A unified pricing of variable annuity guarantees under the optimal stochastic control framework

In this paper, we review pricing of variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework, and review the existing numerical methods. For numerical valuation of these contracts, we develop a direct integration method based on Gauss-Hermite quadrature with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is very efficient when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We also present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a benchmark for practitioners and researchers developing pricing of variable annuity guarantees.Comment: Keywords: variable annuity, guaranteed living and death benefits, guaranteed minimum accumulation benefit, optimal stochastic control, direct integration metho

On the calibration of the 3/2 model

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A Minimal Incentive-based Demand Response Program With Self Reported Baseline Mechanism

In this paper, we propose a novel incentive based Demand Response (DR) program with a self reported baseline mechanism. The System Operator (SO) managing the DR program recruits consumers or aggregators of DR resources. The recruited consumers are required to only report their baseline, which is the minimal information necessary for any DR program. During a DR event, a set of consumers, from this pool of recruited consumers, are randomly selected. The consumers are selected such that the required load reduction is delivered. The selected consumers, who reduce their load, are rewarded for their services and other recruited consumers, who deviate from their reported baseline, are penalized. The randomization in selection and penalty ensure that the baseline inflation is controlled. We also justify that the selection probability can be simultaneously used to control SO's cost. This allows the SO to design the mechanism such that its cost is almost optimal when there are no recruitment costs or at least significantly reduced otherwise. Finally, we also show that the proposed method of self-reported baseline outperforms other baseline estimation methods commonly used in practice

An Institutional Frame to Compare Alternative Market Designs in EU Electricity Balancing

The so-called â electricity wholesale marketâ is, in fact, a sequence of several markets. The chain is closed with a provision for â balancing,â in which energy from all wholesale markets is balanced under the authority of the Transmission Grid Manager (TSO in Europe, ISO in the United States). In selecting the market design, engineers in the European Union have traditionally preferred the technical role of balancing mechanisms as â security mechanisms.â They favour using penalties to restrict the use of balancing energy by market actors. While our paper in no way disputes the importance of grid security, nor the competency of engineers to elaborate the technical rules, we wish to attract attention to the real economic consequences of alternative balancing designs. We propose a numerical simulation in the framework of a two-stage equilibrium model. This simulation allows us to compare the economic properties of designs currently existing within the European Union and to measure their fallout. It reveals that balancing designs, which are typically presented as simple variants on technical security, are in actuality alternative institutional frameworks having at least four potential economic consequences: a distortion of the forward price; an asymmetric shift in the participantsâ profits; an increase in the System Operatorâ s revenues; and inefficiencies

Achieving an optimal trade-off between revenue and energy peak within a smart grid environment

We consider an energy provider whose goal is to simultaneously set revenue-maximizing prices and meet a peak load constraint. In our bilevel setting, the provider acts as a leader (upper level) that takes into account a smart grid (lower level) that minimizes the sum of users' disutilities. The latter bases its decisions on the hourly prices set by the leader, as well as the schedule preferences set by the users for each task. Considering both the monopolistic and competitive situations, we illustrate numerically the validity of the approach, which achieves an 'optimal' trade-off between three objectives: revenue, user cost, and peak demand

An Options Pricing Approach for CO2 Allowances in the EU ETS

If firms are unable to fully control their emissions, the cap in a permit market may be exceeded. Using stochastic aggregate emissions as the underlying I derive an options pricing formula that expresses the permit price as a function of the penalty for noncompliance and the probability of a binding cap. I apply my model to the EU ETS, where rapid market setup made it difficult for firms to adjust their production technology in time for phase 1. The model fits the data well, implying that the permit price was driven by firms hedging against stochastic emissions rather than marginal abatement costs.Permit markets, air pollution, climate change, CO2, options pricing, EU ETS

Semiparametric estimation for a class of time-inhomogenous diffusion processes

Copyright @ 2009 Institute of Statistical Science, Academia SinicaWe develop two likelihood-based approaches to semiparametrically estimate a class of time-inhomogeneous diffusion processes: log penalized splines (P-splines) and the local log-linear method. Positive volatility is naturally embedded and this positivity is not guaranteed in most existing diffusion models. We investigate different smoothing parameter selections. Separate bandwidths are used for drift and volatility estimation. In the log P-splines approach, different smoothness for different time varying coefficients is feasible by assigning different penalty parameters. We also provide theorems for both approaches and report statistical inference results. Finally, we present a case study using the weekly three-month Treasury bill data from 1954 to 2004. We find that the log P-splines approach seems to capture the volatility dip in mid-1960s the best. We also present an application to calculate a financial market risk measure called Value at Risk (VaR) using statistical estimates from log P-splines