We present “The Prayer”, a recipe of ten sequential steps for all portfolio managers, risk managers, algorithmic traders across all asset classes and all investment horizons, to model and manage the P&L distribution of their positions.For each of the ten steps of the Prayer, we introduce all...

There exist two separate branches of finance that require advanced quantitative techniques: the "Q" area of derivatives pricing, whose task is to "extrapolate the present"; and the "P" area of quantitative risk and portfolio management, whose task is to "model the future."We briefly trace the...

After reviewing the parametric and scenario-based approaches to risk management, we discuss a methodology to enhance the flexibility of the scenario-based approach. We change the probability of each scenario, and then we compute the ensuing p&l distribution and all relevant statistics such as...

How can we report returns for a swap that has zero value? How can we perform return optimization for a zero-value long-short portfolio? By introducing a suitable "basis", it is possible to extend the definition of returns to leveraged products in such a way that performance attribution and...

When estimating risk from a window of historical observations, the confidence interval is inverse to the number of scenarios used, which is the length of the window. When estimating risk with exponential decay, where the relative weight of each scenario decreases with time, the confidence...

We show how to mix machine learning signals such as kernel smoothing and fuzzy memberships via the Entropy Pooling approach by Meucci (2008). We illustrate a case study, where we overlay an exponentially time-decayed prior to a pseudo-Gaussian kernel that emphasizes market scenarios where...

Propagating causal stress-tests or contagion on selected risk factors to all the risk drivers is a challenging task. We use Entropy Pooling by Meucci (2008) to address this issue. Our causal stress-tests comprise, but are not restricted to, stress-testing Bayesian networks. We detail the theory...

We extend the Fully Flexible Views generalization of the Black-Litterman approach to effectively handle extreme views on the tails of a distribution. First, we provide a recursive algorithm to process views on the conditional value at risk, which cannot be handled directly by the original...