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Optimal Credit Allocation under Regime Uncertainty with Sensitivity Analysis

GUILLAUME BERNIS Natixis Asset Management, Fixed Income ∗ Paris, France LAURENCE CARASSUS LMR (EA 4535), Universit´ e Reims Champagne Ardenne Reims, France [email protected] ´ GREGOIRE DOCQ Natixis Asset Management, Fixed Income † Paris, France SIMONE SCOTTI LPMA, Universit´ e Paris Diderot Paris, France [email protected]

Received ( 3/11/13) Revised (3/31/14, 7/10/14) We consider the problem of credit allocation in a regime-switching model. The global evolution of the credit market is driven by a benchmark, the drift of which is given by a two-state continuous-time hidden Markov chain. We apply filtering techniques to obtain the diffusion of the credit assets under partial observation and show that they have a specific excess return with respect to the benchmark. The investor performs a simple mean-variance allocation on credit assets. However, returns and variance matrix have to be computed by a numerical method such as Monte Carlo, because of the dynamics of the system and the non-linearity of the asset prices. We use the theory of Dirichlet forms to deal with the uncertainty on the excess returns. This approach provides an estimation of the bias and the variance of the optimal allocation and return. We propose an application in the case of a sectorial allocation with CDS, fully calibrated with observable data or direct input given by the portfolio manager. Keywords: Credit assets’ allocation, Regime-switching model, Error theory based on Dirichlet forms, Continuous-time hidden Markov chain. ∗ The

opinions and views expressed in this document are those of the authors and do not necessarily reflect those of Natixis Asset Management. † The opinions and views expressed in this document are those of the authors and do not necessarily reflect those of Natixis Asset Management. 1

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1. Introduction Optimal allocation is an increasingly important area in credit market modeling. Since the beginning of the subprime crisis in 2007, the financial markets have shown a succession of significant increasing and decreasing trends, in the credit spreads dynamics. This suggests that frequent changes of regime in the credit spreads should be taken into account. Recently, there has been a lot of interest in regime-switching models and hidden Markov chains to represent business cycles and evaluate their influence on optimal portfolio allocation. See e.g. Gregoir & Lenglart (2000), Zhou & Yin (2003), Elliot et al. (2010) or Jeanblanc et al. (2010). However, the research to date has tended to focus on equity markets more than credit markets. So far there has been little attention given to optimal credit allocation under regime uncertainty. A key issue in relation to credit allocation is that the variable monitored by the portfolio manager is not the price of the credit assets (bonds, Credit Default Swaps (CDS), etc.) but their spreads, i.e. the difference between the returns of the asset and the risk-free rate. This results in some non-linearity in the expression of the Profit And Lossa (P&L). We assume that there exists a synthetic asset (the benchmark), which represents the evolution of the global credit market. Other credit assets also exist, in which the manager can invest. This setting is particularly interesting when dealing with large benchmarks which cannot be easily replicated. The benchmark also models the global trend of credit markets, whereas the implemented allocation may reflect sectorial or tactical views, given the global evolution of the credit markets. We consider the classical mean-variance optimisation problem of Markowitz (1952): the investor minimizes the variance of the P&L under constraints on the return. This program is well suited to the credit market case where investment decisions are generally taken for a medium term horizon (say a quarter), which corresponds to the horizon of the portfolio manager’s views. Besides, the natural carry effect of credit instruments advocates the use of a static framework such as Markowitz’s. The optimal strategy is an explicit function of the expected return on the credit assets and the benchmark, the inverse of the variance-covariance matrix and the correlation of the credit assets with the benchmark. In our framework, the mean and variance can only be computed numerically because of the dynamics of the spreads and of the non-linearity of the P&L. On the basis of these observations, we choose to investigate the mean-variance portfolio-selection problem under a hidden Markov regime-switching model. The two-state hidden Markov chain explicitly drives the drift of the benchmark, but not the drift of the other credit assets. This leads to a non-standard filtering problem, which is solved in two steps. First, we use filter methods introduced in Wonham (1965) on the benchmark, in order to obtain the probability of being in one state

a The

P&L is defined as the difference between the (non linear) price of the credit asset and it capitalized cost of acquisition.

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as the solution of a diffusion equation. Then, we assume that the best predictor of the hidden economic state - given the evolution of the benchmark - is unaffected by the knowledge of the other credit assets. Thereafter, under this assumption, we use filtering techniques on the spreads of each asset. We express the drift of these assets as the prediction of the benchmark drift plus an extra drift: the excess return with respect to the benchmark. A major contribution of this paper is to perform a sensitivity analysis, with respect to the random excess returns, on the optimal solution of the portfolio optimisation problem. To do so, we assume that the law of the excess returns is unknown, although their so-called bias and variance are available. This involves the case where the views on the excess returns are approximatively accurate and contain uncertainty in the spirit of Black & Litterman (1992). We apply potential theory and, in particular, Dirichlet forms: the reader can refer to Bouleau (2001) for the general theory and Bouleau (2003) for applications in finance. This allows us to compute the bias and variance of the optimal portfolio estimator as a function of the bias and variance of the extra drifts. Details for the computation as well as a comprehensive presentation of Dirichlet forms applied to error calculus can be found in Carassus & Scotti (2014). We apply the above framework to a concrete case of credit optimal allocation. The benchmark is a theoretical CDS index and represents the global evolution of the credit market. The other credit assets are chosen to be the main sectors of the global economy (financials, consumer discretionary, telecommunication and materials) in which the portfolio manager invests. The state space of the Markov chain is composed of a “good” state, which is the portfolio manager’s view of the market (it is an input of the model) and a “bad” state, which is adverse to the manager view (it is defined as an extreme adverse scenario based on historical data). We do not put any restrictions on the direction considered as the good state: for instance, the portfolio manager can be either “bull” (decreasing spread) or “bear” (increasing spread) and the model can handle both cases. We explicitly perform the sensitivity analysis and find confidence intervals for the optimal allocation. It appears that the variance-covariance matrix magnitude deeply influences the size of those intervals. We also compute and compare the optimal strategies obtained when the extra drifts are constant and when the state is either always “good” or “bad”. In Section 2, we present the credit instruments, the expression of the P&L and the Mean-Variance allocation problem. In Section 3, we study the spread dynamics and investigate the implications of the partial observation of the model. In the case of constant extra drift for the credit spreads, we also derive a lognormal approximation, which provides quasi-closed forms for the returns and covariance. In Section 4, the excess returns are assumed to be random variables for which only the bias and the variance are available: we apply the machinery of error calculus to determine the variance and bias of the allocation. Section 5 is dedicated to numerical applications, especially in the case of a sector allocation with CDS. The appendix

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proposes the proof outline of Theorem 4.1 on error calculus. 2. Optimal Credit Allocation in a Mean-Variance setting In this section, we focus on the credit allocation problem. First, we present the available credit instruments. As CDS and bonds are mainly quoted in spread and not in price, we will express prices as functions of the spread. These functions, presented in Examples 1 and 2, stem from standard models, usually favored by practioners. Then we perform, in a static setup, a classical, widely used, meanvariance allocation. 2.1. Credit instrument and P&L We consider a continuous-time model with time horizon denoted by T . The uncertainty is represented by a filtered probability space (Ω, F, F := {Ft }0≤t≤T , P), satisfying the usual conditions. For any F-measurable, square-integrable random variable A, E {A} denotes its expectation under probability P. We consider a financial model with K credit issuers and one synthetic asset referred to as the benchmark. It represents the global evolution of the credit market. Each credit issuer - as well as the benchmark - is characterized by its spread over the risk-free rate. For 0 ≤ t ≤ T and k ∈ {1, . . . , K}, Xk (t) represents the spread of the credit issuer k at time t and X0 (t) the spread of the benchmark. We assume that it is possible to invest in each credit issuer k ∈ {1, . . . , K} through a debt product, for example a bond or a CDS. The price of the k th debt product is denoted by (P (k) (t, Xk (t)))0≤t≤T . The value of the debt product associated to the benchmark is denoted by (P (0) (t, X0 (t)))0≤t≤T . It is not always possible to trade it. Moreover, let us denote by Cap(t) the deterministic capitalisation factor at time t for the risk-free rate. The framework used in this paper does not take into account the default risk but only the spread risk. This is relevant when the universe of investment is limited to investment grade or when the allocation is achieved through CDS indices (CDS on a basket of issuers). In Examples 1 and 2 we present standard market formulae to infer the price of a bond or an index CDS from the spread. Those formulae are widely used expressions of the price as a function of the credit spread. Indeed, market quotes are generally expressed in terms of spread rather than price, in order to ease the comparison of products. The spread dynamics used in Section 3 are independant from these formulae, as for the Black & Scholes option pricing formula when used with a volatility surface, which is changing over time. More details on the pricing formulae given in Examples 1 and 2 can be found in Duffie (1999) or Liu & Jackel (2005). Note that our theoretical results of Section 4 hold for general price function satisfying the regularity Assumption 4. The framework of Example 2 will be used only in the numerical application, see Section 5. Example 1. Consider a bond paying at future time T1 < · · · < Tn = T the

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deterministic cash flows (Ci )1≤i≤n (interest and capital). In this example, we will drop the superscript k as there is only one bond. Consider a recovery rate Rec ∈ [0, 1[. Set, for any t < T1 ,

P (t, x) =

n X i=1

Ci

Cap(t) −x(Ti −t) e + Rec Cap(Ti )

Z t

Tn

Cap(t) −xu xe du , Cap(u)

which means that the spread x is a default intensity associated with the recovery rate Rec. When Rec = 0, x becomes a (continuous) spread over the risk-free rate. Example 2. Consider a CDS with payment dates and interest periods given by (Ti , δi )1≤i≤n . Set T0 ≤ 0. The recovery rate is Rec ∈ [0, 1[, the coupon is c > 0 and we will assume that there is no up-front payment. If the investor sells protection, the price writes, for any t < T1 , n X

Z Tn (u − Tj(u) ) Cap(t) −xu Cap(t) −x(Ti −t) e +c xe du Cap(Ti ) 360 Cap(u) t i=1 Z Tn Cap(t) −xu − (1 − Rec) xe du Cap(u) t

P (t, x) =c

δi

(2.1)

where j(u) := sup {j ∈ {0, . . . , n − 1} | Tj ≤ u}. The second term in the right hand side of (2.1) represents the payment of accrued interest in case of default, corresponding to the time spent in the current interest period. The P&L, at time t ≤ T , of a buy and hold position on the asset k ∈ {1, . . . , K} is given by P&Lk (t, Xk (t)) = P (k) (t, Xk (t)) − P (k) (0, xk ) × Cap(t) .

(2.2)

The second term of the right hand side in (2.2) represents the cost of acquisition of the debt instrument, which is financed by borrowing at the risk-free rate. As we assume a deterministic capitalisation process, we could incorporate coupon payments in the expression of P&L because there is no default risk. However this would produce more cumbersome notations, so we will stick to the simplest case. In the following, we will consider two alternative portfolio representations G(π, t, X(t)). The first one uses the allocation on the assets to outperform the benchmark (benchmarked allocation). To do that we also define by analogy P &L0 (t, X0 (t)) by P &L0 (t, X0 (t)) = P (0) (t, X0 (t)) − P (0) (0, x0 ) × Cap(t). So for π ∈ RK , the benchmarked allocation is: G(π, t, X(t)) :=

K X k=1

P&Lk (t, Xk (t)) × πk − P&L0 (t, X0 (t)) (Benchmarked).

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The second allocation is a simple allocation on the K assets with no benchmark reference (total return allocation), for π ∈ RK : G(π, t, X(t)) :=

K X

P&Lk (t, Xk (t)) × πk (Total Return).

k=1

This can be summed up in the following formula, where ζ is equal to 0 in the total return case and to 1 in the benchmarked case: G(π, t, X(t)) :=

K X

P&Lk (t, Xk (t)) × πk − ζP&L0 (t, X0 (t)).

(2.3)

k=1

2.2. The Optimisation Program Here our investor aims at minimizing the variance of his/her allocation V{G(π, T, X(T ))} under return and budget constraints. We consider a static allocation problem in the spirit of Markowitz. Our choice is motivated by market practices. Firstly, the credit portfolio managers have views on sectors based on fundamental data, which are not daily but generally quarterly. Even if actions can be taken quickly in case of an adverse market, the allocation is generally scheduled on a rather long time horizon. Secondly, the fixed income products generally benefit from a carry effect (when long credit). It is interesting to get it just by holding the position: if nothing happens on the spread/rates the P&L is increasing just by time effect. Define for any (k, j) ∈ {0, . . . , K}2 , Cov[k, j] := E {P&Lk (t, Xk (t))P&Lj (t, Xk (t))} − Mk Mj , where Mk := E {P&Lk (t, Xk (t))}. Note that for ease of notation we drop the time indexation. We also set M := (Mk )1≤k≤K , Cov := (Cov[i, j])(i,j)∈{1,...,K}2 and Cov[0] := (Cov[0, k])k∈{1,...,K} . Then the variance of the allocation is given by V{G(π, T, X(T ))} = π 0 · Cov · π − 2ζπ 0 · Cov[0] + ζ 2 Cov[0, 0] and its return by E{G(π, T, X(T ))} = π 0 · M − ζM0 . So the mean-variance program (P) solved by the investor can be written    1 0 0   π · Cov · π − ζπ · Cov[0] min   2 π∈RK    (P) : s.t. π 0 · M ≥ r + ζM0        and π 0 · I = 1 where I is the element of RK with all its components equal to 1 and r > 0 is the return budget constraint. It is a slight variation around the classical Markowitz program, because it involves the benchmarked case.

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Remark 2.1. Our optimisation program has two particular features. First, it does not involve cash. Indeed, we mainly focus on a credit portfolio manager problem. Most mutual funds cannot hold a large amount of cash, by regulatory constraints. Besides, the presence of cash is not directly linked to our purpose and should be considered at a higher level in the allocation among big asset classes (equities, commodity, credit etc...). Second, it is not always possible to invest in the benchmark. Generally, credit benchmarks are portfolios containing a large number of bonds. Hence, they may not be completely replicable. Besides, the purpose of the portfolio manager is to obtain a better return than the benchmark and his/her allocation crucially depends, among other factors, on the sectorial allocation. Before solving the optimisation program, we define the following real-valued quantities: z1 := I0 · Cov −1 · I z2 := M 0 · Cov −1 · M z3 := M 0 · Cov −1 · I.

(2.4)

Proposition 2.1. Assume that the following condition holds

z1 r + ζM0 − ζM 0 · Cov −1 · Cov[0] > z3 1 − ζI0 · Cov −1 · Cov[0]

(2.5)

and that M is not colinear to I. Then, the solution of (P) is given by π ∗ = r~h + ~g + ζ~e

(2.6)

where −1 −1 ~h = z1 Cov M − z3 Cov I 2 z1 z2 − z3 −1 z2 Cov I − z3 Cov −1 M ~g = z1 z2 − z32



~e = M0 − M 0 · Cov −1 · Cov[0] ~h − I0 · Cov −1 · Cov[0] ~g + Cov −1 · Cov[0]. Note that in the total return case (ζ = 0) we are back to the classical two funds separation result of Markowitz (1952). Remark 2.2. The quantities involved in the program depend on the expected returns of the credit assets and of the benchmark, the inverse of the variancecovariance matrix and the correlations of the credit assets with the benchmark. In the context of partial observations of Section 3.2, those quantities cannot be computed analytically. In Section 5, we will resort to a Monte Carlo method in order to provide numerical applications. However, for numerical purposes we will use some approximations to recover some tractable formulae. This is the topic of Section 3.3. Proof. The program consists in minimizing a convex quadratic mapping on a closed subset of RK . Therefore, the minimum exists. Since vectors M and I are not colinear, the qualification conditions hold at any point of RK and there exists Lagrange

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multipliers µ and ν. See, for instance, Nocedal & Wright (2006). First order conditions yield π ∗ = Cov −1 · (ζCov[0] + µM − νI) .

(2.7)

Assuming that the first constraint is binding we obtain that (r + ζM0 − ζz5 ) z1 − (1 − ζz4 ) z3 z1 z2 − z32 (r + ζM0 − ζz5 ) z3 − (1 − ζz4 ) z2 ν= z1 z2 − z32

µ=

(2.8) (2.9)

with z4 := I0 · Cov −1 · Cov[0] and z5 = M 0 · Cov −1 · Cov[0].

(2.10)

Condition (2.5) implies that the first constraint is binding (µ > 0). Observe that, as M and I are not colinear, the denominator of µ is strictly positive (by CauchySchwarz inequality). Arranging (2.7) yields (2.6). 3. Credit Spread In this section, we describe the dynamics of the spread in our change-of-regime setting. First, we present the hidden Markov chain which drives explicitly the drift of the benchmark but not the drift of the other credit assets. Then, applying filtering techniques the drift of the K credit assets writes as the drift of the benchmark plus some extra drift which can be interpreted as the views of the investor. Finally in Section 3.3, we propose a lognormal approximation of the spreads’ dynamics, which is presented for computational purposes. 3.1. Spread Dynamics As underlined in the introduction, the financial markets have shown a succession of relevant increasing and decreasing trend. To model this change of regime in the drift, we define on (Ω, F, F, P), a F-adapted, continuous-time, two-state valued Markov chain, Y = (Yt )t≥0 . The state space of the Markov chain is equal to {g, b}. When we will perform the concrete application in Section 5, the state g will correspond to a drift equal to the views of the portfolio manager (“good” state), and b will correspond to a drift opposed to his/her views (“bad” state). Let Λ be the rate matrix of the chain Y , where Λ00 = −Λ01 = −λb and Λ10 = −Λ11 = λg are the constant transition intensities of the chain Y : λb (resp. λg ) is the instantaneous probability to switch from state b to state g (resp. from state g to state b), i.e. P(Yt+∆ = g|Yt = b) = λb ∆ + o(∆). The reader can refer to Last & Brandt (1995) for details about Markov chains. Let (W (t))0≤t≤T = (W0 (t), . . . , WK (t))0≤t≤T be a standard (F, P)-Brownian motion of dimension K + 1. Let us define the K + 1-dimensional stochastic process (X(t))0≤t≤T , with X(t) := (Xk (t))0≤k≤K . The process X0 represents the spread of

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the benchmark, the drift of which, µ0 , is assumed to be a measurable function of the Markov chain Y : dX0 (t) = µ0 (Yt )dt + σ0 dW0 (t), X0 (t) X0 (0) = x0 .

(3.1)

For all k ∈ {1, . . . , K}, Θk = (Θk (t))0≤t≤T are F-adapted processes (thus influenced by Y ), representing the unknown drift of the spread Xk of the credit issuer k and:   K q X dXk (t) = Θk (t)dt + σk ρk dW0 (t) + 1 − ρ2k Lk,j dWj (t) (3.2) Xk (t) j=1 Xk (0) = xk . where L := [Li,j ] is a K ×K lower triangular matrix, such that C := L·L0 is a (nondegenerated) correlation matrix. It means that C := (Ci,j )1≤i,j≤K is a symmetric, semi-definite positive matrix with unit diagonal coefficients. We also denote Zk⊥ (t) = PK j=1 Lk,j Wj (t). It is a standard (F, P)-Brownian motion of dimension 1. Now we state the following assumption which will prevail throughout the paper. Assumption 1. We assume that W0 and Y are independent. We assume that the processes µ0 (Y ) and Θk are uniformly bounded and measurable. Finally σk > 0 for k = 0, . . . , K and −1 < ρk < 1 for k = 1, . . . , K. 3.2. Spread Dynamics Under Partial Observation In our model, we assume that we observe the process X and not the hidden Markov chain Y . Therefore, we will use filtering theory to express the dynamics of the assets and the benchmark credit spreads, according to the partial observation of the system. We will perform, in a first step, a one-dimensional filter on the benchmark spread. This yields a one-dimensional diffusion for the probability of being in the bad state, which is driven by the innovation process of the benchmark. In a second step, we will formally perform the filtering technique on the multi-dimensional spread diffusion. This step permits to identify each drift from the multi-dimensional case as the drift of the benchmark (given by the one-dimensional case) plus an extra drift. In the application of Section 5, these extra drifts will be interpreted as the portfolio manager’s views on the excess returns with respect to the benchmark. To obtain such a result, we will resort to Assumption 2 which states that the best prediction of the hidden Markov chain is unaffected by the knowledge of the K assets’ credit spreads. Let us introduce G := {Gt }0≤t≤T and G0 := {Gt0 }0≤t≤T , the right continuous, complete filtrations generated respectively by the following processes: Gt =σ{X0 (s), X1 (s), . . . , XK (s)|s ≤ t} Gt0 =σ{X0 (s)|s ≤ t}.

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  Assumption 2. Let pt := P {Yt = b} | Gt0 , for any 0 ≤ t ≤ T . We assume that pt = P [{Yt = b} |Gt ] . Remark 3.1. This assumption makes the derivation of Proposition 3.1 tractable. It gathers both one-dimensional and multi-dimensional filtering results. But this assumption has also an economic basis: all the information in the economy about the hidden Markov chain is revealed by the observation of the benchmark. Proposition 3.1. Under Assumptions 1 and 2, we get that  dX0 (t) c0 (t) = E µ0 (Yt ) | Gt0 dt + σ0 dW (3.3) X0 (t)   q 
dXk (t) bk⊥ (t) (3.4) c0 (t) + 1 − ρ2 dZ = E µ0 (Yt ) | Gt0 + ek (t) dt + σk ρk dW k Xk (t) where ek (t) = E {Θk (t)|Gt } − E{µ0 (Yt )|Gt0 } (3.5) Z t 
c0 (t) = W0 (t) + 1 W µ0 (Ys ) − E µ0 (Ys ) | Gs0 ds (3.6) σ0 0 Z t 
1 Zbk⊥ (t) = Zk⊥ (t) + p Θk (s) − E µ0 (Ys ) | Gs0 − ek (s) ds 2 σk 1 − ρk 0 Z t µ0 (Ys ) − E{µ0 (Ys )|Gs0 } ρk ds. (3.7) −p 2 σ0 1 − ρk 0 c0 is a (G0 , P)-Brownian motion and for k = 1, . . . , K, Z b⊥ is a (G, P)-Brownian W k  0 motion. Moreover E µ0 (Yt ) | Gt = (µ0 (b) − µ0 (g))pt + µ0 (g), where (pt )0≤t≤T is solution of dpt = [− (λb + λg ) pt + λg ] dt +

µ0 (b) − µ0 (g) c0 (t). pt (1 − pt )dW σ0

(3.8)

In the following the ek will be interpreted as the G-adapted views of the economic agent on the spread k extra drift. Proof. First step: Filter applied only on the benchmark. This step is based on the “reference probability” method or Zakai’s approach. To the best of our knowledge it appears in the paper of Zakai (1969), see also Pardoux (1991) for a more recent survey. In order to determine the diffusion which drives (pt )0≤t≤T , we will use the Wonham filter. See Wonham (1965). We consider only the benchmark  0 diffusion 0 (3.1) well-defined with respect to its proper filtration F := Ft 0≤t≤T := {σ (X0 (s), Y (s)|s ≤ t)}0≤t≤T . We introduce the risk premium λ0 (t) :=

µ0 (Yt ) σ0

and

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the risk neutral probability Q0 defined by the following Radon-Nikodym derivative with respect to P: ! Z Z t 1 T 0
2 0 0 λ (t) dt . dQ /dP = exp − λ (t)dW0 (t) − 2 0 0 From Assumption 1, the Dolans-Dade exponential is well-defined since the F0 adapted process λ0 is bounded. From Girsanov R t Theorem (see Karatzas & Shreve (1991) or Protter (2004)), W 0 (t) = W0 (t) + 0 λ0 (s)ds is a (F0 , Q0 )-Brownian mo0 (t) 0 0 tion. As dX X0 (t) = σ0 dW 0 (t), X0 is a (F , Q )-martingale. Conversely, W 0 (t) = R t dX (s) 1 0 σ0 0 X0 (s) and Gt0 = σ(X0 (s)|s ≤ t) = σ(W 0 (s)|s ≤ t).

(3.9)

Rt As in Pardoux (1991) we define the innovation process W 0 (t)− 0 E{λ0 (s)|Gs0 }ds. Now we use similar arguments as those of Proposition 2.2.7 in Pardoux (1991). By (3.6), we get that Z W 0 (t)−

t

E{λ

0

(s)|Gs0 }ds

Z

t


c0 (t). (3.10) λ0 (s) − E{λ0 (s)|Gs0 } ds = W

= W0 (t)+

0

0

c0 is a continuous path (G0 , P)-martingale. We also From (3.9) it follows that W remark that the quadratic variation of W0 is unchanged when adding a finite variation process. So using the L´evy characterisation theorem (see Protter (2004)), we c0 is a (G0 , P)-Brownian motion. Moreover recalling (3.1), we find conclude that W easily that (3.3) holds true. From Assumption 1, since W0 and Y are independent, the Wonham filter equation yields to (3.8) for the probability pt . See for example Theorem 9.1 in Lipster & Shiryaev (1974) for the derivation of the Wonham filter equation and the existence of strong solutions for this equation. Second step: Filter applied on the assets and the benchmark. We can rewrite the diffusions (3.1) and (3.2) in the following vectorial form: dX(t) = diag(X)(t) [D(t)dt + ΦdB(t)] , where, diag(X) denotes the diagonal matrix such that diag(X)i,i (t) = Xi (t), 

σ0 q0  W0 (t) 0 µ0 (Yt )  ρ1 σ1 σ1 1 − ρ21  q  Z1⊥ (t)   Θ1 (t)     ρ σ 0 σ2 1 − ρ22    and Φ =  D(t) =   2 2  , B(t) =  . .   . .    . . .   . .  . . .  ⊥ . . . ΘK (t) ZK (t)  ρ K σK 0 0 







...

0



...

0

...

0

     .    

. . q. . . . σK 1 − ρ2K ..

.

We remark that Φ is invertible as soon as σk 6= 0, k ∈ {0, . . . , K} and −1 < ρi < 1

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which is required by Assumption 1. It is easy to see that 1 σ0



−1

Φ

0

 − qρ1  σ 1−ρ2  0 1 ρ   − q2 2  σ 0 1−ρk =  .  .  .  ρK − q 2 σ0

1−ρ

q1 σ1 1−ρ2 1

0

0 σ2

. . . 0

q1 1−ρ2 2

. . . 0

... ...

0 0

...

0

..

. ...

K

. . . q1 σK 1−ρ2 K

      .     

−1

Let λ(t) = Φ D(t) be the associated risk premium and define the matrix L by L0,0 = 1, L0,i = Li,0 = 0 and Li,j = Li,j for 1 ≤ i, j ≤ K. Recall that B is a correlated Brownian motion and that B = LW . In order to apply the “reference probability” method again, we introduce the risk neutral probability Q which Radon-Nikodym derivative with respect to P is given by: ! Z T Z 0 1 T −1 −1 2 dQ/dP = exp − L λ(t) dW (t) − |L λ(t)| dt . 2 0 0 From Assumption 1, this Dolans-Dade exponential is well-defined since the Fadapted process λ is bounded. So from Girsanov Theorem, the process dB(t) = dB(t) + λ(t)dt isR a (F, Q)-Brownian motion. t As X(t) = 0 diag(X)(s)ΦdB(s), we have Gt ⊆ σ{B(s)|s ≤ t}. Conversely, R t −1 B(t) = 0 Φ (diag(X))−1 (s)dX(s), and both filtrations coincide: for any t, we have Gt = σ{B(s)|s ≤ t}

(3.11)

b and B is a (G, Q)-Brownian motion. We now introduce the innovation process B: Z t Z t b := B(t) − B(t) E {λ(s) | Gs } ds = B(t) + (λ(s) − E {λ(s) | Gs }) ds.(3.12) 0

0

As in the previous case, we follow the arguments of Proposition 2.2.7 in Pardoux b is a continuous path (G, P)-martingale. Moreover the matrix (1991). From (3.11), B of cross variation is unchanged if we add a finite variation process. As above we conclude using the L´evy characterization theorem for a correlated Brownian motion, b is a (G, P)-Brownian motion. that the stochastic process B Since by Assumption 2, E{µ0 (Yt )|Gt } = E{µ0 (Yt )|Gt0 }, by (3.12), we get that the first component of vector B fulfills Z t c0 (t), b1 (t) = W0 (t) + 1 (µ0 (Ys ) − E {µ0 (Ys | Gs }) ds = W B σ0 0 by (3.6). For each others components of B (for k = 1, . . . K), using (3.12) again, Z t bk+1 (t) = Zk⊥ (t) + p 1 B (Θk (s) − E{Θk (s)|Gs }) ds σk 1 − ρ2k 0 Z t ρk µ0 (Ys ) − E{µ0 (Ys )|Gs } −p ds 2 σ0 1 − ρk 0 bk⊥ (t), =Z

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b⊥ is given by (3.7) (recall the definition of ek in (3.5)). where Z k c0 and Zb⊥ are (G, P)-Brownian motion. Moreover, we So we have proved that W k have   bt dX(t) = diag(X)(t)ΦdB t = diag(X)(t) ΦE{λ(t)|Gt }dt + ΦdB   bt . = diag(X)(t) E{D(t)|Gt }dt + ΦdB (3.13) We see that the process X0 given by the first coordinate of (3.13) coincides with (3.3) and for k ≥ 1, the process Xk given by the (k + 1)-coordinate of (3.13) is given by   q dXk (t) ⊥ 2 b c = E{Θk (t)|Gt }dt + σk ρk dW0 (t) + 1 − ρk dZk (t) Xk (t) and (3.4) is proved. 3.3. Lognormal Approximation of the Credit Spreads In this section we propose a lognormal approximation for the credit spreads in order to make the computation of the returns and the covariance quasi-closed (see Remark 2.2). This will allow us to make the calibration of our model (σ and C) in the numerical Section 5. We will also see in this section that this approximation can provide reasonable results for the computation of the optimal solution of problem (P). So Assumption 3 below can be seen as a tractable approximation for calibration and also an alternative to Monte-Carlo computation in some cases. But Assumption 3 is not a standing assumption of our model and all the theoretical results of Section 4 are carried out without this assumption. Assumption 3. The processes ek are constant for all k = 1, . . . , K. Under this assumption on extra drifts ek , we propose to approximate the spread (q) processes Xk by lognormal processes Xk . The diffusion of the probability pt of the Markov chain to be in the bad state is given by (3.8). To derive closed form formulae, we will use two successive approximations in order to reduce the drifts in (3.3) and (3.4) to Ornstein-Uhlenbeck processes (See Øksendal (2003)), chap. 5). First approximation: The solution (¯ qu )u≥0 of dyt = [− (λb + λg ) yt + λg ] dt with y0 = p0 = p is given by:  λg  ∀ 0 ≤ u ≤ T, q¯u = pe−(λb +λg )u + 1 − e−(λb +λg )u . λb + λg Second approximation: 0 (g) c0 (t) The solution (qu )u≥0 of dyt = [− (λb + λg ) yt + λg ] dt + µ0 (b)−µ q¯t (1 − q¯t )dW σ0 with q0 = p0 = p is given, for all 0 ≤ u ≤ T by:  λg  qu = pe−(λb +λg )u + 1 − e−(λb +λg )u λb + λZg (3.14) µ0 (b) − µ0 (g) u c0 (s) + q¯s (1 − q¯s )e−(λb +λg )(u−s) dW σ0 0

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Replacing the process (pt )0≤t≤T by (qt )0≤t≤T - solution of (3.14) - in (3.8), we obtain the following approximation of the spread (recall Assumption 3): ! Z t (q) Xk (t) σ2 c0 (u)+ ln = h(t) + (µ(g) + ek )t − k t + (λb + λg )ϕ(u, t)dW xk 2 0 (3.15)   q ⊥ 2 c b +σk ρk W0 (t) + 1 − ρk Zk (t) , where 1 − e−(λb +λg )t λg + t λb + λg λg + λb  2  1 µ0 (b) − µ0 (g)  1 − e−(λb +λg )(t−u) q¯u (1 − q¯u ). ϕ(u, t) := σ0 λb + λg    h(t) := µ0 (b) − µ0 (g) p−

λg λg + λb





(q)

Therefore, the distribution at time t of Xt , solution of (3.15), is lognormal with volatility γk (t) > 0 given by Z t Z t γk2 (t) = σk2 t + (λb + λg )2 ϕ2 (u, t)du + 2ρk σk (λb + λg ) ϕ(u, t)du. 0

0

(q)

The expectation of Xk (t) is equal to   n o γ 2 (t) σk2 (q) (q) − t . X k (t) := E Xk (t) = xk exp h(t) + (µ0 (g) + ek )t + k 2 2 (q)

We can also compute the correlations between the variables Xk (t), k ∈ {0, . . . , K}, using the following equation: Z t  n o  2  (q) (q) (q) (q) E Xk (t)Xj (t) =X k (t)X j (t) exp ϕ (u, t) + (ρk σk + ρj σj )ϕ(u, t) du 0 q n h io × exp σk σj ρk ρj + Ck,j (1 − ρ2k )(1 − ρ2j ) t . It is, therefore, possible to compute the quantities involved in the optimisation program (P) using an integration with respect to two Gaussian variables. This can be achieved through a quadrature method (see Press et al. (1992)). 4. Error Calculus on the Optimal Allocation Let k =

1 T

Z

T

ek (t)dt.

(4.1)

0

In this section, we assume that the k are unknown random variables for which only the so-called bias and covariance are known, see Definition 4.1 below. The typical case is when the k are estimated by the portfolio manager and these estimations contain some uncertainty. This section deals with the problem of computing the influence of the uncertainty (measured through the bias and the variance) on the

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optimal allocation. The methodology developed by Bouleau (2003) provides results on how the bias and the covariance of the random variables are affected by non-linear transformations. Typically with this approach we are able to compute the bias and the covariance of the optimal allocation and thus measure the influence of the excess returns’ uncertainty. Note that even if the excess returns are centred (no bias in the estimation) the optimal allocation will be biased because the optimal allocation is not a linear mapping of the excess returns. These results are summarized in Theorem 4.1. Recall that from (3.4), for k = 1, . . . , K, the spread value at time T is given by: (Z T 
σk T+ Xk (T ) = xk exp E µ0 (Yt ) | Gt0 dt + k T − 2 0 (4.2) ) Z T q ⊥ 2 c0 (t) + 1 − ρ dZbk (t) +σk ρk dW . 0

k

This implies that Xk (T ) is a smooth function of k . We also require the following mild assumption on the price functions (P (k) (·, ·))k∈{0,...,K} of the underlying credit products. Assumption 4. The mappings P (k) (·, ·) from R+ × R+ to (0, ∞) are, at least, twice continuously differentiable, for all k ∈ {0, . . . , K}. (k) We will denote by P˙1 (·, ·) its first order derivative with respect to the first (k) variable, by P˙2 (·, ·) its first order derivative with respect to the second variable, (k) and by P¨2 (·, ·) its second order derivative with respect to the second variable. We present a quick description of the error calculus setting. More details can be found inBouleau (2003)  or in Carassus & Scotti (2014). An error structure is a e D, Γ , where e F, e P, quintuplet Ω,

•



 e is a probability space; e F, e P Ω,

  e such that for any e F, e P • D is a dense, separable sub-vector space of L2 Ω, function F of class C 1 and globally Lipschitz (afterward denoted C 1 ∩ Lip) and U ∈ D, one has F (U ) ∈ D;   e e F, e P • Γ is a positive symmetric bilinear function from D × D into L1 Ω, satisfying the following functional relation: for any functions F and G of class C 1 ∩ Lip and U, V ∈ D Γ [F (U ), G(V )] =

n X ∂G ∂F e a.s.; (U ) (V ) Γ[Ui , Vj ] P ∂U ∂V i j i,j=1

(4.3)

e {Γ[U, V ]} is closed, i.e. D equipped with • the bilinear form E[U, V ] = 12 E    1/2 e U 2 + 1 E[U, U ] the norm |U |D = E is complete; 2 • the constant 1 belongs to D and E[1, 1] = 0.

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An error structure is thus a probability space equipped with a so called “carr´e du champ operator” Γ. We generally write Γ[U ] for Γ[U, U ]. The operator Γ is identified with the covariance operator (see Bouleau & Chorro (2004) and Example 3 below). The Hille-Yosida Theorem guarantees that a semigroup - therefore a generator A e (see for - can be uniquely associated to the Dirichlet form E and the probability P instance Albeverio (2003) Chapter 4). More precisely, the generator A : DA →   2 e e e L Ω, F, P is a self-adjoint operator, its domain DA is included and dense in D, see for instance Remark I.1.2.5 in Bouleau & Hirsch (1991). For all U ∈ DA and V ∈D e {A[U ]V } . E[U, V ] = −E

(4.4)

Even if the proof of the Hille-Yosida theorem is non-constructive (the existence of the semigroup is proved without exhibiting it), the generator satisfies, for F ∈  e : e F, e P C 2 ∩ Lip, U ∈ DA, F (U ) ∈ DA and Γ[U ] ∈ L2 Ω, A [F (U )] =

n n X 1 X ∂2F ∂F e a.s. . (U )A[U ] + (U ) Γ[Ui , Uj ] P ∂Ui 2 i,j=1 ∂Ui Uj i=1

(4.5)

From (4.4), A is a closed operator with respect to the norm | |D , in the sense that DA equipped with the norm | · |D is complete. We conclude with the formal definitions of the bias, the covariance and variance operators in the context of error calculus: Definition 4.1. Let U ∈ DA, A[U ] is the called the bias of U . Let U, V ∈ D, Γ[U, V ] is called the covariance of U and V and Γ[U ] is called the variance of U . We justify the preceding choice of name for A[U ] and Γ[U ] in Example 3 below (see also Bouleau & Chorro (2004)). Example 3. In this example, we present a particular choice of error calculus structure where the interpretation of A and Γ as bias and covariance operators are quite clear. All the computation details can be found in Example 2.2 of Carassus & Scotti (2014). The Ornstein-Uhlenbeck structure is given by (R, B(R), µ, D, Γ), where µ is unidimensional centred Gaussian law and D := H 1,2 (µ), i.e. the first Sobolev space associated to L2 (R, B(R), µ). The definition of Γ will follow from the choice of a particular semi-group. To this end, we introduce an Ornstein-Uhlenbeck process in a probability space (ΩB , F B , PB ) equipped with a Brownian motion B: 1 dX = − X d + dB . 2

(4.6)

We denote by Xx the solution of the preceding equation with X0 = x. We define a semi-group as follows: for all ω ∈ R, P [U ](ω) = EB {U (Xx )|x = ω} ,

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where EB denotes the expectation with respect to PB . The generator A associated to the semi-group P is given by (see Bouleau (2003) chapter II): P [U ](ω) − U (ω)  B E {U (Xx ) − U (x)|x = ω} = lim →0 

A[U ](ω) = lim

→0

(4.7)

Moreover, from Proposition II.1.2.2 in Bouleau & Hirsch (1991) the domain DA of A is equal to the closed extension in L2 (R, B(R), µ) of the set P of polynomial functions. From (4.7) it is clear that A can be interpreted as an asymptotical bias (here bias is used in the classical statistical meaning). We can derive (see Carassus & Scotti (2014) for the computation details) that A[U ](ω) =

1 1 00 U (ω) − ωU 0 (ω) 2Z 2

E[U, U ] = −

(4.8)

A[U ](ω)U (ω)dµ(ω) =

1 2

Z

(U 0 (ω))2 dµ(ω),

(4.9)

where we have used R (4.4) for the first equality of the second equation. As E[U, U ] = 21 Γ[U, U ](ω)dµ(ω), we deduce that Γ(U ) = (U 0 )2 for all U ∈ D. Assuming U ∈ P we have o o n n 2 2 EB (U (Xx )) − U 2 (x) − 2U (x) (U (Xx ) − U (x)) |x = ω EB (U (Xx ) − U (x)) |x = ω = lim lim →0 →0   = A[U 2 ](ω) − 2U (ω)A[U ](ω) 2

= [U 0 (ω)] = Γ[U ](ω), where we have use (4.7), U ∈ P for the second equality and (4.8) for the third one. Recalling that DA is the closure of P and is dense into D, we can extend the previous result to all U ∈ D. From (4.10) it is clear that Γ can be interpreted as an asymptotical mean-square error or variance. We finish with an analogy which will allows us to propose confidence intervals in Section 5. If we assume that conditionally to {x = ω}, U (Xx ) is asymptotically Gaussian as  goes to zero, then we have:   U (Xx ) − U (x) = N A[U ](ω), Γ[U ](ω) , (4.11) where N (m, σ 2 ) denotes the Gaussian law with mean m and variance σ 2 . In the theorem below we compute explicitly the bias and the covariance of the optimal allocation π ∗ and the optimal return R∗ := M 0 · π ∗ as a function of the bias and covariance of (k )k=1,...,K .

(4.10)

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Theorem 4.1 (Sensitivity of the optimal strategy). Let π ∗ be given by Proposition 2.1. Then for all 1 ≤ i, l ≤ K, we get that: X π π Γ[πi∗ , πl∗ ] = δk,i δj,l Γ[k , j ] (4.12a) kj

A[πi∗ ]

=

X

π δk,i A[k ] +

X

k

π αk,i Γ[k ] +

k

X

π βkj,i Γ[k , j ]

! ∗

Γ[R ] =

X

φk πk∗

+

X

! φj πj∗

π δk,i Mi

+

X

i

k,j

(4.12b)

kj π δj,i Mi

Γ[k , j ]

! ∗

A[R ] =

X

φk πk∗

+

X

π δk,i Mi

A[k ] +

i

k

(4.12c)

i

X X k

π αk,i Mi

i

1 + (T φk + ψk )πk∗ 2

! Γ[k ]

! +

X k,j

π δk,j φj

+

X

π βkj,i Mi

Γ[k , j ]

(4.12d)

i

π π π The explicit form of δk,i , αk,i and βkj,i are given in Appendix (see (A.4), (A.5) and (A.6)). Note that all the sums are taken from 1 to K.

The details of the proof of Theorem 4.1 can be found in Carassus & Scotti (2014). A synthetic proof together with the explicit form of the involved coefficient can be found in Appendix A. π π π It is important to note that the coefficients δk,i , αk,i and βkj,i (and thus the sensitivity analysis of the optimal allocation) are functions of φk , ψk , Φk,j , Ψk,j and Υkj , for 1 ≤ k, j ≤ K given below, which can be simply computed through a Monte Carlo method. n o (k) φk := T E Xk (T )P˙2 (T, Xk (T )) (4.13) n o (k) ψk := T 2 E Xk2 (T )P¨2 (T, Xk (T )) (4.14) n o (k) Φk,j := T E Xk (T )P˙2 (T, Xk (T ))P &Lj (T, Xj (T )) (4.15) n o (k) Ψk,j := T 2 E Xk2 (T )P¨2 (T, Xk (T ))P &Lj (T, Xj (T )) (4.16) n o (k) (j) Υkj := T 2 E Xk (T )Xj (T )P˙2 (T, Xk (T ))P˙2 (T, Xj (T )) . (4.17) 5. Application: Optimal allocation of CDS Indices In this section, we will apply the previous analysis to the case of an optimal investment on a portfolio of CDS. We consider, as benchmark, a theoretical CDS representing an equally weighted portfolio of investment grade issuers, with maturity 5 years, but we focus on a Total Return case because CDS are unfunded credit derivatives. Here K = 4 and the credit assets will represent four sub-sectors of the economy: Financials, Consumer Discretionary, Telecommunications and Materials. We use the formula given by Example 2. The coupon is 1% paid quaterly, and the

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maturity is 20/06/2017. The risk-free discount curve is the EUR swap-rate curve at this date. Its influence is marginal on the CDS pricing. The as-of date will be 18/05/2012, the horizon of investment T is equal to 15/09/2012, which represents 0.33 year. The observations of the spreads, used for the estimations, represent a sample of 1.5 years. We first comment on the benchmark diffusion parameters (see (3.1)). In May 2012, the credit spreads were wide, close to their highest level of 2011. So, it seems reasonable to postulate a tightening of credit spreads. We take µ0 (g) = −1 and µ0 (b) = 1.5 per year. In this application we assume that the drift µ0 (g) is an input of the model and is given by the view of the portfolio manager. The quantity µ0 (b) is considered as a worst case scenario observed on the data sample. The historical evolution of the drift is analysed by the mean of simple techniques, such as moving average, or more sophisticated, such as wavelet filtering (see e.g. Mallat (1999)). When the historical evolution of the drift is obtained, it is possible to estimated the parameters λb and λg , see Last & Brandt (1995) for the estimation of transitions probabilities. We obtain: λb = 6 and λg = 2 per year. We also give the initial value X(0) and compute the empirical variance σ0 of the benchmark (see Table 1 below). In order to estimate the parameter of the sector spreads (see (3.2) and (3.4)), we work under Assumption 3. We choose purely illustrative value for the sector excess returns , given by Table 1. Then, we compute empirical variance and correlation matrix of the spreads, in the lognormal framework given by (3.15). It yields σ and C (see Table 1 and 2 below). Let us take a budget constraint r = 1% on a three month period, which represents a target of return equal to 3% per year over the risk-free. Components Benchmark Financials Consumer Discretionary Telecommunications Materials

X(0) 2.9% 3.3% 2.4% 4.9% 3.2%

ε -10% 10% -10% 10%

σ 39% 37% 32% 48% 39%

Table 1 Data for the diffusion of X.

Benchmark Financials Consumer Discretionary Telecommunication Services Materials

1.0 0.9 0.9 0.8 0.7

0.9 1.0 0.8 0.7 0.7

0.9 0.8 1.0 0.6 0.9

0.8 0.7 0.6 1.0 0.4

0.7 0.7 0.9 0.4 1.0

Table 2 Correlation matrix C. First, we compute the optimal allocation denoted by π ˆ ∗ with  constant given by Table 1.

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Then, to check the pertinence of the approximation described in Section 3.3, we have implemented the approximated allocation, denoted by π e∗ , associated to the approximated diffusion (3.15). Finally, in order to see how the drift uncertainty influences the optimal allocation, we also perform the simulation when there is no hidden Markov chain. We consider the “good” case i.e. µ0 ≡ µ0 (g) = −1 and denote by π ∗ this allocation. We also consider the “bad” case i.e. µ0 ≡ µ0 (b) = 1.5 and denote by π ∗ this allocation. For the computation of M , Cov −1 and Cov[0], we have used a Monte Carlo method with a Euler Scheme (see Talay & Tubaro (1990)). We need to choose a rule to be sure that the discretisation of p (see (3.8)) remains in [0, 1]. If we denote by s and s + ∆, two consecutive steps of the scheme, we keep p(s + ∆) equal to p(s) if the simulated increment δp(s) is such that p(s) + δp(s) > 1 or p(s) + δp(s) < 0. There exist alternative methods to this simple truncated version of the diffusion. For instance, more sophisticated methods taking into account the reflexion on the boundary are developed in Kushner (1996). The number of simulations is equal to 100000, providing a small empirical confidence interval on M (smaller than 0.06%). The whole computation time for 100000 simulations is about 30 seconds on a computer with a dual-core processor of 3GHz. The optimal allocations are given in Table 3.

Financials Consumer Discretionary Telecommunications Materials

π ˆ∗ -83% 314% -37% -93%

π e∗ -77% 318% -40% -101%

π∗ -87% 341% -43% -111%

π∗ 33% 389% -87% -235%

Table 3 Optimal allocations for 100000 Monte Carlo simulations. We can see, in Table 3, that π ˆ ∗ and π e∗ are similar, but show relatively large differences in relative value (more than 8%) on certain coordinates. Note that the difference between π ˆ ∗ and π e∗ would be smaller for smaller intensities λg and λb . We will provide such an example at the end of the section. The influence of the Markov chain can be seen when we compare π ˆ ∗ , π ∗ and π ∗ . In the “good” case, the fact that the drift µ0 (b) is a worst case scenario accounts for the differences between π ˆ ∗ and ∗ π . Our model yields an allocation with smaller amounts invested on each asset, which is a natural consequence of the possible occurrence of an adverse state in the credit spreads’ drifts. We can also observe that the Markov chain differs widely from a diffusion with the sole bad state, by comparing π ˆ ∗ and π ∗ . This effect is strengthened by the fact that µ0 (b) is large (extremely adverse scenario). Now, we turn to the error calculus on the extra drifts presented in Section 4. Let us consider non-biased uncorrelated errors with the same standard deviation of 1 1%, i.e. A[] = 0, Γ[i , j ] = 0 for 1 ≤ i < j ≤ 4 and Γ[] 2 = 1%. Now the expected returns M , the optimal allocation π ∗ and the optimal return R∗ := M 0 · π ∗ are

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random variables. It is possible to make statistical interpretation of error calculus theory and constructed confidence interval for π ∗ using A[π ∗ ] and Γ[π ∗ ]. This analogy is developed in Bouleau & Chorro (2004) (see also (4.11)). In Table 6 we have computed some confidence intervals (CI) at level 0.95.

Financials Consumer Discretionary Telecommunications Materials

π ˆ∗ -83% 314% -37% -93%

CI for π ∗ [-93.1%, -72.1%] [292.6 %, 332.6%] [-41.5 %, -33.3%] [-97.9 %, -87.6%]

Bias A[π ∗ ] 0.6% -1.5% 0.0% 0.7%

Table 6 Influence of centred uncorrelated errors with the same standard deviation of 1%. Calculus made for 100000 Monte Carlo simulations. We can see that even with centred errors the optimal allocation is biased, with important influence on some of its components. Concerning the estimation of the variance and bias of the optimal return, we find out that A[R∗ ] = −0.011% and that Γ[R∗ ] ≈ 0. This is not surprising since, in this case, the return constraint is binding, i.e. R∗ = r = 1%. In order to illustrate the influence of the intensities λb and λg on the approximation, let us take λb = 1.5 and λg = 0.5, which means that the expected time spent in each state is multiplied by 4. We obtain the results displayed in Table 5. The difference between π ˆ ∗ and π e∗ is relatively small. Financials Consumer Discretionary Telecommunications Materials

π ˆ∗ -87% 331% -41% -104%

π e∗ -83% 332% -42% -107%

π∗ -87% 341% -43% -111%

1

Std-Dev Γ[π ∗ ] 2 5.3% 10.2% 2.1% 2.6%

π∗ 33% 389% -87% -235%

Table 7 Optimal allocations for 100000 Monte Carlo simulations, with λb = 1.5 and λg = 0.5. 6. Conclusion This paper defines a new approach for optimising credit portfolios, which reflects the specific features of credit products. The non-linearity of the pay-off is fully taken into account. Two distinct sources of uncertainty are considered. First, the drift of the spread of the credit assets are influenced by a two-state hidden Markov chain. Second, the credit assets excess returns are afflicted by uncertainty and treated in a statistical way. Filtering techniques are used to deal with the hidden Markov chain. The uncertainty on the extra drifts is treated with the tools of Dirichlet forms: the non-linearity of the pay-off induces a bias on the optimal allocation even if the extra drifts are assumed to be unbiased.

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The optimal allocation is handled through a mean-variance framework. The inputs of this program, i.e returns and variance matrix, can be computed by a Monte Carlo method or using a lognormal approximation. The expressions of the variance and the bias of the optimal credit allocation and optimal return are derived. They can be expressed in terms of simple elements which can be computed by the Monte Carlo method. A numerical application is provided in the case of a credit allocation with CDS. One of the interesting properties of our model is that it can be fully calibrated with observable data or direct input given by the portfolio manager. Note finally that the transition intensities of the hidden Markov chain could be linked to the skills of the portfolio manager on directional strategies, and could be calibrated with objective data, such as the “hit ratio” in the directional allocation. The random variables representing the views on the excess return of the assets could also be used to parameterise the skills of the portfolio manager in sector allocation. Appendix A. Details on error calculus In this appendix, we present a quick proof of Theorem 4.1. The missing details as well as a comprehensive presentation of Dirichlet forms applied to error calculus can be found in Carassus & Scotti (2014). All proofs are based on the same approach. We start with the analysis of the variance since the related operator Γ has an autonomous chain rule, see (4.3), while the chain rule for the bias A depends on Γ too, see (4.5). However, as the covariance operator Γ is bilinear, it is generally awkward to compute it directly. Therefore, to overcome this problem, we introduce an auxiliary linear operator called gradient. Recall that we have assumed that D is separable. Let H be a Hilbert space with inner product < ·, · >H , then there exists a gradient operator denoted by (·)# (see Bouleau & Hirsch (1991) p. 242) which is linear and satisfied ∀U, V ∈ D, Γ[U, V ] = < U # , V # >H n X ∂F ∀U ∈ D, ∀F ∈ C 1 ∩ Lip, (F (U ))# = ( (U ))Ui# . ∂x i i=1

(A.1) (A.2)

Now all our proofs follow the same path: we start with the computation of the gradient operator, we then deduce the covariance operator thanks to the inner product on H (see (A.1)), we finally compute the bias using the chain rule (see (4.5)). In order to prove Theorem 4.1, we first apply the gradient operator to the optimal portfolio. Note that, for sake of tractability, we will use (2.7) instead of (2.6). From (2.7), we see that we can split the computation of (π ∗ )# and apply iteratively the gradient operator (·)# to the expected return M (see Proposition Appendix A.1), the vector Cov[0], the covariance matrix Cov, its inverse Cov −1 and the quantities zi , µ and ν (see (2.4), (2.8 and 2.9) and (2.10) for definitions). Some long, tedious but simple algebra gives the gradient of the optimal portfolio,

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the details of the algebraic derivations are available in Carassus & Scotti (2014). When the computation of gradient is achieved, we deduce the variance operator Γ (see (4.12a)) using (A.1). In a similar way, we can deduce (4.12c). As before, to obtain the bias of the optimal portfolio, we split the computation applying the operator A to M (see Proposition Appendix A.1), Cov[0], Cov, Cov −1 , zi , µ and ν. Again all the details of the algebraic derivations can be founded in Carassus & Scotti (2014). This allow to obtain the bias of the optimal portfolio π ∗ (see (4.12b)) and, by the same way, of its return R∗ (see (4.12d)). In order to illustrate our methodology, we present an explicit computation of the covariance and the bias of the expected value of P &L (the vector M ) in Proposition Appendix A.1. The explicit computations for Cov[0], Cov, Cov −1 and quantities zi , µ and ν are available in Carassus & Scotti (2014). Proposition Appendix A.1. For any 1 ≤ k, j ≤ K, we have Γ[Mk , Mj ] = φk φj Γ[k , j ]   1 1 T φk + ψk Γ[k ] A[Mk ] = φk A[k ] + 2 2 where φk and ψk are defined in (4.13) and (4.14) Proof. of Proposition Appendix A.1. By direct application of the gradient chain rule (see (A.2)) to the spread, we have Xk# (T ) = T Xk (T )# k . From (A.2) again applied to (2.2), we get that (k)

# ˙ P &L# k (T, Xk (T )) = P2 (T, Xk (T ))T Xk (T )k .

As the gradient operator is closed and linear, we can exchange the gradient operator and the integral sum (see Lemma 3.6 in Carassus & Scotti (2014) for more details) # # and we get Mk# = (E[P &L(T, Xk (T ))]) = E[(P &L(T, Xk (T ))) ]. Using (A.1) we get Γ[Mk , Mj ] = < Mk# , Mj# >H = φk φj Γ[k , j ]. Now, applying the chain rule of A to the P&L (recall (4.5)), we obtain 1 (k) (k) A[P &Lk (T, X(T ))] = P˙2 (T, Xk (T ))T Xk (T )A[k ] + P˙2 (T, X(T ))T 2 Xk (T )Γ[k ] 2 1 ¨ (k) + P2 (T, Xk (T ))T 2 Xk2 (T )Γ[k ] 2 Similarly to the gradient operator, A is closed and linear and we can exchange A and the integral sum to obtain the last equality. A.1. Explicit formulae In this section, we give the explicit formulae of the quantities introduced (4.12). Note that

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• the quantities φk and ψk appear in the bias and variance of the expected returns M (see Proposition Appendix A.1); C C C • δk,j , αk,j and βkj appear in the bias and variance of the matrix Cov (see Proposition 3.7 in Carassus & Scotti (2014)); Cov −1 Cov −1 Cov −1 • δk,il , αk,il and βkj,il appear in the bias and variance of the matrix Cov −1 (see Proposition 3.8 in Carassus & Scotti (2014)); zi • δkzi , αkzi and βkj are used for the bias and variance of the quantities zi for all i = 1, . . . , 5 (see Proposition 3.9 in Carassus & Scotti (2014)); µ ν • δkµ , αkµ , βkj and δkν , αkν , βkj are used for the bias and variance of the quantities µ and ν respectively (see Corollary 3.11 in Carassus & Scotti (2014)); π π π • δk,i , αk,i and βkj,i appear in the bias and variance of the optimal portfolio π (see Theorem 3.12 in Carassus & Scotti (2014)).

For any 1 ≤ k, j, l ≤ K, those quantities are given by (the sums are from 1 to K)

C δk,j := Φk,j − φk Mj 1 C αk,j := [T Φk,j + Ψk,j − (T φk + ψk ) Mj ] 2 C βkj := Υkj − φk φj X
C −1 −1 −1 −1 Cov −1 δk,il := − Covik Covml + Covim Covkl δk,m m Cov αk,il

−1

:= −

X


C −1 −1 −1 −1 Covik Covml + Covim Covkl αk,m

m Cov −1 βkj,il

−1 −1 C := −Covik Covjl βkj −

 X −1 C −1 C Cov −1 Cov −1 Covik δk,m δj,ml + Covim δj,m δk,jl . m

Then define

δkµ :=

∂µ za δk ∂z a a=1,...,5

αkµ :=

∂µ za αk ∂z a a=1,...,5

µ βkj :=

∂µ za βkj ∂z a a=1,...,5

δkν :=

∂ν za δ ∂za k a=1,...,5

αkν :=

∂ν za α ∂za k a=1,...,5

ν βkj :=

∂ν za β ∂za kj a=1,...,5

χµkj :=

X

X

1 2

X a,b=1,...,5

∂ 2 µ za zb δ δ ∂za ∂zb k j

X

X

χνkj :=

1 2

X

X

X a,b=1,...,5

∂ 2 ν za zb δ δ , ∂za ∂zb k j

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where the derivatives of µ and ν are straightforward and

δkz1 :=

X

δkz2

X

Cov δk,il

−1

X

αkz1 :=

il

:= :=

X

:=

X

X

δkz3 :=

X

z1 βkj :=

+2

X

−1

Cov βkj,il

il

−1 φk Covki Mi

i Cov −1 Mi αk,il Ml

+ (T φk + ψk )

X

−1 Mi Covik

i

il z2 βkj

−1

il Cov −1 Mi δk,il Ml

il

αkz2

Cov αk,il

Cov −1 Mi βkj,il Ml

+2

X

Cov −1 φj Mi δk,ij

−1 + Covjk φj φk

i

il −1 φk Covki +

X

i

Cov δk,il

−1

Mi

il −1

X1

αkz3 :=

X

z3 βkj

:=

X

:=

X

:=

X

z4 βkj :=

X

δkz5

:=

X

αkz5

X 1 −1 C −1 Cov := Mi αk,il Cov[0]l + Mi Covik αk,0 + (T φk + ψk ) Covki Cov[0]i 2 i i il X X X −1 Cov −1 C Cov −1 Cov −1 C Mi δj,0 δk,ij + δk,ji φj Cov[0]i + Covjk φj δk,0 . := Mi βkj,il Cov[0]l +

Cov Mi αk,il

+

i

il Cov −1 Mi βkj,il

+

X

Cov φk δj,ki

Cov −1 δk,il Cov[0]l

+

X

+

X

Cov βkj,il Cov[0]l +

X

Cov −1 αk,il Cov[0]l −1

Cov −1 Mi δk,il Cov[0]l

C Cov δj,0 δk,ij

+

X i

il

il

−1 C αk,0 Covik −1

i

il

z5 βkj

−1 C δk,0 Covik

i

il

X

−1

i

il

αkz4

−1 (T φk + ψk )Covki

i

il

δkz4

2

−1

−1 C Mi Covik δk,0 +

X i

X

i

−1 φk Covki Cov[0]i

i

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Finally, we can state the coefficients used in Theorem 4.1. X π Cov −1 δk,i := δk,il (ζCov[0]l + µMl − ν)

(A.4)

l

+

X

−1 −1 C Covil (δkµ Ml − δkν ) + Covik ζδk,0 + µφk




l

X

−1

X

Covil−1 (Ml αkµ − αkν )

(A.5)

  1 −1 C +Covik ζαk,0 + µ (T φk + ψk ) 2 X X
π Cov −1 Cov −1 βkj,i := βkj,il (ζCov[0]l + µMl − ν) + δk,il δjµ Ml − δjν

(A.6)

π αk,i :=

Cov αk,il

(ζCov[0]l + µMl − ν) +

l

l

l

+

l

X

 
µ ν Cov −1 C Covil−1 Ml χµkj + Ml βkj − χνkj − βkj + δk,ij ζδj,0 + µφj

l −1 +Covik φk δjµ .

Acknowledgments Part of this work was carried out while L. Carassus was affiliated to the University of Paris 7. L. Carassus thanks LPMA (UMR 7599) for support. We are especially grateful to Giorgia Callegaro, Monique Jeanblanc, Ying Jiao and Nathalie Pistre for stimulating discussions and for bringing to our attention some important references. We also thank two anonymous referees for their helpful remarks and comments which allowed us to improve significantly the quality of the paper. References S. Albeverio, (2003) Theory of Dirichlet forms and applications. In Lectures on Probability Theory and Statistics, 1816, Berlin: Springer-Verlag. N. Bouleau & F. Hirsch (1991) Dirichlet Forms and Analysis on Wiener Space. Berlin: De Gruyter. N. Bouleau (2001) Calcul d’erreur complet et Lipschitzien et formes de Dirichlet, Journal de Math´ematiques Pures et Appliqu´ees 80 (9), 961–976. N. Bouleau (2003) Error Calculus for Finance and Physics. Berlin: De Gryuter. N. Bouleau & C. Chorro (2004) Error Structures and Parameter Estimation, C.R. Acad. Sci. Paris Ser. I 338, 305–310. F. Black & R. Litterman (1992) Global Portfolio Optimization, Financial Analysts Journal September/October, 28–43. L. Carassus & S. Scotti (2014) Stochastic Sensitivity Study for Optimal Credit Allocation. In: Arbitrage, Credit and Informational Risks (C. Hillairet, M. Jeanblanc & Y. Jiao, eds), 147–167. Peking University Series in Mathematics: Volume 5. D. Duffie (1999) Credit Swap Valuation, Financial Analysts Journal January/February, 73–87. R.J. Elliott,T.K. Siu & A. Badescu (2010) On mean-variance portfolio selection under a hidden Markovian regime-switching model, Economic Modelling 27, 678–686. S. Gregoir & F. Lenglart (2000) Measuring the probability of a business cycle turning point

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