Portfolio Choice with Market-Credit Risk Dependencies

# Portfolio Choice with Market-Credit Risk Dependencies

Lijun Bo Email: lijunbo@ustc.edu.cn, School of Mathematics and Statistics, Xidian University, Xi’an 710071, China and School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China.    Agostino Capponi E-mail: ac3827@columbia.edu, Department of Industrial Engineering and Operations Research, Columbia University, New York, 10027, NY, USA.
###### Abstract

We study an optimal investment/consumption problem in a model capturing market and credit risk dependencies. Stochastic factors drive both the default intensity and the volatility of the stocks in the portfolio. We use the martingale approach and analyze the recursive system of nonlinear Hamilton-Jacobi-Bellman equations associated with the dual problem. We transform such a system into an equivalent system of semi-linear PDEs, for which we establish existence and uniqueness of a bounded global classical solution. We obtain explicit representations for the optimal strategy, consumption path and wealth process, in terms of the solution to the recursive system of semi-linear PDEs. We numerically analyze the sensitivity of the optimal investment strategies to risk aversion, default risk and volatility.

AMS 2000 subject classifications. 91G10, 91G40, 60J20

Key words. investment/consumption problem, stochastic factors, martingale method, recursive system of PDEs

## 1 Introduction

Portfolio optimization problems, originating from the seminal work of Merton (1971), have been the subject of considerable investigation. Important developments include the impact of trading constraints (see Cvitanić (2001) for a survey), the inclusion of stochastic volatility, see for instance Fouque et al. (2017), and the forward utility approach (see Musiela and Zariphopoulou (2006)) to model the time changing preferences of an agent. The most direct extension of the log-normal assumption made by Merton (1971) is the stochastic factor model. Such a model is able to capture empirically observed features of price processes and has been successfully used in several contexts, including stochastic interest rates (e.g. Brennan and Xia (2000)), mean returns of individual stocks (e.g. Bielecki and Pliska (1999)), and stochastic volatility (e.g. Castaneda-Leyva and Hernandez-Hernandez (2005); Fouque et al. (2017)). We also refer to Zariphopoulou (2009) for an excellent overview of the stochastic factor model.

The objective of the present paper is to introduce a credit portfolio framework to assess the joint impact of systemic and macroeconomic factors on the optimal portfolio strategy of an investor. To the best of our knowledge, ours is the first model to incorporate in a tractable manner the impact of both stochastic volatility and systemic risk on optimal portfolio allocations. Prior work has focused on stochastic volatility models as surveyed above, direct contagion models based on interacting intensities (e.g. Bo and Capponi (2016) and Bo and Capponi (2017) and Jarrow and Yu (2001)), and default correlation through exposure to systematic factors (e.g. Callegaro et al. (2012)). None of the above studies take into account the joint impact of market and credit risk on optimal investment. Empirical studies, however, suggest that the interplay of these two risks plays a critical role. It is well documented that stock return volatility is stochastic (e.g. Bollerslev et al. (1994); Ghysels et al. (1996)); moreover, there is evidence that credit spreads of a company are positively related to the equity return volatilities (e.g Campbell et al. (2003); Bakshi et al. (2006)). In the pricing space, model specifications featuring the dependence of default intensities on asset volatilities have been proposed by Bayarakhtar and Yang (2011), Carr and Wu (2009), Carr and Linetsky (2006) and Mendoza et al. (2010), but only for a security underwritten by an individual entity. Mendoza and Linetsky (2016) extend the analysis to a multi-name credit-equity model. The importance of credit-market risk dependencies has also been highlighted by Basel (2009), which provides empirical evidence for their interaction both at the macro level, and at the micro level (sensitivity of individual bank risk to different risk factors). A related branch of the literature has analyzed the optimal investment problem in a portfolio consisting of default risk sensitive assets. Bielecki and Jang (2006) consider a portfolio consisting a stock and a bond, but assume their price processes to be independent, thus ignoring market-credit risk dependencies. Pham (2010) and Jiao et al. (2013) study optimal investment in a portfolio model consisting of multiple securities subject to default risk. They decompose the original control problem defined under the enlarged filtration, inclusive of default event information, into classical stochastic control problems under the reference filtration, using a finite backward induction procedure. Iftimie et al. (2016) use the dual approach to solve the portfolio optimization problem in a market environment where the risk-free interest rate process can experience sudden jumps. Di Nunno and Sjursen (2014) consider optimal investment in defaultable assets when the investor has access to different sets of information. They find necessary and sufficient conditions for the existence of a portfolio which locally maximizes the expected investor’s utility from terminal wealth.

We consider a risk averse investor with power utility, who allocates his wealth across defaultable stocks and a bank account. The default intensity of a stock depends not only on its volatility, but also on common factors that influence the volatility processes of other stocks in the portfolio. These factors model the evolution of macro-economic variables which influence both the market and the credit risk of the portfolio. Moreover, the default intensity of each stock exhibit jumps when other stocks in the portfolio default. Empirically, it has been shown that for many financial sectors, e.g commercial banks, the default likelihood of an entity is likely to abruptly increase if some of its major counterparties default, see also Yu (2007).

There are several mathematical contributions in our efforts. Our portfolio analysis employs the martingale approach to deal with market incompleteness, as in Karatzas and Shreve (1998) (Chapter 5) and Kramkov and Schachermayer (1999). Because of the default risk, we need to introduce an additional control process to establish the dual stochastic control problem. We show that the value function of the dual problem satisfies a recursive system of default-state dependent nonlinear PDEs. Using the power transform method developed by Zariphopoulou (2001), we transform the original system into a recursive system of semi-linear PDEs whose nonlinear coefficients are still non-Lipschitz continuous. We then employ a two-steps approach to establish existence and uniqueness of a smooth solution to the system. We first construct a system of truncated PDEs using stopping time arguments. Such a system falls within the class of semi-linear PDEs analyzed by Becherer and Schweizer (2005), for which existence and uniqueness of a bounded classical solution can be guaranteed. Using probabilistic representations of classical solutions to PDEs, we show the equivalence between the truncated system and the original system. We further prove that the gradient of the solution to the recursive system is bounded, and obtain a closed-form representation for the optimal admissible investment strategy, consumption path and wealth process. Our paper is also related to Pham (2002), who studies an optimal investment problem under stochastic volatility. He discusses existence of classical solutions and provides gradient estimates for the solution to the HJB equation of the primal problem. His methodology cannot deal with the additional jump-to-default term appearing in our Hamiltonian.

We develop a numerical study for a special case of our model setup, in which the stochastic factor is constant and the portfolio consists of two stocks subject to credit risk. We find that the signs associated with the sensitivities of the investment strategies to the model parameters are in line with economic intuition. The investor reduces his holdings in the stock if (i) the default probability of the stock increases, (ii) his risk aversion increases, (iii) the planning horizon gets shorter, and (iv) the volatility parameter of the stocks’ price processes increases.

The rest of the paper is organized as follows. Section 2 introduces the portfolio model and formulates the primal problem. Section 3 develops the dual formulation and provides a verification result. Section 4 gives the optimal investment/consumption strategy. Section 5 develops a numerical analysis. Section 6 concludes.

## 2 The Model and Investor’s Problem

We describe the model in Section 2.1. We set up the primal problem of the investor in Section 2.2.

### 2.1 The Model

The portfolio model consists of defaultable stocks and a risk-free bank account with dynamics , where is the constant interest rate.111Throughout the paper, we consider a constant interest rate since this is not the main focus of our analysis. Our results can be easily extended to the case of a stochastic interest rate as long as is in for each default state . We fix to be the finite target horizon and consider a complete filtered probability space , where . Two independent -dimensional standard Brownian motions and generate a filtration , where . We use to denote the transpose operator. The default state is described by a -dimensional default indicator process with state space , where if the asset has defaulted by time and otherwise. The default time of the -th security is given by for . For , the sigma-algebra , where , contains information about default events of the stocks up to time . The filtration contains all information about default events till the target horizon . Our model consists of three blocks: a stochastic factor, the price processes and the credit model. The stochastic factor influences not only the returns and volatility of the prices, but also the credit risk of the stocks.

Stochastic factor. It is a reduced form model for the evolution of macroeconomic variables. Examples of these variables are interest rates, broad share price indices, and measures of economic activity or growth. Such a factor drives the drift, volatility, and the default intensities of the stock price processes. Consider a domain (open connected subset) . The process is referred to as the stochastic factor and has dynamics given by, , and

 dYt =μ0(Yt)dt+σ0(Yt)[ρdWt+√1−ρ2d¯Wt], (1)

where the correlation coefficient . The drift coefficient is an -valued column vector of functions and is an -valued matrix of functions. For each , let , , be the solution of Eq. (1), with the constraint at time .

Credit risk model. We assume that the bivariate process is Markovian with state space . The default indicator process transits from a state in which the stock is alive () to the neighbouring state in which the stock has defaulted at a stochastic rate . Notice that the default intensity of the -th stock may depend on the default state of other stocks in the portfolio, but is defined on the event that . By construction, simultaneous defaults are precluded in the model because transitions from can only occur to a state differing from in exactly one of the entries. The intensity function is assumed to be strictly positive for all . The default intensity of the -th stock may change if (i) a stock in the portfolio defaults (counterparty risk effect), and (ii) there are fluctuations in the macro-economic environment. Our default model thus belongs to the rich class of interacting Markovian intensity models, introduced by Frey and Runggaldier (2010).

Price processes. The vector of price processes of the stocks is denoted by . For , the price process of the -th defaultable stock is given by

 ~Pit=(1−Hit)Pit,i=1,…,n. (2)

In other words, the price of the -th stock is given by the predefault price up to , and jumps to at time , where it remains forever afterwards. The dynamics of the pre-default price process of the defaultable stocks is given by

 dPt =diag(Pt)[(μ(Yt)+λ(Yt,Ht))dt+σ(Yt)dWt]. (3)

In the above expression, is the diagonal -dimensional square matrix whose -th entry is . The vector is an -valued function and the matrix is an -valued function. Further, is assumed to be invertible and its inverse is denoted by . The vector is the vector of default intensities. Eq. (3) indicates that the investor holding the credit sensitive stock is compensated for the incurred default risk at the premium rate . Using (2), (3) and integration by parts, we obtain the dynamics given by

 d~Pt=diag(~Pt−)[μ(Yt)dt+σ(Yt)dWt−dMt], (4)

where is a pure jump -martingale given by, for ,

 Mit :=Hit−∫t∧τi0λi(Ys,Hs)ds,      t∈[0,T]. (5)

The joint process satisfying (4) and (5) can be constructed in an iterative manner following a procedure similar to that in Lemma A.1 of Capponi and Frei (2016). For completeness, we next give the construction of the process : Let be independent standard exponentially distributed random variables. We assume that these are also independent of the Brownian motions . We first consider the following SDE given by, for ,

 d~P0,it~P0,it =(μi(Yt)+λi(Yt,0))dt+n∑j=1σij(Yt)dWjt,~P0,i0=~Pi0>0. (6)

Then is a geometric Brownian motion whose coefficients depend on stochastic factors. Assume that no default has occurred at inception, i.e., . We define as the first time that either of the stocks defaults, i.e.,

 ^τ1 :=mini=1,…,nτ1i,τ1i:=inf{t>0; ∫t0λi(Ys,0)ds≥ξ1i},  i=1,…,n.

For , we set and when . Further, define and define for . For , consider the following SDE: on ,

 ~P1,it =~P0,iτ1+∫t^τ1~P1,is(μi(Ys)+λi(Ys,0i1))ds+n∑j=1∫t^τ1~P1,isσij(Ys)dWjs. (7)

We use to denote the -dimensional row vector whose entries are except for the -th entry which is set to . It can be easily seen that Eq. (7) admits a unique positive strong solution on . Further, define the second default time

 ^τ2 :=mini∈{1,…,n}∖{i1}τ2i,τ2i:=inf{t≥^τ1; ∫t^τ1λi(Ys,0i1)ds≥ξ2i},  i∈{1,…,n}∖{i1}.

Proceeding similarly to the construction of the process for , we set for all and if . Moreover, let . More generally, for , the -th default time is specified by

 ^τk :=mini∈{1,…,n}∖{i1,…,ik−1}τki, τki :=inf{t≥^τk−1; ∫t^τk−1λi(Ys,0i1,…,ik−1)ds≥ξki},  i∈{1,…,k}∖{i1,…,ik−1}.

The defaulted names in the equation above are defined following an iterative procedure, in a similar way to and . For , and for , consider the process

 ~Pk−1,it =~Pk−2,i^τk−1+∫t^τk−1~Pk−1,is(μi(Ys)+λi(Ys,0i1,…,ik−1))ds+n∑j=1∫t^τk−1~Pk−1,isσij(Ys)dWjs.

In the expression above, we use to denote the -dimensional row vector whose -th entries are equal to , and the remaining entries are set to . Iterating the recursive procedure described above, we can construct the Markov process on , until . At time , all stocks in the portfolio have defaulted. If we set and , then the process counts the number of defaults in the interval . For , recall that the random variable denotes the identity of the stock defaulting at . Then, the default indicator process may be represented as a marked point process via the sequence , i.e., for (we use to denote the indicator function of the event ). This concludes the construction of the process . Using the above construction of the credit risk model, the filtration is included in the filtration since the Brownian motions are independent of the random variables . Thus implies that any -martingale is a -martingale and hence a -martingale. In particular, is also a -Brownian motion.

In the special case of a portfolio consisting of one risk-free stock () and a one-dimensional stochastic factor (), our model reduces to the one considered by Castaneda-Leyva and Hernandez-Hernandez (2005), see equations (2.1) and (2.2) therein. Before proceeding further, define the -dimensional column vector by

 ξ(y) :=σ−1(y)(μ(y)−ren),     y∈D. (8)

In the above expression, denotes the -dimensional column vector with entries identically equal to . The vector is the market price of risk, i.e., the excess compensation demanded by the investor to bear the risk coming from uncertainty in the stocks’ returns. We also define the space of equivalent local martingale measures (henceforth p.m. for short) as

 Q :={p.m. Q: Q∼P on GT,and (B−1t~Pt)t∈[0,T] is a (Q,G)-local martingale}. (9)

Let denote the set of all bounded -functions on , which also admit bounded first-order partial derivatives in . Throughout the paper, we make the following assumptions.

• There exists a sequence of bounded domains with -boundary and closure such that . Moreover, for all , .

• The vector function with bounded first-order partial derivatives in , and . For each , the vector function .

• For , let . Here for , and denotes the set of functions which are continuous in and are in . Define

 Θz ={(θ(⋅,z),h(⋅,z))∈C×B; θ(t,y,z)∈\mathdsRn, h(t,y,z)∈(−1,∞)n, and σ(y)(ξ(y)−θ(t,y,z))=diag((1−z)λ(y,z))h(t,y,z), for (t,y)∈[0,T]×D}.

The set is nonempty for each . Above, represents the set of all Borel functions on . Moreover, we set and .

###### Remark 2.1.

If the coefficients and are Lipschitz continuous on , then the assumption (A1) is satisfied with . This setting covers the case of a stochastic factor given by an Ornstein-Uhlenbeck (OU) process, also considered by Castaneda-Leyva and Hernandez-Hernandez (2005). Fix and assume . Under the parameter choices of and , where are positive constants satisfying the Feller’s condition , the assumption (A1) is satisfied choosing . This specification corresponds to a stochastic factor model given by a Cox-Ingersson-Ross (CIR) process. Assumptions (A1) and (A2) guarantee that Eq. (1) admits a unique strong non-exploding solution. We will use the elements of the set in assumption (A3) to establish the equivalent martingale measure in terms of the Radon-Nikodym density, when we consider the dual of the optimal investment-consumption problem. In most situations of practical interest, the set in the assumption (A3) is nonempty, see also the example below.

The proposed framework is rich enough to include several stochastic volatility models considered in the literature as special cases. In addition, it allows for systemic effects through the dependence of the default intensity on the common factor and the default states. To illustrate this, we next present a two-dimensional factor model featuring stochastic volatility and default contagion.

###### Example 2.1.

Consider a two-dimensional stochastic factor process of the OU type:

 dY1t =(u1−μ01Y1t)dt+2∑i=1σ0i[ρdWit+√1−ρ2d¯Wit], dY2t =(u2−μ02Y2t)dt+2∑i=1¯σ0i[ρdWit+√1−ρ2d¯Wit],

where and for . The state space of is given by with sub-domains , , hence satisfying the assumption . We can view as a vector of economic state variables such as growth of real returns, or deflator/inflation processes for the factors of production, which have been shown to exhibit mean reversion (see Jensen and Liu (2006)). The price dynamics of the two defaultable stocks are given by

 d~P1t~P1t− =μ1dt+√ϑ1(Y1t)dW1t−dM1t, d~P2t~P2t− =μ2dt+√ϑ2(Y2t)[¯ρdW1t+√1−¯ρ2dW2t]−dM2t.

In the above expressions, , and ’s are positive and . We recall that , , are -martingales given in Eq. (5). We have the following coefficients:

 μ0(y) =[u1−μ01y1u2−μ02y2],       σ0(y)≡[σ01σ02¯σ01¯σ02],      λ(y,z)=[λ1(y,z)λ2(y,z)], μ(y) =[μ1μ2],       σ(y)=[√ϑ1(y1)0¯ρ√ϑ2(y2)√1−¯ρ2√ϑ2(y2)].

Then the inverse of the volatility matrix is given by

 σ−1(y)=⎡⎢ ⎢⎣1√ϑ1(y1)0−¯ρ√1−¯ρ2√ϑ1(y1)1√1−¯ρ2√ϑ2(y2)⎤⎥ ⎥⎦,

and the vector is given by

 ξ(y)=⎡⎢ ⎢ ⎢⎣¯μ1√ϑ1(y1)¯μ2√1−¯ρ2√ϑ2(y2)−¯ρ¯μ1√1−¯ρ2√ϑ1(y1)⎤⎥ ⎥ ⎥⎦.

Moreover, for , we have

 N(f;y,z)=⎡⎢ ⎢⎣√ϑ1(y1)λ1(y,z)(1−z1)(ξ1(y)−f1(y))√ϑ2(y2)λ2(y,z)(1−z2){¯ρ(ξ1(y)−f1(y))+√1−¯ρ2(ξ2(y)−f2(y))}⎤⎥ ⎥⎦.

Above, for , . We can further compute

 ∂ξ1∂y1 =−ϑ′1(y1)¯μ12ϑ321(y1),∂ξ1∂y2=0,∂ξ2∂y1=−¯ρ√1−¯ρ2∂ξ1∂y1,∂ξ2∂y2=−1√1−¯ρ2ϑ′2(y2)¯μ22ϑ322(y2).

We next consider two choices for the volatility function , , previously considered in the literature. Under both choices, we can see that .

• Uniformly elliptic Scott volatility, i.e. for . Then we have , and . Hence .

• Uniformly elliptic Stein-Stein volatility, i.e. for . Then , and . Hence .

Let and for . The defining equation of the set in the assumption (A3) may be rewritten as

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1−z1)λ1(y,z)√ϑ1(y1)h1(t,y,z)=ξ1(y)−θ1(t,y,z)1√1−¯ρ2(1−z2)λ2(y,z)√ϑ2(y2)h2(t,y,z)−¯ρ√1−¯ρ2(1−z1)λ1(y,z)√ϑ1(y1)h1(t,y,z)=ξ2(y)−θ2(t,y,z). (10)

We next verify that the setup satisfies the assumption (A3). Consider the solution to Eq. (10) in the different default states . When , we deduce from Eq. (10) that for , i.e. is the market price of risk. Meanwhile and hence . Thus, given for any Borel function , we have . When , we deduce from Eq. (10) that, for , and we choose . Then . Let and take values in a interval with and . Then . Thus for any Borel function , we have . Similar we can discuss the case . For the last case , a direct solution is to take and hence . Then . Obviously it holds that . We can also take the same as in the case . Then, it holds that . Hence, for each default state , is nonempty, i,e. the assumption (A3) is satisfied.

### 2.2 The Optimal Investment/Consumption Problem

We consider a power investor who wants to maximize his expected utility from consumption plus wealth at the target horizon . He dynamically allocates his wealth into the risk-free money market account, and defaultable stocks.

For , denote by the number of shares of the risk-free bank account held by the investor at time . We use to denote the number of shares of the -th defaultable stock, , at time . The wealth process associated with the -predictable portfolio process , , and with a nonnegative consumption rate process , is given by

 Xt=ϕBtBt+ϕ⊤t~Pt−∫t0csds.

If the wealth process is positive, we may define the fractions of wealth invested in the stocks and money market account as

 πit:=ϕit~P