Brenda López Cabrera

Localising temperature risk

ABSTRACT On the temperature derivative market, modeling temperature volatility is an important issue for pricing and hedging. To apply the pricing tools of financial mathematics, one needs to isolate a Gaussian risk factor. A conventional model for temperature dynamics is a stochastic model with seasonality and intertemporal autocorrelation. Empirical work based on seasonality and autocorrelation correction reveals that the obtained residuals are heteroscedastic with a periodic pattern. The object of this research is to estimate this heteroscedastic function so that, after scale normalization, a pure standardized Gaussian variable appears. Earlier works investigated temperature risk in different locations and showed that neither parametric component functions nor a local linear smoother with constant smoothing parameter are flexible enough to generally describe the variance process well. Therefore, we consider a local adaptive modeling approach to find, at each time point, an optimal smoothing parameter to locally estimate the seasonality and volatility. Our approach provides a more flexible and accurate fitting procedure for localized temperature risk by achieving nearly normal risk factors. We also employ our model to forecast the temperaturein different cities and compare it to a model developed in 2005 by Campbell and Diebold. Supplementary materials for this article are available online.

Implied Market Price of Weather Risk

Abstract Weather derivatives (WD) are end-products of a process known as securitization that transforms non-tradable risk factors (weather) into tradable financial assets. For pricing and hedging non-tradable assets, one essentially needs to incorporate the market price of risk (MPR), which is an important parameter of the associated equivalent martingale measure (EMM). The majority of papers so far has priced non-tradable assets assuming zero or constant MPR, but this assumption yields biased prices and has never been quantified earlier under the EMM framework. Given that liquid-derivative contracts based on daily temperature are traded on the Chicago Mercantile Exchange (CME), we infer the MPR from traded futures-type contracts (CAT, CDD, HDD and AAT). The results show how the MPR significantly differs from 0, how it varies in time and changes in sign. It can be parameterized, given its dependencies on time and temperature seasonal variation. We establish connections between the market risk premium (RP) and the MPR.

Calibrating CAT Bonds for Mexican Earthquakes

The study of natural catastrophe models plays an important role in the prevention and mitigation of disasters. After the occurrence of a natural disaster, the reconstruction can be financed with catastrophe bonds (CAT bonds) or reinsurance. This paper examines the calibration of a real parametric CAT bond for earthquakes that was sponsored by the Mexican government. The calibration of the CAT bond is based on the estimation of the intensity rate that describes the earthquake process from the two sides of the contract, the reinsurance and the capital markets, and from the historical data. The results demonstrate that, under specific conditions, the financial strategy of the government, a mix of reinsurance and CAT bond, is optimal in the sense that it provides coverage of USD 450 million for a lower cost than the reinsurance itself. Since other variables can affect the value of the losses caused by earthquakes, e.g. magnitude, depth, city impact, etc., we also derive the price of a hypothetical modeled-index loss (zero) coupon CAT bond for earthquakes, which is based on the compound doubly stochastic Poisson pricing methodology from BARYSHNIKOV, MAYO and TAYLOR (2001) and BURNECKI and KUKLA (2003). In essence, this hybrid trigger combines modeled loss and index trigger types, trying to reduce basis risk borne by the sponsor while still preserving a nonindemnity trigger mechanism. Our results indicate that the (zero) coupon CAT bond price increases as the threshold level increases, but decreases as the expiration time increases. Due to the quality of the data, the results show that the expected loss is considerably more important for the valuation of the CAT bond than the entire distribution of losses. The study of natural catastrophe models plays an important role in the prevention and mitigation of disasters. After the occurrence of a natural disaster, the reconstruction can be financed with catastrophe bonds (CAT bonds) or reinsurance. This paper examines the calibration of a real parametric CAT bond for earthquakes that was sponsored by the Mexican government. The calibration of the CAT bond is based on the estimation of the intensity rate that describes the earthquake process from the two sides of the contract, the reinsurance and the capital markets, and from the historical data. The results demonstrate that, under specific conditions, the financial strategy of the government, a mix of reinsurance and CAT bond, is optimal in the sense that it provides coverage of USD 450 million for a lower cost than the reinsurance itself. Since other variables can affect the value of the losses caused by earthquakes, e.g. magnitude, depth, city impact, etc., we also derive the price of a hypothetical modeled-index loss (zero) coupon CAT bond for earthquakes, which is based on the compound doubly stochastic Poisson pricing methodology from BARYSHNIKOV, MAYO and TAYLOR (2001) and BURNECKI and KUKLA (2003). In essence, this hybrid trigger combines modeled loss and index trigger types, trying to reduce basis risk borne by the sponsor while still preserving a nonindemnity trigger mechanism. Our results indicate that the (zero) coupon CAT bond price increases as the threshold level increases, but decreases as the expiration time increases. Due to the quality of the data, the results show that the expected loss is considerably more important for the valuation of the CAT bond than the entire distribution of losses.

Pricing of Asian temperature risk

Weather derivatives (WD) are different from most financial derivatives because the underlying weather cannot be traded and therefore cannot be replicated by other financial instruments. The market price of risk (MPR) is an important parameter of the associated equivalent martingale measures used to price and hedge weather futures/options in the market. The majority of papers so far have priced non-tradable assets assuming zero MPR, but this assumption underestimates WD prices. We study the MPR structure as a time dependent object with concentration on emerging markets in Asia. We find that Asian Temperatures (Tokyo, Osaka, Beijing, Teipei) are normal in the sense that the driving stochastics are close to a Wiener Process. The regression residuals of the temperature show a clear seasonal variation and the volatility term structure of CAT temperature futures presents a modified Samuelson effect. In order to achieve normality in standardized residuals, the seasonal variation is calibrated with a combination of a fourier truncated series with a GARCH model and with a local linear regression. By calibrating model prices, we implied the MPR from Cumulative total of 24- hour average temperature futures (C24AT) for Japanese Cities, or by knowing the formal dependence of MPR on seasonal variation, we price derivatives for Kaohsiung, where weather derivative market does not exist. The findings support theoretical results of reverse relation between MPR and seasonal variation of temperature process.

Forecasting Generalized Quantiles of Electricity Demand: A Functional Data Approach

ABSTRACT Electricity load forecasts are an integral part of many decision-making processes in the electricity market. However, most literature on electricity load forecasting concentrates on deterministic forecasts, neglecting possibly important information about uncertainty. A more complete picture of future demand can be obtained by using distributional forecasts, allowing for more efficient decision-making. A predictive density can be fully characterized by tail measures such as quantiles and expectiles. Furthermore, interest often lies in the accurate estimation of tail events rather than in the mean or median. We propose a new methodology to obtain probabilistic forecasts of electricity load that is based on functional data analysis of generalized quantile curves. The core of the methodology is dimension reduction based on functional principal components of tail curves with dependence structure. The approach has several advantages, such as flexible inclusion of explanatory variables like meteorological forecasts and no distributional assumptions. The methodology is applied to load data from a transmission system operator (TSO) and a balancing unit in Germany. Our forecast method is evaluated against other models including the TSO forecast model. It outperforms them in terms of mean absolute percentage error and mean squared error. Supplementary materials for this article are available online.

Time-Adaptive Probabilistic Forecasts of Electricity Spot Prices with Application to Risk Management.

The increasing exposure to renewable energy has amplied the need for risk management in electricity markets. Electricity price risk poses a major challenge to market participants. We propose an approach to model and fore- cast electricity prices taking into account information on renewable energy production. While most literature focuses on point forecasting, our method- ology forecasts the whole distribution of electricity prices and incorporates spike risk, which is of great value for risk management. It is based on func- tional principal component analysis and time-adaptive nonparametric density estimation techniques. The methodology is applied to electricity market data from Germany. We nd that renewable infeed eects both, the location and the shape of spot price densities. A comparison with benchmark methods and an application to risk management are provided.

Financial Time Series Models

This section deals with financial time series analysis. The statistical properties of asset and return time series are inuenced by the media (daily news on the radio, television and newspapers) that informs us about the latest changes in stock prices, interest rates and exchange rates. This information is also available to traders who deal with immanent risk in security prices. It is therefore interesting to understand the behavior of asset prices. Economic models on the pricing of securities are mostly based on theoretical concepts which involve the formation of expectations, utility functions and risk preferences. In this section we concentrate on answering the empirical questions. Firstly, given a data set we aim to specify an appropriate model reecting the main characteristics of the empirically observable stock price process and we wish to know whether the assumptions underlying the model are fulfilled in reality or whether the model has to be modified. A new model on the stock price process could possibly effect the function of the markets. To this end we apply statistical tools to empirical data and start with considering the concepts of univariate analysis before moving on to multivariate time series.

Models for the Interest Rate and Interest Rate Derivatives

Pricing interest rate derivatives fundamentally depends on the underlying term structure. The often made assumptions of constant risk free interest rate and its independence of equity prices will not be reasonable when considering interest rate derivatives. Just as the dynamics of a stock price are modeled via a stochastic process, the term structure of interest rates is modeled stochastically. As interest rate derivatives have become increasingly popular, especially among institutional investors, the standard models for the term structure have become a core part of financial engineering. It is therefore important to practice the basic tools of pricing interest rate derivatives. For interest rate dynamics, there are one-factor and two-factor short rate models, the Heath Jarrow Morton framework and the LIBOR Market Model.

Portfolio Credit Risk

Financial institutions are interested in loss protection and loan insurance. Thus determining the loss reserves needed to cover the risk stemming from credit portfolios is a major issue in banking. By charging risk premiums a bank can create a loss reserve account which it can exploit to be shielded against losses from defaulted debt. However, it is imperative that these premiums are appropriate to the issued loans and to the credit portfolio risk inherent to the bank. To determine the current risk exposure it is necessary that financial institutions can model the default probabilities for their portfolios of credit instruments appropriately. To begin with, these probabilities can be viewed as independent but it is apparent that it is plausible to drop this assumption and to model possible defaults as correlated events.