Estimation of Value-at-Risk with Minimal Specification Error

Abstract

In this paper we provide a new statistical approach to estimate VaR with minimal specification error. In our approach we use current market information obtained from traded options to infer risk-neutral moments of the underlying asset returns over different horizons. For each horizon, the moments are fitted to a general 4-moment Variance Gamma risk-neutral probability distribution of future market prices. We also compute a Radon-Nikodym derivative and its probability distribution, and use this to transform the risk-neutral distribution of the underlying to its empirical distribution. The empirical distribution is then used to find the VaR. The novelty of our approach is that it avoids unnecessary mis-specification bias as we do not need to assume a particular empirical distribution of the underlying asset returns. As option prices trade frequently, the risk-neutral distribution can be updated frequently for transformation into the empirical distribution. This provides for timely updating of the VaR measures. The fitted risk-neutral Variance Gamma distribution is popular in finance due to its ability to model heavy tails and skewness. Our key contributions are in providing an alternative new approach to modeling VaR, and also in showing that underestimation of risk is largely not due to VaR itself but perhaps due to mis-specification errors which we minimize in our approach. We show that our method of measuring VaR clearly captures large tail risk in the empirical examples on S&P 500 index.