Sums of a Random Number of Random Variables and their Approximations with ν- Accompanying Infinitely Divisible Laws

* Research supported by NATO GRANT CRG 900 798 and by Humboldt Award for U.S. Scientists.In this paper a general theory of a random number of random variables is constructed. A description of all random variables ν admitting an analog of the Gaussian distribution under ν-summation, that is, the summation of a random number ν of random terms, is given. The v-infinitely divisible distributions are described for these ν-summations and finite estimates of the approximation of ν-sum distributions with the help of v-accompanying infinitely divisible distributions are given. The results include, in particular, the description of geometrically infinitely divisible and geometrically stable distributions as well as their domains of attraction

Exploring Implied Certainty Equivalent Rates in Financial Markets: Empirical Analysis and Application to the Electric Vehicle Industry

In this paper, we mainly study the impact of the implied certainty equivalent rate on investment in financial markets. First, we derived the mathematical expression of the implied certainty equivalent rate by using put-call parity, and then we selected some company stocks and options; we considered the best-performing and worst-performing company stocks and options from the beginning of 2023 to the present for empirical research. By visualizing the relationship between the time to maturity, moneyness, and implied certainty equivalent rate of these options, we have obtained a universal conclusion -- a positive implied certainty equivalent rate is more suitable for investment than a negative implied certainty equivalent rate, but for a positive implied certainty equivalent rate, a larger value also means a higher investment risk. Next, we applied these results to the electric vehicle industry, and by comparing several well-known US electric vehicle production companies, we further strengthened our conclusions. Finally, we give a warning concerning risk, that is, investment in the financial market should not focus solely on the implied certainty equivalent rate, because investment is not an easy task, and many factors need to be considered, including some factors that are difficult to predict with models

A profit model for spread trading with an application to energy futures

This paper proposes a profit model for spread trading by focusing on the stochastic movement of the price spread and its first hitting time probability density. The model is general in that it can be used for any financial instrument. The advantage of the model is that the profit from the trades can be easily calculated if the first hitting time probability density of the stochastic process is given. We then modify the profit model for a particular market, the energy futures market. It is shown that energy futures spreads are modeled by using a meanreverting process. Since the first hitting time probability density of a mean-reverting process is approximately known, the profit model for energy futures price spreads is given in a computable way by using the parameters of the process. Finally, we provide empirical evidence for spread trades of energy futures by employing historical prices of energy futures (WTI crude oil, heating oil, and natural gas futures) traded on the New York Mercantile Exchange. The results suggest that natural gas futures trading may be more profitable than WTI crude oil and heating oil due to its high volatility in addition to its long-term mean reversion, which offers supportive evidence of the model prediction. --futures spread trading,energy futures markets,mean-reverting process,first hitting,time probability density,profit model,WTI crude oil,heating oil,natural gas

The Implied Views of Bond Traders on the Spot Equity Market

By using the Black-Derman-Toy (BDT) model, we predict the future trend of the riskless rate, and then we build an equation that relates the market price of zero-coupon bonds and the theoretical price of zero-coupon bonds calculated using a binomial option pricing model. Based on this, we can find the implied daily return $\mu$, implied natural upturn probability $p$, and implied daily volatility $\sigma$ with respect to different time-to-maturity values of zero-coupon bonds. With these results, we can give some suggestions to bond traders

Estimation of operational value-at-risk in the presence of minimum collection threshold: An empirical study

The recently finalized Basel II Capital Accord requires banks to adopt a procedure to estimate the operational risk capital charge. Under the Advanced Measurement Approaches, that are currently mandated for all large internationally active US banks, require the use of historic operational loss data. Operational loss databases are typically subject to a minimum recording threshold of roughly $10,000. We demonstrate that ignoring such thresholds leads to biases in corresponding parameter estimates when the threshold is ignored. Using publicly available operational loss data, we analyze the effects of model misspecification on resulting expected loss, Value-at-Risk, and Conditional Value-at-Risk figures and show that underestimation of the regulatory capital is a consequence of such model error. The choice of an adequate loss distribution is conducted via in-sample goodness-of-fit procedures and backtesting, using both classical and robust methodologies. --

CVaR sensitivity with respect to tail thickness

We consider the sensitivity of conditional value-at-risk (CVaR) with respect to the tail index assuming regularly varying tails and exponential and faster-than-exponential tail decay for the return distribution. We compare it to the CVaR sensitivity with respect to the scale parameter for stable Paretian, the Student's t, and generalized Gaussian laws and discuss implications for the modeling of daily returns and marginal rebalancing decisions. Finally, we explore empirically the impact on the asymptotic variability of the CVaR estimator with daily returns which is a standard choice for the return frequency for risk estimation. --fat-tailed distributions,regularly varying tails,conditional value-at-risk,marginal rebalancing,asymptotic variability

Asymptotic distribution of linear unbiased estimators in the presence of heavy-tailed stochastic regressors and residuals

Under the symmetric á-stable distributional assumption for the disturbances, Blattberg et al (1971) consider unbiased linear estimators for a regression model with non-stochastic regressors. We consider both the rate of convergence to the true value and the asymptotic distribution of the normalized error of the linear unbiased estimators. By doing this, we allow the regressors to be stochastic and disturbances to be heavy-tailed with either finite or infinite variances, where the tail-thickness parameters of the regressors and disturbances may be different. --Asymptotic distribution,rate of convergence,stochastic regressor,stable non-Gaussian,finite or infinite variance,heavy tails

Modeling catastrophe claims with left-truncated severity distributions (extended version)

In this paper, we present a procedure for consistent estimation of the severity and frequency distributions based on incomplete insurance data and demonstrate that ignoring the thresholds leads to a serious underestimation of the ruin probabilities. The event frequency is modelled with a non-homogeneous Poisson process with a sinusoidal intensity rate function. The choice of an adequate loss distribution is conducted via the in-sample goodness-of-fit procedures and forecasting, using classical and robust methodologies.Natural catastrophe; Property insurance; Loss distribution; Truncated data; Ruin probability;