$${{\texttt {openIRM}}}$$: publicly accessible internal risk model of an artificial life insurer for analyzing and benchmarking actuarial methods in the solvency II setting

This is the first comprehensive study of the SABR (stochastic alpha‐beta‐rho) model (Hagan, Kumar, Lesniewski, & Woodward, 2002) on the pricing and hedging of interest rate caps. I implement several versions of the SABR interest rate model and analyze their respective pricing and hedging performance using two years of daily data with seven different strikes and ten different tenors on each trading day. In‐sample and out‐of‐sample tests show that the fully stochastic version of the SABR model exhibits excellent pricing accuracy and, more importantly, captures the dynamics of the volatility smile over time very well. This is further demonstrated through examining delta‐hedging performance based on the SABR model. My hedging result indicates that the SABR model produces accurate hedge ratios that outperform those implied by the Black model. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 32:773‐791, 2012

This is the first comprehensive study of the SABR (Stochastic Alpha-Beta-Rho) model (Hagan et. al (2002)) on the pricing and hedging of interest rate caps. We implement several versions of the SABR interest rate model and analyze their respective pricing and hedging performance using two years of daily data with seven different strikes and ten different tenors on each trading day. In-sample and out-of-sample tests show that in addition to having stochastic volatility for the forward rate, it is essential to recalibrate daily either the “vol of vol” or the correlation between forward rate and its volatility, although recalibrating both further improves pricing performance. The fully stochastic version of the SABR model exhibits excellent pricing accuracy and more importantly, captures the dynamics of the volatility smile over time very well. This is further demonstrated through examining delta hedging performance based on the SABR model. Our hedging result indicates that the SABR model produces accurate hedge ratios that outperform those implied by the Black model.

Solvency 2 requires insurance companies to compute a Best-Estimate of their Liabilities (BEL) as well as a Solvency Capital Requirement (SCR). Life insurance companies being in the business of selling participating contracts with financial guarantees have to rely on a Monte-Carlo approach to appropriately value their BEL, which is the source of a first Monte-Carlo statistical error. In addition, several insurance companies rely on a (partial) internal model to derive their SCR. In this context, insurance companies rely again on a Monte-Carlo approach to value their SCR, which is the source of a second Monte-Carlo statistical error. These later computations require evaluating the BEL several thousand times, which is not possible in practice, since one single Monte-Carlo BEL evaluation can be a computational burden. The BEL has therefore to be approximated by an analytic proxy function, which introduces an additional source of numerical approximation error. In this paper, we show how these three sources of error (statistical and numerical) are intrinsically related to one another. We show that to obtain the best possible SCR accuracy, the computing power invested in assessing the Monte-Carlo SCR should be directly related to that invested in computing the Monte-Carlo BEL. Interestingly, and to achieve these results, we introduce a novel proxy method, which is highly practical, modular, smooth and naturally relates the approximation errors to the Monte-Carlo statistical errors. Furthermore, our approach allows insurance companies to naturally and transparently start reporting confidence levels on their prudential reporting, which is not disclosed so far by insurance companies and would be a relevant information within solvency disclosures for the industry.

Neural Correlates of Internal-Model Loading

Dynamic term structure models price interest rate options based on the model-implied fair values of the yield curve, ignoring any pricing residuals on the yield curve that are either from model approximations or market imperfections. In contrast, option pricing in practice often takes the market observed yield curve as given and focuses exclusively on the specification of the volatility structure of forward rates. Thus, if any errors exist on the observed yield curve, they will be carried over permanently. In this paper, we propose an m+n model structure that bridges the gap between the two approaches and consistently prices both interest rates and interest rate options. The first m factors capture the systematic movement of the yield curve, whereas the latter n factors capture the impacts of the yield curve residuals on option pricing. We estimate a 3+3 Gaussian affine example using eight years of data on U.S. dollar LIBOR, swap rates, and interest rate caps. The model performs well in pricing both interest rates and interest rate caps. Furthermore, estimation shows that small residuals on the yield curve can have large impacts on pricing interest rate caps. Under the estimated model, the three yield curve factors explain over 99 percent of the variation on the yield curve, but account for less than 50 percent of the variation on cap implied volatilities. Incorporating the three residual factors improves the explained variance on cap implied volatility to over 99 percent.

The volatility “smile” or “skew” observed in the S&P 500 index options has been one of the main drivers for the development of new option pricing models since the seminal works of Black and Scholes (J Polit Econ 81:637–654, 1973) and Merton (Bell J Econ Manag Sci 4:141–183, 1973). The literature on interest rate derivatives, however, has mainly focused on at-the-money interest rate options. This paper advances the literature on interest rate derivatives in several aspects. First, we present systematic evidence on volatility smiles in interest rate caps over a wide range of moneyness and maturities. Second, we discuss the pricing and hedging of interest rate caps under dynamic term structure models (DTSMs). We show that even some of the most sophisticated DTSMs have serious difficulties in pricing and hedging caps and cap straddles, even though they capture bond yields well. Furthermore, at-the-money straddle hedging errors are highly correlated with cap-implied volatilities and can explain a large fraction of hedging errors of all caps and straddles across moneyness and maturities. These findings strongly suggest the existence of systematic unspanned factors related to stochastic volatility in interest rate derivatives markets. Third, we develop multifactor Heath–Jarrow–Morton (HJM) models with stochastic volatility and jumps to capture the smile in interest rate caps. We show that although a three-factor stochastic volatility model can price at-the-money caps well, significant negative jumps in interest rates are needed to capture the smile. Finally, we present nonparametric evidence on the economic determinants of the volatility smile. We show that the forward densities depend significantly on the slope and volatility of LIBOR rates and that mortgage refinance activities have strong impacts on the shape of the volatility smile. These results provide nonparametric evidence of unspanned stochastic volatility and suggest that the unspanned factors could be partly driven by activities in the mortgage markets.

A New Proposal on How Motor Memory Is Consolidated

In the context of Solvency II, insurance companies access and manage economic capital across various techniques requiring generally heavy Monte Carlo simulation procedures in order to derive an appropriate loss distribution. However, very little attention has been addressed to model Solvency Capital Requirements (SCR) based on the derivation of a closed-form formula. While the current industry standard is to make use of sophisticated algorithms requiring a significant calculation time given the current state-of-the-art computer technology and the amount of computational resources required for some techniques such as the stochastic-in-stochastic valuation that is particularly difficult to implement in practice for large participating Life insurance portfolios, a flexible and realistic modeling structure is proposed. This paper explores a convenient and straightforward solution for quantifying economic capital that aims at investigating notably the limited attempt that has been addressed so far at modeling economic capital via an adequate closed-form formula. The analytical formula proposed has the advantage of avoiding the use of simplified approaches such as the replicating portfolio for instance that has been largely criticized for its poor ability to replicate appropriately liabilities without introducing a significant error in the loss distribution covering against market risk. This paper contradicts the recent trend of highly complex modeling frameworks developed by some insurance companies for producing economic capital that can be hardly produced with high frequency and is in clear opposition with directives provided in CEIOPS. We aim to support the idea of developing a type of partial internal model allowing to better know the risks an insurance company is exposed to while keeping the attractive advantage of a closed-form solution. The approach adopted shows that SCR can be obtained based on the knowledge of systematic risks and conditional moments of order 1 and 2 of the distribution of net cash flows. An explicit solution for SCR is provided based on the derivation of a partial internal economic capital model that is general and flexible enough to further assess SCR via the inclusion of various assumptions largely detailed in this paper and allows to refine the approach. The attractive features of the modeling structure are its analytical formula, accuracy and encompass the interactions between assets and liabilities. Importantly, from our belief, the proposed modeling structure appears in accordance with CEIOPS, the general principles that aim to ensure a harmonized approach to the use of internal models throughout insurance companies.

This article examined the relationship between internal working models of self and other (Bowlby, 1969) and expectations for satisfaction in a future relationship, and how that relationship is moderated by the accessibility of one's internal models. Study one showed that the model of self was predictive of expected satisfaction, but the model of other was not. In study two, the results of study one were replicated. However, using a reaction time task to measure the chronic accessibility of internal models, it was shown that the relationship between model of self and expected satisfaction existed only for people with highly accessible internal models. The implications of these findings for a more cognitive view of attachment‐processes is discussed.

Chapter 15 - Beyond “Model Risk”: A Practice Perspective on Modeling in Insurance

In order to avoid the risks brought by blind investment, it is necessary to forecast the trading price of gold and bitcoin on that trading day before investment. To aim at this prediction problem, this paper puts forward the price prediction model of the BP neural network. The experimental results show that the model can reasonably predict the trading price of gold and bitcoin on the next trading day, and the relative error between them is less than 10%.However, predicting the trading price of gold and bitcoin can roughly determine the future direction, and it cannot provide scientific and reasonable trading strategies for traders, so it is necessary to design a trading strategy planning model. Considering that every trading activity except holding financial products will charge additional trade costs, the trade costs of gold and bitcoin are 1% and 2% of the trade amount, respectively. By comparing various classic trading models, we propose to use a dynamic programming model to provide traders with effective trading strategies, which takes the trading changes of gold and bitcoin in the next three trading days as decision variables. Meanwhile, it takes the highest value of the Sharp ratio in the next trading day as the objective function. It takes the proportion of each part as equal to or greater than 0 as the constraint condition to establish a trading strategy planning model to obtain the maximum profit.

The Solvency II Directive mandates insurance firms to value their assets and liabilities using market consistent valuation. For many types of insurance business Economic Scenario Generators (ESGs) are the only practical way to determine the market consistent value of liabilities. The directive also allows insurance companies to use an internal model to calculate their solvency capital requirement. In particular, this includes use of ESG models. Regardless of whether an insurer chooses to use an internal model, Economic Scenario Generators will be the only practical way of valuing many life insurance contracts. Draft advice published by the Committee of European Insurance and Occupational Pensions Supervisors (CEIOPS) requires that insurance firms who intend to use an internal model to calculate their capital requirements under Solvency II need to comply with a number of tests regardless of whether the model (or data) is produced internally or is externally sourced. In particular the tests include a ‘use test’, mandating the use of the model for important decision making within the insurer. This means that Economic Scenario Generators will need to subject themselves to the governance processes and that senior managers and boards will need to understand what ESG models do and what they don't do. In general, few senior managers are keen practitioners of stochastic calculus, the building blocks of ESG models. The paper therefore seeks to explain Economic Scenario Generator models from a non-technical perspective as far as possible and to give senior management some guidance of the main issues surrounding these models from an ERM/Solvency II perspective.

Internal model control design based on approximation of linear discrete dynamical systems

Prediction of soil organic carbon stock by laboratory spectral data and airborne hyperspectral images

This study investigated the ability to use an internal model of the environmental dynamics when the dynamics were predictable but unstable. Subjects performed goal-directed movements using a robot manipulandum while counteracting a force field, which created instability by assisting the movement in proportion to hand velocity. Subjects' performance was better on the last trial than on the first trial in the force field for all four movement directions tested: out, in, right and left. Subjects adapted to the force field primarily by increasing muscle co-contraction, compared to null field movements, during all phases of movement. This co-contraction generally declined for both the deceleration and stabilization phases during the course of the first 25 movements in each direction, but tended not to decrease significantly thereafter. Catch trials at the end of the learning period suggested that increased viscoelastic impedance due to muscle co-contraction was used to counteract the force field. Only in the case of outward movements were aftereffects observed that were consistent with formation of an accurate internal model of the force field dynamics. Stabilization of the hand for outward movements required less muscle co-contraction than for movements in other directions due to stability conferred by the geometry of the arm. The results suggest that the accuracy of an internal model depends critically on the stability of the coupled dynamics of the limb and the environment.