Sebastien Lleo

Risk-sensitive benchmarked asset management

This paper extends the risk-sensitive asset management theory developed by Bielecki and Pliska and by Kuroda and Nagai to the case where the investors objective is to outperform an investment benchmark. The main result is a mutual fund theorem. Every investor following the same benchmark will take positions, in proportions dependent on his/her risk sensitivity coefficient, in two funds: the log-optimal portfolio and a second fund which adjusts for the correlation between the traded assets, the benchmark and the underlying valuation factors.

Stock market crashes in 2007–2009: were we able to predict them?

We investigate the stock market crashes in China, Iceland, and the US in the 2007-2009 period. The bond stock earnings yield difference model is used as a prediction tool. Historically, when the measure is too high, meaning that long bond interest rates are too high relative to the trailing earnings over price ratio, then there usually is a crash of 10% or more within four to twelve months. The model did in fact predict all three crashes. Iceland had a drop of fully 95%, China fell by two thirds and the US by 57%.

Jump-Diffusion Risk-Sensitive Asset Management I: Diffusion Factor Model

This paper considers a portfolio optimization problem in which asset prices are represented by SDEs driven by Brownian motion and a Poisson random measure, with drifts that are functions of an auxiliary diffusion factor process. The criterion, following earlier work by Bielecki, Pliska, Nagai, and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constraint on variance). By using a change of measure technique introduced by Kuroda and Nagai we show that the problem reduces to solving a certain stochastic control problem in the factor process, which has no jumps. The main result of this paper is to show that the risk-sensitive jump-diffusion problem can be fully characterized in terms of a parabolic Hamilton-Jacobi-Bellman PDE rather than a partial integro-differential equation, and that this PDE admits a classical

Jump-Diffusion Risk-Sensitive Asset Management II: Jump-Diffusion Factor Model

(C^{1,2})

Risk-Sensitive Investment Management

solution.

Black-Litterman in Continuous Time: The Case for Filtering

In this article we extend our earlier work on the jump-diffusion risk-sensitive asset management problem in a factor model [SIAM J. Financial Math., 2 (2011), pp. 22--54] by allowing jumps in both the factor process and the asset prices, as well as stochastic volatility and investment constraints. In this case, the Hamilton--Jacobi--Bellman (HJB) equation is a partial integro-differential equation (PIDE). We are able to show that finding a viscosity solution to this PIDE is equivalent to finding a viscosity solution to a related PDE, for which classical results give uniqueness. With this in hand, a policy improvement argument and classical results on parabolic PDEs show that the HJB PIDE admits a unique smooth solution. The optimal investment strategy is given by the feedback control that minimizes the Hamiltonian function appearing in the HJB PIDE.

Can Warren Buffett Forecast Equity Market Corrections

Over the last two decades, risk-sensitive control has evolved into an innovative and successful framework for solving dynamically a wide range of practical investment management problems. This book shows how to use risk-sensitive investment management to manage portfolios against an investment benchmark, with constraints, and with assets and liabilities. It also addresses model implementation issues in parameter estimation and numerical methods. Most importantly, it shows how to integrate jump-diffusion processes which are crucial to model market crashes. With its emphasis on the interconnection between mathematical techniques and real-world problems, this book will be of interest to both academic researchers and money managers. Risk-sensitive investment management links stochastic control and portfolio management. Because of its distinct emphasis on integrating advanced theoretical concepts into practical dynamic investment management tools, this book stands out from the existing literature in fundamental ways. It goes beyond mainstream research in portfolio management in a traditional static setting. The theoretical developments build on contemporary research in stochastic control theory, but are informed throughout by the need to construct an effective and practical framework for dynamic portfolio management. This book fills a gap in the literature by connecting mathematical techniques with the real world of investment management. Readers seeking to solve key problems such as benchmarked asset management or asset and liability management will certainly find it useful. Contents: Diffusion Models: The Merton Problem Risk-Sensitive Asset Management Managing Against a Benchmark Asset and Liability Management Investment Constraints Infinite Horizon Problems Jump-Diffusion Models: Jumps in Asset Prices General Jump-Diffusion Setting Fund Separation and Fractional Kelly Strategies Managing Against a Benchmark: Jump-Diffusion Case Asset and Liability Management: Jump-Diffusion Case Implementation: Factor and Securities Models Case Studies Numerical Methods Factor Estimation: Filtering and Black-Litterman Readership: Professionals, researchers, academics and graduate students in the field of investment management, stochastic optimization, stochastic analysis and probability, and quantitative finance.

Jump-diffusion asset---liability management via risk-sensitive control

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.

Stock Market Crashes: Predictable and Unpredictable and What to do About Them

In a 2001 interview, Warren Buffett suggested that the ratio of the market value of all publicly traded stocks to the Gross National Product could identify potential overvaluations and undervaluations in the US equity market. In this paper, we investigate whether this ratio is a statistically significant predictor of equity market downturns.

Risk-sensitive investment in a finite-factor model

In this paper, we use risk-sensitive control methods to solve a jump-diffusion asset–liability management (ALM) problem. We show that the ALM problem admits a unique classical (