Hedging Error Research Articles

Pricing and hedging of inflation-indexed bonds in an affine framework

This study deals with the pricing and hedging of inflation-indexed bonds. Under foreign exchange analogy we model the nominal short rate, real short rate and logarithm of the price index with an affine Gaussian process. Using the underlying affine property, we compute the nominal and inflation-indexed bond prices explicitly. We derive no-arbitrage drift conditions for the factor process. Then, we perform a novel hedging analysis where our objective is to replicate an indexed bond of a given maturity by trading a portfolio of nominal bonds. This analysis leads to a hedging criterion based on a set of restrictions on the eigenvalues and the eigenvectors of mean reversion speed matrix of the factor process. We fit the model to the US bond data and perform an in-sample hedging analysis. Having relatively small in-sample hedging errors, we validate the theoretical hedging result for the considered dataset.

Pricing and Hedging of Inflation-Indexed Bonds in an Affi ne Framework

This study deals with the pricing and hedging of inflation-indexed bonds. Under foreign exchange analogy we model the nominal short rate, real short rate and logarithm of the price index with an affine Gaussian process. Using the underlying affine property, we compute the nominal and inflation-indexed bond prices explicitly. We derive no-arbitrage drift conditions for the factor process. Then, we perform a novel hedging analysis where our objective is to replicate an indexed bond of a given maturity by trading a portfolio of nominal bonds. This analysis leads to a hedging criterion based on a set of restrictions on the eigenvalues and the eigen vectors of mean reversion speed matrix of the factor process. We fit the model to the U.S. bond data and perform an in-sample hedging analysis. Having relatively small in-sample hedging errors, we validate the theoretical hedging result for the considered dataset.

On the performance of delta hedging strategies in exponential Lévy models

We consider the performance of non-optimal hedging strategies in exponential Lévy models. Given that both the payoff of the contingent claim and the hedging strategy admit suitable integral representations, we use the Laplace transform approach of Hubalek et al. [Ann. Appl. Probab., 2006, 16(2), 853–885] to derive semi-explicit formulas for the resulting mean-squared hedging error in terms of the cumulant generating function of the underlying Lévy process. In two numerical examples, we apply these results to compare the efficiency of the Black–Scholes hedge and the model delta with the mean–variance optimal hedge in a normal inverse Gaussian and a diffusion-extended CGMY Lévy model.

HEDGING MORTALITY CLAIMS WITH LONGEVITY BONDS

AbstractWe study mean–variance hedging of a pure endowment, a term insurance and general annuities by trading in a longevity bond with continuous rate payments proportional to the survival probability. In particular, we discuss the introduction of a gratification annuity as an interesting insurance product for the life insurance market. The optimal hedging strategies are determined via their Galtchouk–Kunita–Watanabe decompositions under specific, yet sufficiently general model assumptions. The results are then further illustrated by assuming a general affine structure of the mortality intensity process. The optimal hedging strategies as well as the residual hedging error of a gratification annuity and a simple life annuity are finally investigated with numerical simulations, which illustrate the nice features of the gratification annuity for the insurance industry.

Pricing and Hedging Asian-Style Options in Energy

We solve the problem of pricing and hedging Asian-style options on energy with a quadratic risk criterion when trading in the underlying future is restricted. Liquid trading in the future is only possible up to the start of a so-called delivery period. After the start of the delivery period, the hedge positions can not be adjusted anymore until maturity. This reflects the trading situation at the Nordic energy market Nord Pool for example. We show that there exists a unique solution to this combined continuous-discrete quadratic hedging problem if the future price process is a special semimartingale with bounded mean-variance tradeoff. Additionally, under the assumption that the future price process is a local martingale, the hedge positions before the averaging period are inherited from the market specification without trading restriction. As an application we consider three models and derive their quadratic hedge positions in explicit form, a simple Black Scholes model with time-dependent volatility, the stochastic volatility model of Barndorff-Nielsen and Shephard and an exponential additive model. Based on an exponential spot price model driven by two NIG \levy processes, we determine an exponential additive model for the future price by moment matching techniques. We calculate hedge positions and determine the quadratic hedge error in a simulation study.

Minimal variance hedging of natural gas derivatives in exponential Lévy models: Theory and empirical performance

We consider the problem of hedging European options written on natural gas futures, in a market where prices of traded assets exhibit jumps, by trading in the underlying asset. We provide a general expression for the hedging strategy which minimizes the variance of the terminal hedging error, in terms of stochastic integral representations of the payoffs of the options involved. This formula is then applied to compute hedge ratios for common options in various models with jumps, leading to easily computable expressions. As a benchmark we take the standard Black–Scholes and Merton delta hedges. We show that in natural gas option markets minimal variance hedging with underlying consistently outperform the benchmarks by quite a margin.

Equity-linked products: evaluation of the dynamic hedging errors under stochastic mortality

In this paper, we analyze the effect of unpredicted changes in mortality rates on the risk incurred by the sale of equity-indexed annuities (EIAs). Jumps in mortality indices due to catastrophes (for example, the 1918 flu pandemic) may occur in the future, causing important financial losses to insurers selling products offering death benefits. Thus, we analyze the distribution of hedging errors extracted from the dynamic hedging strategy underlying the fair valuation. To model mortality jumps stochastically, we use a regime-switching model introduced by Milidonis et al. (North Am Actuarial J 15(2):266–289, 2011). We then employ Esscher transforms to obtain closed-form expressions for the price of a term-end EIA at any time between the inception of the contract and its maturity date. The hedging strategy is derived from that valuation, and hedging errors are extracted from this strategy since it is not self-financing. A detailed numerical analysis is performed for a term-end EIA.

Real World Pricing of Long Term Contracts

Long dated contingent claims are relevant in insurance, pension fund management and derivative pricing. This paper proposes a paradigm shift in the valuation of long term contracts, away from classical no-arbitrage pricing towards pricing under the real world probability measure. In contrast to risk neutral pricing, the long term excess return of the equity market, known as the equity premium, is taken into account. Further, instead of the savings account, the numeraire portfolio is used, as the fundamental unit of value in the analysis. The numeraire portfolio is the strictly positive, tradable portfolio that when used as benchmark makes all benchmarked nonnegative portfolios supermartingales, which means intuitively that these are downward trending or at least trendless. Furthermore, the benchmarked real world price of a benchmarked claim is deflned to be its real world conditional expectation. This yields the minimal possible price for its hedgable part and minimizes the variance of the benchmarked hedge error. The pooled total benchmarked replication error of a large insurance company or bank essentially vanishes due to diversiflcation. Interestingly, in long term liability and asset valuation, real world pricing can lead to signiflcantly lower prices than suggested by classical no-arbitrage arguments. Moreover, since the existence of some equivalent risk neutral probability measure is no longer required, a wider and more realistic modeling framework is available for exploration. Classical actuarial and risk neutral pricing emerge as special cases of real world pricing.

Estimating Residual Hedging Risk with Least-Squares Monte Carlo

Frequently, dynamic hedging strategies minimizing risk exposure are not given in closed form, but need to be approximated numerically. This makes it difficult to estimate residual hedging risk, also called basis risk, when only imperfect hedging instruments are at hand. We propose an easy to implement and computationally efficient least-squares Monte Carlo algorithm to estimate residual hedging risk. The algorithm approximates the variance minimal hedging strategy within general diffusion models. Moreover, the algorithm produces both high-biased and low-biased estimators for the residual hedging error variance, thus providing an intrinsic criterion for the quality of the approximation. In a number of examples we show that the algorithm delivers accurate hedging error characteristics within seconds.

Futures Cross-Hedging with a Stationary Basis

AbstractWhen managing risk, frequently only imperfect hedging instruments are at hand. We show how to optimally cross-hedge risk when the spread between the hedging instrument and the risk is stationary. For linear risk positions we derive explicit formulas for the hedge error, and for nonlinear positions we show how to obtain numerically efficient estimates. Finally, we demonstrate that even in cases with no clear-cut decision concerning the stationarity of the spread, it is better to allow for mean reversion of the spread rather than to neglect it.

Minimal Variance Hedging of Natural Gas Derivatives in Exponential Levy Models: Theory and Empirical Performance

We consider the problem of hedging European options written on natural gas futures, in a market where prices of traded assets exhibit jumps, by trading in the underlying asset. We provide a general expression for the hedging strategy which minimizes the variance of the terminal hedging error, in terms of stochastic integral representations of the payoffs of the options involved. This formula is then applied to compute hedge ratios for common options in various models with jumps, leading to easily computable expressions. As a benchmark we take the standard Black-Scholes-Merton delta hedges. We show that in natural gas option markets minimal variance hedging with underlying consistently outperform the benchmarks by quite a margin.

NUMERICAL HEDGING OF ELECTRICITY CONTRACTS USING DIMENSION REDUCTION

The basic contracts traded on energy exchanges involve fixed-rate payments for the delivery of electricity over a certain period of time. It has been shown that options on these electricity swaps can be priced efficiently using a Hilbert space-valued time-inhomogeneous jump-diffusion model for the forward curve. We consider the mean-variance hedging problem for European options under this model. The computation of hedging strategies leads to quadratic optimization problems whose parameters depend on the solution of an infinite-dimensional partial integro-differential equation. The main objective of this article is to find an efficient numerical algorithm for this task. Using proper orthogonal decomposition (a dimension reduction method), approximately optimal strategies are computed. We prove convergence of the corresponding hedging error to the minimal achievable error in the electricity market. Numerical experiments are performed to analyze the resulting hedging strategies.

Psychological Barriers and Option Pricing

Psychological barriers are prevalent among various asset classes, and it is important to consider their impact on the prices of derivative securities. This paper demonstrates the potential existence of such barriers on the SP this paper also evaluates this model's empirical performance relative to the Black-Scholes and constant elasticity of variance (CEV) models. The in-sample calibration result of the threshold model is found to be superior. Furthermore, it is found that the model provides an efficient hedging method in terms of dollar-value hedging errors.

Optimal hedging of demographic risk in life insurance

A Markov chain model is taken to describe the development of a multi-state life insurance policy or portfolio in a stochastic economic–demographic environment. It is assumed that there exists an arbitrage-free market with tradeable securities derived from demographic indices. Adopting a mean-variance criterion, two problems are formulated and solved. First, how can an insurer optimally hedge environmental risk by trading in a given set of derivatives? Second, assuming that insurers perform optimal hedging strategies in a given derivatives market, how can the very derivatives be designed in order to minimize the average hedging error across a given population of insurers? The paper comes with the caveat emptor that the theory will find its prime applications, not in securitization of longevity risk, but rather in securitization of catastrophic mortality risk.

Advances in cointegration and subset correlation hedging methods

Hedging methods are divided into single-period and multiperiod forms. After reviewing some well-known hedging algorithms, two new procedures called the Dickey-Fuller optimal (DFO) method and the minimax subset correlation (MMSC) method are introduced. The former is a multiperiod, cointegration- based hedging method that estimates the holdings that are most likely to deliver a hedging error with no unit root. The latter is a single-period method that studies the geometry of the hedging errors and estimates a hedging vector such that subsets of its components are as orthogonal as possible to the error. We test each method for stability and robustness of the derived hedged portfolio. Results indicate that the DFO method produces estimates that are similar to those for the error correction method, but more stable. Likewise, MMSC estimates are similar to principal component analysis, but more stable. A generalized Box-Tiao canonical decomposition (BTCD) method, which is of the multiperiod class, is proposed. The BTCD estimates are also very stable, and cannot be related to any of the aforementioned methodologies. Finally, it is found that all three advanced hedging methods (MMSC, BTCD and DFO) perform well.

Advances in Cointegration and Subset Correlation Hedging Methods

We divide hedging methods between single-period and multi-period. After reviewing some well-known hedging algorithms, two new procedures are introduced, called Dickey-Fuller Optimal (DFO), Mini-Max Subset Correlation (MMSC). The former is a multi-period, cointegration-based hedging method that estimates the holdings that are most likely to deliver a hedging error absent of unit root. The latter is a single-period method that studies the geometry of the hedging errors and estimates a hedging vector such that subsets of its components are as orthogonal as possible to the error. We test each method for stability and robustness of the derived hedged portfolio. Results indicate that DFO produces estimates similar to the Error Correction Method, but more stable. Likewise, MMSC estimates are similar to Principal Component Analysis but more stable. Finally, a generalized Box-Tiao Canonical Decomposition (BTCD) method is proposed, which is of the multi-period class. BTCD estimates are also very stable, and cannot be related to any of the aforementioned methodologies. Finally, we find that all three advanced hedging methods (MMSC, BTCD, DFO) perform well.

Convergence and optimality of BS-type discrete hedging strategy under stochastic interest rate

We focus on the asymptotic convergence behavior of the hedging errors of European stock option due to discrete hedging under stochastic interest rates. There are two kinds of BS-type discrete hedging differ in hedging instruments: one is the portfolio of underlying stock, zero coupon bond, and the money market account (Strategy BSI); the other is the underlying stock, zero coupon bond (Strategy BSII). Similar to the results of the deterministic interest rate case, we show that convergence speed of the discounted hedging errors is 1/2-order of trading frequency for both strategies. Then, we prove each of the BS-type strategy is not only locally optimal, but also globally optimal under the corresponding measure. Finally, we give some numerical examples to illustrate the results. All the discussion is based on non-arbitrage condition and zero transaction cost.

Hedging under model misspecification: All risk factors are equal, but some are more equal than others …

AbstractIt is often difficult to distinguish among different option pricing models that consider stochastic volatility and/or jumps based on a cross‐section of European option prices. This can result in model misspecification. We analyze the hedging error induced by model misspecification and show that it can be economically significant in the cases of a delta hedge, a minimum‐variance hedge, and a delta‐vega hedge. Furthermore, we explain the surprisingly good performance of a simple ad‐hoc Black‐Scholes hedge. We compare realized hedging errors (an incorrect hedge model is applied) and anticipated hedging errors (the hedge model is the true one) and find that there are substantial differences between the two distributions, particularly depending on whether stochastic volatility is included in the hedge model. Therefore, hedging errors can be useful for identifying model misspecification. Furthermore, model risk has severe implications for risk measurement and can lead to a significant misestimation, specifically underestimation, of the risk to which a hedged position is exposed. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark

On the Convergence of Higher Order Hedging Schemes: The Delta-Gamma Case

Hedging errors induced by discrete rebalancing of the hedge portfolio of a delta-gamma hedging strategy are investigated. The rate of convergence of the expected squared hedging error as the number of adjustments of the hedge portfolio goes to infinity is analyzed. It is found that the delta-gamma strategy produces higher convergence rates than the usual delta strategy.

On the Effect of Skewness and Kurtosis Misspecification on the Hedging Error

Using a result in Angelini and Herzel (2009a), we measure, in terms of variance, the cost of hedging a contingent claim when the hedging portfolio is re-balanced at a discrete set of dates. We analyse the dependence of the variance of the hedging error on the skewness and kurtosis as modeled by a Normal Inverse Gaussian model. We consider two types of strategies, the standard Black–Scholes Delta strategy and the locally variance-optimal strategy, and we perform some robustness tests. In particular, we investigate the effect of different types of model misspecification on the performance of the hedging, like that of hedging without taking skewness into account. Computations are performed using a Fast Fourier Transform approach.