### Historical Scenarios with Fully Flexible Probabilities

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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 VaR and volatility. The probabilities can be changed to reflect specific market conditions, advanced estimation techniques, or partial information, using the entropy-based Fully Flexible Views technique in Meucci (2008). The implementation of this approach is trivial, as no costly repricing is needed.

Consider a market driven by *N* risk drivers *X* ≡ {*X _{n}*}

_{n=1,…,N}, i.e. a market where the p&l Π of each security is fully determined by

*X*through a known pricing function π:

For instance, in equity options the risk drivers are the log-changes of the implied volatility smile, of the underlying, and of the interest rates; and the p&l depends on the risk drivers through the Black-Scholes-Merton formula. For corporate bonds the drivers are the interest rates changes of a reference curve such as swap or government and the spreads changes over that curve; and the pricing function is the usual discount formula. We refer to Meucci (2010) for more details.

To manage risk we must estimate the joint distribution of the risk drivers, as represented by the probability density function (pdf) ƒ_{X} , and analyze its impact on the distribution of the prospective p&l (1) of a given portfolio. There exist two approaches to achieve this, namely parametric and scenario-based, but only the latter is reliable upon across all asset classes and investment horizons.

In Section 1 we review these two approaches and we introduce the generalized empirical distribution, which allows us to greatly enhance the scenario-based approach by associating non-equal probabilities to diﬀerent scenarios. In Section 2 we discuss techniques to specify exogenously all the probabilities of the generalized empirical distribution. These techniques include rolling window, exponential smoothing and market conditioning by kernel smoothing. In Section 3 we rely on the Fully Flexible Views approach in Meucci (2008) to specify only some features of the probabilities of the generalized empirical distribution, and then ﬁll in the missing information by relative entropy minimization. This allows us for instance to perform market conditioning without having to specify the shape of the smoothing kernel.

**1 PARAMETRIC VERSUS SCENARIO-BASED RISK MANAGEMENT**

The ﬁrst approach to risk management is parametric: the distribution of the risk drivers is modeled by a parametric distribution , where *Θ* is a set of parameters. The most classical example in this direction is the normal distribution, where *Θ* represent the expected values and the covariances of the risk drivers. Other examples are elliptical distributions, skew-*t* distributions, parametric marginal-copula decompositions, etc. The parametric approach is computationally cheap. However, for a wide spectrum of instruments the p&l (1) is a non-linear function of the risk drivers. In such cases, the distribution of the p&l cannot be computed exactly and one must resort to approximations.

A second approach to risk management represents the distribution of the risk drivers ƒ_{X} in terms of a *T* × *N* panel of joint scenarios {*χ _{t,n}*}. Each

*N*dimensional row

(2)

is an independent joint scenario from the distribution ƒ_{X} , and each scenario occurs with constant probability *p*_{t} ≡ 1/*T*, see Figure 1. If the risk drivers are "invariants", i.e. if they can be assumed independently and identically distributed across time, the scenarios can be historical realizations. Alternatively, the scenarios can be Monte Carlo simulations. In the sequel we focus on historical scenarios, and *t* is to be interpreted as time.

**Figure 1: Distribution of p&l and risk-drivers as scenarios/probabilities**

Using the scenario-based approach it is easy to generate scenarios for the portfolio p&l {*π*_{t}} that represent the p&l distribution. Indeed, as in Figure 1, it suﬃces to apply the pricing function (1) to each scenario

From these p&l scenarios it is then trivial to extract the relevant statistics, such as volatility and VaR, in terms of their sample counterparts.

Notice that the computational burden of pricing each scenario can be handled by parallel computing across both securities and scenarios. Therefore, even the computational challenge of the scenario-based approach can be overcome. As a result, the scenario-based approach has become the reference methodology for risk management in most ﬁnancial institutions.

Assume that a panel of historical scenarios for the risk drivers {χ* _{t,n}*} is available, and that the respective p&l scenarios {

*π*

_{t}} for the securities and the portfolio have been generated as in (3), possibly by a vendor during an overnight batch process. We can greatly enhance the ﬂexibility of the scenario-based approach by suitably changing the probabilities

*p*

_{t}associated with the scenarios.

In other words, we represent the distribution of the risk drivers ƒ_{X} by means of the generalized empirical distribution

In this expression *δ*^{(}^{z}^{)} denotes the Dirac delta centered in the generic point , i.e. a spike of probability mass equal to one in *z*, refer to the appendix for more details. The standard empirical distribution is the special case of (4) where for all times t the probability is set as *p*_{t} ≡ 1/*T* .

Then, using the simple recipes in the appendix, we can recompute the p&l distribution with the new probabilities *p*_{t} and extract all the relevant statistics such as volatility and VaR. From an implementation perspective, this approach is trivial. Once the new probabilities have been deﬁned, all the above computations can be performed in real time, without calling again the costly pricing functions (3).

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**–Attilio Meucci, PhD, CFA, is the chief risk officer at Kepos Capital LP and an adjunct professor at Baruch College. He is also the author of Risk and Asset Allocation and a regular contributor to publications, including Risk Magazine and to GARP Risk Professional Magazine.**

Excerpted from an article published in *GARP Risk Professional*, Dec 2010, p 47-51.

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