R to be the same as the maturity T of the defaultable bond, and equal to one year.

5.2 Numerical results and strategy analysis

We present the results obtained under the simulation scenario detailed in Section 5.1. We use a fixed point algorithm to

solve the coupled system introduced in Proposition 4.8. Namely, the system consists of (1) a system of three ordinary

differential equations for the time component of the pre-default value function and (2) a system of three nonlinear

equations for the defaultable bond strategy. This algorithm initially sets the pre-default value function equal to the

post-default counterpart. Then, it keeps iterating between solving for the time component of the pre-default value

function and the bond investment strategy until a desired level of convergence is achieved. For convenience, we also

provide the pseudo-code of the Algorithm 3 in Appendix C.

We start showing the behavior of the bond investment strategy for the power utility investor under three different

levels of risk aversions, and for the logarithmic investor. Figure 3 shows that the investor always shorts the bond

security unless the macro-economy is in the high risk regime, and the time to maturity is not too small. This is in

agreement with the right bottom graph of the figure, showing that the power and logarithmic investor go long only if

the economy is in the high risk macro-economic regime and there are still about 2.4 months left to maturity.

In the high risk regime, the corporate bond returns are positive and of larger magnitude than the negative instan-

taneous forward rate. This is because, any transition from this regime will be towards a safer regime and will occur

with larger probability under the historical measure, see Table 3 and 4. On the contrary, in the low risk regime, the

corporate bond returns are negative and of smaller magnitude than the positive instantaneous forward rate, because,

any transition from this regime will be towards a riskier regime and will occur with higher probability under the risk

neutral measure. Therefore, in both of these cases the right hand side of Eq. (67) and Eq. (69) will be positive. How-

ever, in the high risk regime, historical transitions towards the low risk regime occurs with probability large enough to

guarantee that the long condition is satisfied when the time to maturity is not too small. On the contrary, in the low

risk regime, the risk neutral transition probabilities of moving towards the riskier regimes are small, and thus do not

generate instantaneous forward rates which are large enough to satisfy the long condition. In the middle risk regime,

the historical probability of transitioning to the low risk regime is very high, thus generating a positive corporate bond

return. However, the (adjusted) expected recovery is the largest in the middle risk regime (from the right panel in

Table 3, we can see that h2(1 − L2) > hi(1 − Li), i = 1, 3), and the long bond condition is never satisfied. All this

appears to indicate that, in an unfavorable market situation (which in our model corresponds to the macro-economy

being in the low default regime, from there the macro-economy can only get worse), it is preferable to go short in

defaultable assets. These results are in agreement with Callegaro et al. (2010), who come to similar conclusions with

22

0 0.2 0.4 0.6 0.8 1−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

0

Time

πP 1

γ=0.1γ=0.3γ=0.5log

0 0.2 0.4 0.6 0.8 1−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

Time

πP 2

γ=0.1γ=0.3γ=0.5log

0 0.2 0.4 0.6 0.8 1−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Time

πP 3

γ=0.1γ=0.3γ=0.5log

0 0.2 0.4 0.6 0.8 1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

Time

Long

/Sho

rt D

ista

nce

γ=0.5, Regime 1γ=0.5, Regime 2γ=0.5, Regime 3

Figure 3: Optimal bond strategy πP1 , πP2 and πP3 versus time, for different levels of risk aversion γ, and for the loga-

rithmic investor. The bottom right panel shows the long/short distance for the square root (γ = 0.5) and logarithmic

investor, respectively defined as EPi (t) + gi(t)− hi

(K(t,i)J(t,i) − Li

)and EP

i (t) + gi(t)− hi (1− Li).

23

0 0.2 0.4 0.6 0.8 11

1.05

1.1

1.15

1.2

1.25

Time

Pre

−D

efau

lt R

atio

s

J1/J

2

J1/J

3

J2/J

3

0 0.2 0.4 0.6 0.8 10.991

0.992

0.993

0.994

0.995

0.996

0.997

0.998

0.999

1

1.001

Time

Pos

t/Pre

−D

efau

lt R

atio

s

K1/J

1

K2/J

2

K3/J

3

Figure 4: The left panel reports the behavior over time of the pre-default value functions ratio. The right panel reports

the behavior over time of the post/pre default value functions ratio. The risk aversion level is γ = 0.5.

a different model.

As the time approaches maturity, the bond prices in the different regimes will get closer, see also the bond price

plots in Figure 1, and consequently the expected return will decrease, until reaching a point where the sum of expected

bond return and instantaneous forward rate becomes smaller than the (adjusted) expected recovery, which triggers the

investor decision to go short. The bottom left graph of Figure 3 shows that, in the high risk regime, the logarithmic

investor changes the directionality of his strategy from long to short before the power utility investor. The reason

for that can be understood from Figure 4. Here, we can see from the left graph that the pre-default value function

is higher in safer regimes. This means the power utility investor is more “optimistic” than the history, because his

expected corporate bond return, computed under the equivalent measure P given in Eq. (66), is always larger than the

one computed by the logarithmic investor under the historical measure P. Moreover, the right graph of Figure 4 shows

that the ratio of post vs pre-default value function is always smaller than one in all regimes. This implies that the

adjusted expected recovery is smaller than the expected recovery. The conclusion is that when the long bond condition

is satisfied for the logarithmic investor, it will be surely satisfied for the the power investor, and consequently the latter

will keep going long for larger times.

It is evident from Eq. (56) and (68) that the optimal investment strategy in the defaultable bond is time dependent

for both the power and logarithmic investor. The graphs of Figure 3 further illustrate that the investor buys (sell) a

larger (smaller) number of bond units when the time to maturity is higher. This happens because, for a given level of

default probability, the bond price appreciates in value as maturity approaches. As in our scenario, the risk neutral

generator is time invariant, the likelihood of a default event happening within a given interval remains the same as

time progresses. Therefore, the investor should buy more (sell less) in the defaultable bond when its price is low, that

is for longer time to maturity, all else being equal. Similar findings are also obtained from Bielecki and Jang (2006) in

a different framework.

Moreover, we can see from Figure 3 that the larger the risk aversion level of the investor, the smaller the number of

bond units traded. This is expected because an investor who goes long in the bond security is exposed to the default

risk, and therefore buys a smaller number of units with respect to a less risk averse investor. An investor who goes

short is instead exposed to regime switching risk, and thus sells a shorter number of units because this would result in

a mark-to-market loss in case the macro-economy transitions to a safer regime.

We conclude the section with an analysis of the behavior of the bond investment strategy as a function of the loss

given default in the high risk regime. Figure 5 shows that as the loss increases, the short (long) investor will sell (buy)

a smaller (higher) number of bond units. This is expected because, all else being equal, larger losses will translate into

cheaper bond prices, and therefore, following the buy low sell high rule, the long investor will buy more and the short

investor will sell less. As expected, more risk averse investors will trade a smaller number of bond units to reduce

exposure to default or regime switching risk.

24

0.5 0.6 0.7 0.8 0.9−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

L3

πP 1

γ=0.1γ=0.3γ=0.5log

0.5 0.6 0.7 0.8 0.9−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

L3

πP 3

γ=0.1γ=0.3γ=0.5log

Figure 5: Optimal bond strategy πP1 , and πP3 versus the loss given default, for different levels γ of risk aversion, and

for the logarithmic investor. The time t is fixed to 0.5.

6 Conclusions

We analyzed pricing and portfolio optimization problems in regime switching markets, and provided novel characteri-

zations of option prices and optimal portfolio strategies. First, we developed a novel methodology for pricing claims,

which can depend on the full path of the underlying Markov chain. We demonstrated the algorithm on defaultable

bonds and on barrier options on the stock volatility, showing that it achieves a high level of accuracy, and, at the same

time, it is computationally fast when compared to other existing methods. The methodology performs an equivalent

change of measure to make the generator of the Markov chain time homogenous, and then expands the price of the

claim in terms of the Laplace transform of the symmetric Dirichlet distribution. Under a reduced form credit risk model

with the recovery of market value assumption, we illustrated how the algorithm can be used to price vulnerable claims

on the stock price. We have then analyzed portfolio optimization problems under our defaultable regime switching

framework, and obtained explicit construction for pre/post default value functions as well as stock investment strategies

for the case of a CRRA investor. We provided a detailed economic interpretation of the conditions determining the

long/short directionality of the investment strategy in the defaultable asset. We have then analyzed the behavior of the

strategies under a realistic three-regime switching model. We found that investors go long when the macro-economy

is in the least favorable regime, where bond prices are cheap, and thus produce positive future returns. Additionally,

we found that the investor invests more in the defaultable bond if he is less risk averse, or if the planning horizon is

higher.

A Proofs related to Section 3

Proof of Lemma 3.1.

From the assumed time-invariance of the parameters, we have the following:

Φi,m[Ξ](ζ) = EQi

[Ξ(Xm)e−

∑mn=0 ζ(U(n+1)−U(n))r

′Xn−∑mn=1 X

′n−1KXn

]Let Λn := U(n)−U(n−1) for n = 1, . . . ,m+ 1 and note that

∑m+1n=1 Λn = 1. It is well known Kendall and Moran (1963)

that the distribution of

Λ := (Λ1, . . . ,Λm)

is the symmetric Dirichlet distribution with parameter ~α := (1, . . . , 1) ∈ Rm+1. We recall that a random vector

Λ := (Λ1, . . . ,Λm) follows a Dirichlet distribution with parameters ~α := (α1, . . . , αm+1) ∈ Rm+1 such that mini αi > 0

25

if its density is given by

D(λ1, . . . , λm) = B(α)−1

m∏i=1

λαi−1i (1− λ1 − · · · − λm)αm+1−11Tm(λ1, . . . , λm),

where B(α) :=∏m+1i=1 Γ(αi)/Γ

(∑m+1i=1 αi

)and Tm := (λ1, . . . , λm) ∈ Rm : λi ≥ 0 &

∑mi=1 λi ≤ 1. Next, condition-

ing on (X1, . . . , Xm) and using independence between U and X,

Φi,m[Ξ](ζ) = EQi

[ ∫Tm

e−∑mn=0 ζr

′Xnλn+1D(λ1, . . . , λm)dλ1 . . . dλmΞ(Xm)e−∑mn=1 X

′n−1KXn

],

where λm+1 = 1−∑mn=0 λn. The following simplification can be made:∫

Tm

e−∑mn=0 ζr

′Xnλn+1D(λ1, . . . , λm)dλ1 . . . dλm =

∫Tm

e−∑m−1n=0 ζr′(Xn−Xm)λn+1−ζr′XmD(λ1, . . . , λm)dλ1 . . . dλm

= e−ζr′XmLm

(ζr′(X0 − Xm), . . . , ζr′(Xm−1 − Xm)

),

where Lm is given as in (18). Finally, note that, by construction, Q(Xi = ek|Xi−1 = em) = 1/(N − 1) for any k 6= m

and, thus,

Q(X1 = e1, . . . , Xm = em|X0 = ei) =1

(N − 1)m,

for all (e1, . . . , em) ∈ e1, . . . , eNm such that ei 6= ei−1. In that case, it is clear that

Φi,m[Ξ](ζ) =∑

(e1,...,em)

ei 6=ei+1

Ξ(em)e−ζr′em−

∑mn=1 e

′n−1KenLm (ζr′(e0 − em), . . . , ζr′(em−1 − em))

1

(N − 1)m.

Proof of Proposition 3.5.

For simplicity, we write Ψi[Ξ](T ) instead of Ψi[Ξ](0;T ). Fix πk := δ1, . . . , δk such that δi > 0 and δ1 + · · ·+ δk < T .

Let Ψi,πk [Ξ](T ) be the approximation of Ψi[Ξ](T ) when using the δi’s to approximate the option prices. Specifically,

we define Ψi,πk [Ξ](T ) iteratively as follows. Set

Ψi,∅[Ξ](ζ) :=

M−1∑m=0

e−ζζm

m!Φi,m[Ξ](ζ), Ψi,∅[Ξ](ζ) :=

∞∑m=0

e−ζζm

m!Φi,m[Ξ](ζ).

Note that, for an absolute constant B (depending on maxi |Ξ(ei)|),

maxi

∣∣∣Ψi,∅[Ξ](t)− Ψi,∅[Ξ](t)∣∣∣ ≤ BtM , (t ≤ T ). (71)

Now, fixing πk−1 = (δ1, . . . , δk−1), suppose that Ψi,πk−1[Ξ] has already been defined. Let δk > 0 be such that

δ1 + · · ·+ δk < T . Then, we define

Ψi,πk [1](T ) := Ψi,∅

[Ψ·,πk−1

[Ξ](T − δk)]

(δk),

where πk := δ1, . . . , δk. Then, by an argument similar to (22), it will follow that

Ψi[Ξ](T ) = EQi

[eI0,δkΨCδk

[Ξ](T − δk)]

= EQi

[eI0,δk ΨCδk ,πk−1

[Ξ](T − δk)]

+ EQi

[eI0,δk

(ΨCδk

[Ξ](T − δk)− ΨCδk ,πk−1[Ξ](T − δk)

)]= Ψi,∅

[Ψ·,πk−1

[Ξ](T − δk)]

(δk) +O(δMk ) +O(δM1 + · · ·+ δMk−1 + (T −k∑1

δj)M ),

26

where we have used (71) and induction in the last equality above. Therefore,∣∣∣Ψi[Ξ](T )− Ψi,πk [Ξ](T )∣∣∣ ≤ B[δM1 + · · ·+ δMk + (T −

k∑1

δj)M ].

One can check that the previous bound is minimized when δ1 = · · · = δk = T/(k + 1). Thus,∣∣∣Ψi[Ξ](T )− Ψi,πk [Ξ](T )∣∣∣ ≤ B(k + 1)

TM

(k + 1)M= B

TM

(k + 1)M−1.

Proof of Lemma 3.7.

Let us consider the (F,Q)-martingale φ(t) := EQ[e−

∫ T0

(rs+hsLs)ds|Ft]. Since φ(t) = e−

∫ t0

(rs+hsLs)ds 〈ψ(t), Xt〉, where

ψ(t) is defined in (13), Ito’s formula leads to

dφ(t) = −(rt + htLt)e−

∫ t0

(ru+huLu)du 〈ψ(t), Xt〉 dt+ e−∫ t0

(ru+huLu)du(⟨dψ(t)

dt,Xt

⟩+⟨ψ(t), (AQ)′tXt

⟩ )dt

+ e−∫ t0

(ru+huLu)du⟨ψ(t), dMQ(t)

⟩,

where we have used the fact that ψ(t) is differentiable in t and the semi-martingale representation formula of our

Markov chain Xt under the pricing measure Q (see, e.g., Appendix B in Capponi and Figueroa-Lopez (2011)). As φ(t)

is a (F,Q)-(local) martingale, its drift term must be zero. When Xt = ei, such condition translates into the following

differential equation

dψi(t)

dt− (ri + hiLi)ψi(t) +

N∑j=1

aQi,j(t)ψj(t) = 0. (72)

As Xt can range in the set e1, . . . , eN, it follows that the vector of bond prices ψ(t) = (ψ1(t), . . . , ψN (t)) satisfies the

system (25).

B Proofs related to Section 4

We first recall a useful lemma, which will be needed for following proofs.

Lemma B.1 (Coddington and Levinson (1955), Kaczorek (2001)). Let F : R+ → RN×N and b : R+ → RN×1. For a

fix ς ∈ RN , consider the time varying linear system

xt(t) = F (t)x(t) + b(t), x(t0) = ς. (73)

Then, the following statements hold true:

(1) The system (73) admits a unique solution given by

x(t) = φF (t, t0)ς +

∫ t

t0

φF (s, t0)b(s)ds, (74)

where φF (t, t0) is defined by the Peano-Baker series

φF (t, t0) = IN +

∫ t

t0

F (s)ds+

∫ t

t0

F (s)

∫ s

t0

F (y)dyds+

∫ t

t0

F (s)

∫ s

t0

F (y)

∫ y

t0

F (z)dzdyds+ . . . ,

with IN denoting the N dimensional identity matrix.

(2) For all t ≥ t0, φF (t, t0) has all nonnegative entries if and only if the off-diagonal entries of F (t) satisfy the

condition∫ tt0

[F (s)]i,jds ≥ 0, for i 6= j and all t ≥ t0.

(3) If the matrix F (t) is time invariant (i.e. F (t) ≡ F for any t and some matrix F ), then φF (t, t0) = exp ((t− t0)F ),

where exp(B) denotes the exponential of a square-matrix B.

27

B.1 Proofs related to Section 4.2.1

Proof of Lemma 4.4. A simple calculation shows that

∂ψi(t, T )

∂T

∣∣∣∣T=t

= lim∆t→0

ψi(t, T )− ψi(t+ ∆t, T )

∆t= lim

∆t→0

∑j 6=i

pQi,j(t, t+ ∆t)

∆t(ψj(t+ ∆t, T )− ψi(t+ ∆t, T ))

=∑j 6=i

aQi,j (ψj(t, T )− ψi(t, T )) .

Using the previous equation and the definition of instantaneous forward rate given in Eq. (48), we obtain Eq. (49).

Proof of Lemma 4.5. Let pΥi,j(t, s) denote the probability that the chain with generator AΥ(t) transits to regime j

at time s > t, given that it is in regime i at time t. Then,

EΥi (t) = lim

s→t+1

s− tEΥ

[ψXs(s, T )− ψi(t, T )

ψi(t, T )

∣∣∣∣Xt = ei

]= lims→t+

∑j 6=i

pΥi,j(t, s)

s− tψj(s, T )− ψi(t, T )

ψi(t, T )

=∑j 6=i

aΥi,j(t)

ψj(s, T )− ψi(t, T )

ψi(t, T ).

B.2 Proofs related to Section 4.2.2

Proof of Proposition 4.6.

(i) It can be checked by direct substitution that ϕRt

(t, v, i) = K(t, i)vγ solves the Dirichlet problem (37) with

terminal condition U(v) = vγ

γ , if and only if the functions K(t, i), i = 1, . . . , N , 0 ≤ t ≤ R, satisfy the system of

ODE’s given by Eq. (51). Using the substitution s = R − t, we have that the solution K(s) of the initial value

problem given by

Ks(s) = −F (R− s)K(s) (0 ≤ s ≤ R), K(0, i) =1

γ, (i = 1, . . . , N), (75)

is such that K(t) = K(R−t). Using Lemma B.1, part (1), we have that the unique solution of system (75) may be

written as K(s) = φF (R−s,R) 1γ1. Therefore, using that K(t) = K(R− t), we obtain that K(t) = φF (t, R) 1

γ1 =

φ−F (R, t) 1γ1. As for all i 6= j, and for all t, we have [F (t)]i,j ≤ 0, then

∫ t0[−F (s)]i,jds ≥ 0. Therefore using

Lemma B.1, part (2), we obtain that φ−F (R, t) has all nonnegative entries, and consequently K(t, i) ≥ 0 for all

0 ≤ t ≤ R and i = 1, . . . , N . Therefore, ϕRt

(t, v, i) ∈ C01,2 due that K(t, i) ≥ 0 and vγ is concave and increasing

in v. Under the choice D(t) = 0, E(t) = maxi=1,...,N K(t, i) and G(t) = | 1γ−1 |, the function ϕR

t(t, v, i) satisfies

the conditions in (38). Therefore, applying Theorem 4.2, we can conclude that, for each i = 1, . . . , N , ϕRt

(t, v, i)

is the optimal post-default value function.

(ii) Plugging the expression for ϕRt

(t, v, i) inside Eq. (40), we obtain immediately Eq. (53).

Proof of Corollary 4.7.

(i) It follows directly from Lemma B.1, part (3), using the expression K(s) = φF (R− s,R).

28

(ii) It is enough to prove that the time derivative vector K ′(t) consists of all negative entries. From Eq. (54), we

obtain that K ′(t) = Fe(t−R)F 1γ1′. Using the well known fact that if two matrices A and B commute, then

AetB = etBA, we get K ′(t) = e(t−R)FF 1γ1′. We notice that F 1

γ1′ is a vector whose entries are negative and

given by [F

1

γ1′]i

= −γri +η2i

2

γ

γ − 1

Since (t − R)F consists of positive off-diagonal entries, from lemma B.1, part (2), we have that e(t−R)F has all

nonnegative entries, and consequently K ′(t, i) ≤ 0 for all i and t, thus completing the proof.

Proof of Lemma 4.9.

For fixed i and s, consider the functions

fi(t, p, J) = θi(t)Ji − hiK(t, i)(1− pi)γ−1 +∑j 6=i

ai,j(t)Jj

(ψj(t)

ψi(t)− 1

)(1 + pi

(ψj(t)

ψi(t)− 1

))γ−1

, (76)

where J = [J1, . . . , JN ]′ and p = [p1, . . . , pN ]′. We observe that fi(t, p, J) is a continuous function of pi in the interval

(Mi(t), 1). Moreover, we know by Proposition 4.6 that K(t, i) ≥ 0 and, by assumption Jj ≥ 0, thus implying that

pi → fi(t, p, J) is strictly decreasing in pi ∈ (Mi, 1). We consider two cases: Mi ∈ (−∞, 0) and Mi = −∞. In the

first case, it is easy to check that limp→M+ifi(t, p, J) = ∞ and limp→1− fi(t, p, J) = −∞. Therefore, applying the

Intermediate Value Theorem, there exists unique p(t, J) = [p1(t, J), . . . , pN (t, J)]′ such that fi(t, p(t, J), J) = 0, for

i = 1, . . . , N . The case Mi = −∞ means that ψj(t)/ψi(t) ≤ 1 for all j 6= i. Then, we have limp→−∞ fi(t, p, J) = θi(t)Ji.

By the definition (41), θi > 0 (as hi, Li > 0) and, hence, Intermediate Value Theorem implies again the existence of

a unique p(t, J) = [p1(t, J), . . . , pN (t, J)]′. The differentiability of p(t, J) follows directly from the implicit function

theorem.

Proof of Proposition 4.10.

Let us denote f1(t, J(t)) and f2(t, J(t)), with J(t) = [J(t, 1), J(t, 2)]′, the right-hand side of the differential equations

(61) and (62), respectively. Similarly, g1(t, p1(t, J), J) and g2(t, p2(t, J), J), with J = [J1, J2]′, denote the right-hand

side of the equations (64) and (65), respectively. First, we show that for any fixed b > 1 and t ∈ [0, R], the functions

f1, f2 are bounded and Lipschitz on the domain R = [0, R] × [b−1, b]2. Let us assume ψ2(t)/ψ1(t) > 1 (the case

ψ2(t)/ψ1(t) < 1 can be treated similarly). In light of (64), p1(t, J) will take values on (−(ψ2(t)/ψ1(t) − 1)−1, 1) and

hence, f1(t, J) is uniformly bounded as follows:

supb−1<J1,J2<b

|f1(t, J)| ≤ |ξ1(t)|+ b

(|a12||ψ2/ψ1|γ +K(t, 2)| ψ2

ψ2 − ψ1|γ).

where for simplicity we had omitted the dependence on t of the functions appearing on the right-hand. We conclude

that for a bounded function C1(t) > 0,

supb−1<J1,J2<b

|f1(t, J)| ≤ C1(t)b.

In order to estimate, f2 on the given domain, let us proceed as follows. Noticed the following inequality (still assuming

ψ2(t)/ψ1(t) > 1), valid for p2 < 0:

g2(t, p, J) ≥ θ2J2 +

[a21J1

(ψ1

ψ2− 1

)− h2K2

](1 + p2

(ψ1

ψ2− 1

))γ−1

.

Therefore, we can lower bound the root of g(t, p, J) with the root of the right-hand side in the previous inequality.

Then,

p2(t, J) ≥ − ψ2

ψ2 − ψ1

([a1,2(1− ψ1

ψ2)J1 + h2K2]/θ2J2

)1/(1−γ)

. (77)

29

In terms of b, we have p2(t, J) ≥ − C(t)b2/(1−γ), for some bounded function C(t) > 0. This fact will imply that

supb−1<J1,J2<b

|f2(t, J)| ≤ C2(t)b(3−γ)/(1−γ),

for a bounded function C2(t). In order to show that f1, f2 satisfy a Lipschitz condition in the Region R, it suffices to

show that each pi(t, J) remains bounded away from Mi(t) and 1 when (t, J) ∈ R. Indeed, suppose for instance that

ψ2(t)/ψ1(t) > 1. By definition,

0 = θ1(t)J1 − h1K(t, 1)(1− p1(t, J))γ−1 + a1,2(t)J2

(ψ2(t)

ψ1(t)− 1

)(1 + p1(t, J)

(ψ2(t)

ψ1(t)− 1

))γ−1

.

Hence, if p1 is either close to 1 or M1(t) = −(ψ2(t)/ψ1(t) − 1)−1, J1 will be arbitrarily large when J2 is bounded.

Similarly, p2(t, J) cannot be arbitrarily close to 1. Also, when J1, J2 ∈ (b−1, b), p2 > C > −∞ for some C in light

of (77). Finally, we use the classical existence theorems (see, e.g., Theorem I-1-4 in Hsieh (2009)) to conclude the

existence of the solution for t ∈ [R− α,R].

Remark B.2. One of the consequences of the Theorem I-1-4 in Hsieh (2009) is that one can take α = minR, b/M,where M(b) := maxi sup(t,J)∈R |fi(t, J)|, where as before R := [0, R] × (b−1, b)2. Typically, M(b) increases when b

increase and hence, α might decrease. As seen in the proof, M(b) seems to increase as b(3−γ)/γ , and hence, M(b)/b→ 0

as b→∞. We can however take α = 0 if, e.g., K, ai,j, |ψ2/ψ1|, etc. are made small enough.

B.3 Proofs related to Section 4.2.3

Proof of Lemma 4.12. First, we show that pi(t) > 0 if and only if the following relation holds.∑j 6=i

ai,j(t)J(t, j)ψj(t)− ψi(t)

ψi(t)> hi(K(t, i)− LiJ(t, i)) +

∑j 6=i

aQi,j(t)J(t, i)ψj(t)− ψi(t)

ψi(t)(78)

We may rewrite Eq. (56) from Proposition 4.8 as

∑j 6=i

ai,j(t)J(t, j)

ψi(t)ψj(t)−ψi(t)

(1 + pi(t)

ψj(t)−ψi(t)ψi(t)

)1−γ =hi

(1− pi(t))1−γK(t, i)− hiLiJ(t, i) +∑j 6=i

aQi,j(t)J(t, i)ψi(t)

ψj(t)−ψi(t)

, (79)

where we have used the expression for θi(t) given in Eq. (41). It can be easily checked that the left hand side of

Eq. (79) is a decreasing function of pi(t) from (Mi, 1) to (−∞,∞). The right hand side, instead, is a strictly increasing

function of pi(t), defined from (Mi, 1) to (0,∞). Since we are assuming that there exists a unique solution pi(t) to the

nonlinear equation (79), then we can evaluate both left and right hand side of Eq. (79) at pi(t) = 0, and obtain that

pi(t) > 0 if and only if Eq. (78) holds. From the definition of expected instantaneous corporate bond return given in

Eq. (47), computed under the measure Υ = P, we obtain that EPi (t) =

∑j 6=i a

Pi,j(t)(ψj(t) − ψi(t))/ψi(t). Using the

latter relationship and Eq. (49), we may rewrite Eq. (80) as in Eq. (67).

Proof of Lemma 4.14. First, we establish that pi(t) > 0 if and only if the following relation holds.∑j 6=i

(ψj(t)

ψi(t)− 1

)(ai,j(t)− aQi,j(t)

)> hi(1− Li) (80)

Using Lemma 4.13 and Eq. (41), we have the fraction of wealth invested in the bond at time t satisfies the following

equation ∑j 6=i

ai,j(t)1

pi(t) + ψi(t)ψj(t)−ψi(t)

=hi

1− pi(t)− hiLi +

∑j 6=i

aQi,j(t)1

ψi(t)ψj(t)−ψi(t)

(81)

30

Notice that, for each fixed t, the left hand side is a strictly decreasing function of pi(t) from (Mi, 1) to (−∞,∞).

The right hand side, instead, is a strictly increasing function of pi(t) defined from (Mi, 1) to (0,∞). Moreover, we

know from lemma 4.13 that there exists a unique pi(t) satisfying Eq. (81). Evaluating both left and right hand side

at pi(t) = 0 leads to the conclusion that pi(t) > 0 if and only if Eq. (80) holds. From the definition of expected

instantaneous corporate bond return given in Eq. (47), computed under the historical measure Υ = P, and using

Eq. (3), we obtain that EPi (t) =

∑j 6=i ai,j(t)(ψj(t) − ψi(t))/ψi(t). Using this relation and Eq. (49), we may rewrite

Eq. (80) as in Eq. (69).

Proof of Corollary 4.15. Both in the case when N = 1 and in the case when aQi,j(t) = ai,j(t), we have that

EPi (t) = gi(t). Therefore, the long condition in (69) will be never satisfied. Moreover, in case when N = 1, we

can see directly from Eq. (68) that pi(t) = 1− 1L1

. This ends the proof.

Proof of Lemma 4.16. Using Eq. (56), we obtain that

p1(t) = 1−(L1J(t, 1)

K(t, 1)

)−2

. (82)

Plugging Eq. (82) into Eq. (55), we obtain

Jt(t, 1) =

(η2

1

2

γ

γ − 1− r1γ + h1

)J(t, 1) + h1L1J(t, 1)

(L1J(t, 1)

K(t, 1)

)−2

(γ − 1)− h1L1γJ(t, 1). (83)

Moreover, notice that the post-default time component of the solution K(t, 1) satisfies

Kt(t, 1) = −(γr1 −

η21

2

γ

γ − 1

)K(t, 1), K(R, 1) =

1

γ, (84)

leading to K(t, 1) = 1γ exp

γ1r1 − η2

1

2γγ−1

. Let us denote by

a :=η2

1

2

γ

γ − 1− r1γ + h1 − h1L1γ, b(t) :=

h1

L1(γ − 1)K2(t, 1). (85)

Then, we find that Eq. (83) admits two solutions, given by

J(t, 1) = ±

√√√√− 1

γ2e2a(t−R)

(−1 + 2e2aRγ2

∫ R

t

e−2asb(s)ds

)(86)

which can be evaluated, using Eq. (85), to obtain

J(t, 1) = ±

√√√√exp− (R−t)γ(η2−2r1(γ−1))

γ−1 − 2h1tLγ

L1γ2Lγ

√exp 2h1tLγ (γ − 1) + exp 2h1RLγ (L1 − 1)(Lγ + γ).

Note that the expression under the square root is nonnegative, thus J(t, 1) is well defined. Moreover, we know that the

post-default value function is concave, which means that we need to only consider the positive solution in (86). Plugging

Eq. (84) and Eq. (87) into Eq. (82), we can conclude that p1(t) evaluates to the expression given in Eq. (70).

C Pseudo-codes

Note: In the Algorithm 3, the function Compute Bond Strategy solves the nonlinear system of equations given by

Eq. (56) for a given J(t). The function Compute Pre-Default Value Fn solves the in-homogenous system of differential

equations given by Eq. (55) for a given πP .

31

Algorithm 1 [Prices] = PriceClaim(T , Ξ, δ, M)

ζ = minT, δfor i = 1, . . . , N

Prices(i, 1) = e−ζ∑M−1m=0

ζm

m! Φi,m[Ξ](ζ)

endfor

if T ≤ δReturn Prices(:,1)

else

for j = 2, 3, . . . , dTδ efor i = 1, . . . , N

Prices(i, j) = e−δ∑M−1m=0

δm

m! Φi,m[Prices(:, j− 1)](δ)

endfor

endfor

endif

Return Prices(:, :)

Algorithm 2 [Price] = ComputeVulnOption(T , δ, ∆,M ,k)

if T ≤ δfor i = −B,−B + 1, . . . , B

for j = 1, 2, . . . , N, . . . , B

Ξi(j) = BS(T ; sei∆+Trj , σ2

j , 0,K)

endfor

[Price(i, 1), . . . ,Price(i,N)] = PriceClaim(T,Ξi, T/k,M)

endfor

else

Pricenew = ComputeVulnOption(T − δ, δ,∆,M, k)

for i = −B,−B + 1, . . . , B

for j = 1, 2, . . . , N, . . . , B

Ξi(j) =∑Bk=−B Pricenew(k, j)

∫ zi,jkzi,jk−1

1√2πe−z

2/2dz

endfor

[Price(i, 1), . . . ,Price(i,N)] = PriceClaim(δ, Ξi, δ/k,M)

endfor

endif

D Computational speed comparisons

Table 5 compares the computational times for our proposed method (Algorithm 1) with respect to both the Runge-

Kutta and a matrix exponential approximation method based on the representation (26) using the parameter setup

of Tables 3 and 4 given in Section 5.1. For each given pair (δ, T ), the bond prices (ψ1(ζ), ψ2(ζ), ψ3(ζ)) are computed

for all time-to-maturity ζ ∈ δ, 2δ, . . . , kδ := T. The entries in Table 5 represent the ratio of the new algorithm’s

computational time with M = 2 to the respective ODE Method computational time and the matrix exponential

computational time. As we can see, our method tends to outperform ODE Method for maturities shorter than 15 years

and also for mesh sizes larger than 0.05 years; however, even in the worst case our method operates at a speed which

is very comparable to the faster of the two.

32

Algorithm 3 [π, J ] = FixedPoint(γ,R, ε,Ξ, δ,M)

ψ = PriceClaim(R,Ξ, δ,M)

J = K

πPprev = Compute Bond Strategy(ψ, J, γ,R)

Jprev = Compute Pre-Default Value Fn(ψ, πPprev, γ, R)

repeat

πP = Compute Bond Strategy(ψ, Jprev, γ, R)

J = Compute Pre-Default Value Fn(ψ, πP , γ, R)

distπ = ||πP − πPprev||2distJ = ||J − Jprev||2πPprev = πP

Jprev = J

until (distJ < ε) ∧ (distπ < ε)

πP = πP

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0.025 0.5757 0.6770 0.9240 1.1613 1.0855 1.2903 1.4646 1.1325

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