Excess of loss reinsurance under joint survival optimality

Explicit expressions for the probability of joint survival up to time x of the cedent and the reinsurer, under an excess of loss reinsurance contract with a limiting and a retention level are obtained, under the reasonably general assumptions of any non-decreasing premium income function, Poisson claim arrivals and continuous claim amounts, modelled by any joint distribution. By stating appropriate optimality problems, we show that these results can be used to set the limiting and the retention levels in an optimal way with respect to the probability of joint survival. Alternatively, for fixed retention and limiting levels, the results yield an optimal split of the total premium income between the two parties in the excess of loss contract. This methodology is illustrated numerically on several examples of independent and dependent claim severities. The latter are modelled by a copula function. The effect of varying its dependence parameter and the marginals, on the solutions of the optimality problems and the joint survival probability, has also been explored

The Garman-Klass volatility estimator revisited

The Garman-Klass unbiased estimator of the variance per unit time of a zero-drift Brownian Motion B, based on the usual financial data that reports for time windows of equal length the open (OPEN), minimum (MIN), maximum (MAX) and close (CLOSE) values, is quadratic in the statistic S1=(CLOSE-OPEN, OPEN-MIN, MAX-OPEN). This estimator, with efficiency 7.4 with respect to the classical estimator (CLOSE-OPEN)^2, is widely believed to be of minimal variance. The current report disproves this belief by exhibiting an unbiased estimator with slightly but strictly higher efficiency 7.7322. The essence of the improvement lies in the observation that the data should be compressed to the statistic S2 defined on W(t)= B(0)+[B(t)-B(0)]sign[(B(1)-B(0)] as S1 was defined on the Brownian path B(t). The best S2-based quadratic unbiased estimator is presented explicitly. The Cramer-Rao upper bound for the efficiency of unbiased estimators, corresponding to the efficiency of large-sample Maximum Likelihood estimators, is 8.471. This bound cannot be attained because the distribution is not of exponential type. Regression-fitted quadratic functions of S2 (with mean 1) markedly out-perform those of S1 when applied to random walks with heavy-tail-distributed increments. Performance is empirically studied in terms of the tail parameter