Black–Litterman in continuous time: the case for filtering

In this article, we extend the Black–Litterman approach to a continuous time setting. We model analyst views jointly with asset prices to estimate the unobservable factors driving asset returns. The key in our approach is that the filtering problem and the stochastic control problem are effectively separable. We use this insight to incorporate analyst views and non-investable assets as observations in our filter even though they are not present in the portfolio optimisation.


Introduction
propose a one-period mean-variance optimisation in which the expected risk premia of the assets incorporate views formulated by securities and market analysts. This model can be viewed as the Bayesian updating of a prior distribution of risk premia (without views) into a posterior distribution reflecting the views. Black and Litterman derive a prior distribution through a reverse optimisation of the market portfolio under the assumption that it corresponds to the equilibrium portfolio. Then they derive the posterior distribution of the risk premia using Theil's mixed estimation model (see 1971). Finally, they use the expected (posterior) risk premia in a mean-variance optimisation to obtain an updated asset allocation.
The Black-Litterman model is very popular. It has been extended and reformulated by He and Litterman (1999), Litterman and the Quantitative Research Group (2003), Idzorek (2004), and Walters (2011), among others. A search on JSTOR for the terms 'Black' and 'Litterman' generated 184 results and a search on Econlit returned 26 entries between 2007 and 2012. SSRN lists 57 papers for the more restrictive search on 'Black-Litterman'. However, none of these articles or papers propose an implementation of the Black-Litterman model in a continuous time setting.
In this article, we show how standard filtering arguments can be used to incorporate views in a continuous time asset allocation. Filtering theory has developed considerably since the seminal work by Kalman (1960) and Kalman and Bucy (1961). Readers should refer to Chapter 6 in Øksendal (2003) for a concise introduction to filtering and to Bain and Crisan (2009) for an excellent treatment of filtering theory and applications. Filtering techniques quickly gained acceptance in stochastic control, as evidenced by Bucy and Joseph (1987), Davis (1977) or Bensoussan (2004). In a portfolio management context, Brennan (1998) and Xia (2001) used filtering to estimate the parameters of their models. Nagai and Peng (2002) and Davis and Lleo (2011) also applied filtering techniques to risk-sensitive asset management models.
The key is that the filtering problem and the stochastic control problem are effectively separable. We use this insight to incorporate analyst views and non-investable assets as observations in our filter even though they are not present in the portfolio optimisation. The paper is organised as follows. We introduce the asset market in Section 2. The financial market comprises investable and non-investable assets and we assume that unobservable factors drive the evolution of asset prices. Asset managers can hold and trade investable assets, but not non-investable assets. We treat the latter as an additional source of observation to estimate the factors as part of the filtering. In Section 3, we propose a model for analyst views and demonstrate how to estimate the factors via a Kalman filter on the assets and views. Then, we show how to incorporate these estimates in a risk-sensitive asset management model.

The financial market: asset prices are driven by unobservable factors
We start by considering an asset market comprising M = m 1 + m 2 , m 1 ≥ 1, m 2 ≥ 0, risky securities S i , i = 1, . . . , M , and a money market account process S 0 . The growth rates of the assets depend on n unobservable factors X 1 (t), . . . , X n (t) which follow the affine dynamics given in Equation (1).
Let ( , F, P) be the underlying probability space. On this space is defined an R N -valued (F t )-Brownian motion W (t) with components W j (t), j = 1, . . . , N , and N := n + M + k. We are in an incomplete market setting with n sources of risk corresponding to the factors, M sources of risk related to the assets and k sources of uncertainty about the analyst views.
The asset returns and risk premia are subject to the evolution of the n-dimensional vector of unobservable factors X (t) modelled using an affine dynamics We derive an estimateX (t) for the factor process X (t) using filtering in the next section. Once we have obtained the estimate, we can solve the optimisation problem.
The dynamics of the money market asset S 0 is given by and that of the M risky assets follows the SDEs or alternatively the SDE We also assume that no two assets have the same risk profile.
Assumption 2.1 The matrix is positive definite.
The investor is allowed to trade in the first m 1 securities but not in the next m 2 securities. In the case when m 2 = 0, M = m 1 , which means that the investor can trade on the entire market. When m 2 = 0, we can decompose the asset price vector S(t) as where S 1 (t) is the m 1 -dimensional process of investable asset prices and S 2 (t) is the m 2 -dimensional process of non-investable asset prices. We further define vectors and matrices of parameters for each segment of the market We focus on the discounted prices and the risk premia. The money market rate is the base rate or break-even rate for the portfolio, and it does not affect the optimisation problem. This observation, which probably dates back to the development of the CAPM in the 1960s, is at the heart of the Black and Litterman (1992) model. Therefore, the portfolio construction process is akin to selecting an optimal exposure to various risk premia.
The discounted asset priceS i (t) arẽ and the dynamics ofS(t) are whereã = a − a 0 1,Ã = A − A 0 1, and 1 ∈ R M denotes the M -element column vector with entries equal to 1. Alternatively, we could express the asset market dynamics more synthetically as The relation between discounted pricesS i (t) and risk premium π i (t) is simply Note that the dynamics of the risk premia is Gaussian (conditional on X t ) which enables us to use linear filtering. Nagai and Peng (2002) took a similar road: they defined the log returns log S i (t), i = 1, . . . , m 1 , as their observation vector.

The asset allocation model: incorporating analyst views and market information into the estimation process
The continuous time asset allocation model we propose follows three steps which we detail in the next subsections: (i) express the analyst views, (ii) filter the views and asset prices to estimate the factors, (iii) solve the stochastic control problem.

Express the analyst views
We ask analysts to formulate today views about risk premia or the spread between risk premia over a time horizon. A typical analyst statement would be my research leads me to believe that the spread between 10-year Treasury Notes and 3-month Treasury Bills will remain low over the next year before gradually widening over the following 2 years to 200 basis points in response to improving macroeconomic conditions. Mathematically, we can translate the k views Z(t) expressed by the analysts into a system of ordinary differential equations (ODEs) Because analyst predictions are not fully accurate, we introduce a white noise term to construct a dynamic confidence interval around the views: where W is a k-dimensional white noise process and ϕ is a k × k matrix. If we assume that the analysts formulate their views independently from each other, then ϕ will be a diagonal matrix. If we chose to model herding behaviour among the analysts, ϕ will reflect the correlation between the forecasting error terms. Finally, we express Equation (7) as a stochastic differential equation (8) where W (t) is the N -dimensional Brownian motion and is a k × N matrix with zeroes on its first (n + M ) rows.
This entire construction takes place at initial time t = 0. We are neither modelling the arrival of new opinions nor the evolution of the views over time. Indeed, we do not believe that either can be predicted: any attempt to model these aspects would defeat the purpose of the analysis. Rather, we are modelling the view formulated today about the evolution of the risk premia.

Filter the views and asset prices to estimate the factors
The money market rate r(t) = a 0 + A 0 X (t) is observed directly, which generates some complication for the estimation process. We start by solving the case A 0 = 0 before sketching the argument required in the case when A 0 = 0.

The pair of processes
takes the form of the 'signal' and 'observation' processes in a Kalman filter system, and consequently, the conditional

distribution of X (t) is normal N (X (t), P(t)), whereX (t) = E[X (t)|F Y t ] satisfies the Kalman filter equation and P(t) is a deterministic matrix-valued function.
The dynamics of the elements Y i (t), i = 1, . . . , M , of the observation vector Y (t) satisfy We express the dynamics of Y (t) succinctly as Next, we define processes Y 1 (t), Y 2 (t) ∈ R m as follows: In the present case, we need to assume that X 0 is a normal random vector N (m 0 , P 0 ) with known mean m 0 and covariance P 0 , and that X 0 is independent of the Brownian motion W . The processes (X (t), Y 1 (t)) satisfying Equations (1) and (12) and the filtering equations, which are standard, are stated in the following proposition.
Proposition 1 (Kalman filter) The conditional distribution of X (t) given F Y t is N (X (t), P(t)) and calculated as follows: (i) The innovations process U (t) ∈ R M +k defined by is a vector Brownian motion.
(ii)X (t) is the unique solution of the SDE (iii) P(t) is the unique non-negative definite symmetric solution of the matrix Riccati equatioṅ Now the Kalman filter has replaced our initial state process X (t) by an estimateX (t) with dynamics given in Equation (15). To recover the asset price process, we use Equations (9) and (10) together with Equation (14) to obtain the dynamics of Y (t) : and from there, we recover the dynamics of Z(t), π(t),S(t), and finally S(t). We observe that This implies that and as a result The filtering problem is unrelated to the subsequent stochastic control problem: the dynamics ofX (t) will be the same for all investors regardless of their risk aversion or time horizon.
We solve this case similarly to our earlier article (Davis and Lleo 2011). We observe the short rate r(t) = a 0 + A 0 X (t), and hence, the one-dimensional statistic Y 0 (t) ≡ A 0 X (t), exactly. We need to assume that this observation contains positive 'noise', that is, A 0 A 0 > 0. Changing coordinates if necessary, we can assume that A 0 = (0, 0, . . . , 1) and hence Y 0 (t) = X n (t).
Our 'observation' is now the (M + k + 1)-dimensional processȲ = (Y 0 , . . . , Y M ) and we can set up a Kalman filter system to estimate the unobserved statesX = (X 1 , . . . , X n−1 ) ∈ R n−1 . Ultimately, our optimal strategy will take the form h(t,X (t), X n (t)), whereX (t) is the Kalman filter estimate forX (t) given {Ȳ (u), u ≤ t}. The details are left to the reader.

Solve the stochastic control problem
By using the idea developed in the previous steps we will express and solve the stochastic control problem in which X (t) is replaced byX (t) and the dynamic equation (1) by the Kalman filter. Optimal strategies take the form h(t,X (t)). This very old idea in stochastic control goes back at least to Wonham (1968). To illustrate this point, we solve a risk-sensitive asset management problem (see, for instance, Bielecki andPliska 1999 or Nagai andPeng 2002), although the estimation method presented in this article would apply to any continuous time investment problem including the Merton model with consumption and HARA utility.
The factor process X (t) is not directly observed and the asset allocation strategy h t must be adapted to the filtration F Y t = σ {S i (u), Z j , 0 ≤ u ≤ t, i = 0, . . . , M , j = 1, . . . , k} generated by the asset price processes and the views.  The wealth, V (t) of the investor in response to an investment strategy h(t) ∈ C(T ), follows the dynamics: with initial endowment V (0) = v. Without loss of generality we will assume that v = 1.
is an exponential martingale, that is, E[χ h T ] = 1.