Perfect Hedging Research Articles

Quantile hedging for guaranteed minimum death benefits

Quantile hedging for contingent claims is an active topic of research in mathematical finance. It plays a role in incomplete markets when perfect hedging is not possible. Guaranteed minimum death benefits (GMDBs) are present in many variable annuity contracts, and act as a form of portfolio insurance. They cannot be perfectly hedged due to the mortality component, except in the limit as the number of contracts becomes infinitely large. In this article, we apply ideas from finance to derive quantile hedges for these products under various assumptions.

On Break-Even Correlation: The Way to Price Structured Credit Derivatives by Replication

We consider the pricing of European structured products under a 'static' framework, particularly the Gaussian copula model (GCM). Being hedged continuously against individual spread moves with single name Credit Default Swaps, we calculate the associated replication errors. Therefore, we introduce the concept of 'break-even' correlation (and more generally 'break-even' correlation matrix), that allow a perfect hedging under the GCM if and only if the underlying single names follow a particular family of dynamics, and when no jump-to-default events occur. We exhibit a class of Merton-type models that are consistent with this result. We explain why the GCM has not a lot of competitors among one-period static models, except the Clayton copula. Finally, we illustrate the theory by the empirical features of break-even correlations on true quotes (standard credit indices) and on simulated spread trajectories.

The 21st Century Solutions to Hedging Low-Basis Stock

Confusion among practitioners on just what can and can’t be done to hedge and monetize a low-cost-basis stock without triggering a taxable sale is widespread. Investors can still hedge and monetize without concern for triggering a sale, although the task is more difficult than it needs to be. Hedging was fairly straightforward until 1997 when the constructive sale rules (IRC Section 1259) were enacted. The constructive sale rules force owners of low-basis shares to retain some upside or downside when hedging. Before these rules a perfect hedge was permissible, before 1997 we recommended a short-against-the-box, others advocated equity swaps, and others, deep-in-the money calls (all would now trigger a gain under IRC Section 1259). Although it seems daunting, hedging and monetization can still be safely executed without negative tax concerns if investors are careful to avoid the traps that are possible in such an endeavor. <b>TOPICS:</b>Security analysis and valuation, derivatives, legal/regulatory/public policy

Dynamic Hedging in Incomplete Markets: A Simple Solution

Despite much work on hedging in incomplete markets, the literature still lacks tractable dynamic hedges in plausible environments. In this article, we provide a simple solution to this problem in a general incomplete-market economy in which a hedger, guided by the traditional minimum-variance criterion, aims at reducing the risk of a non-tradable asset or a contingent claim. We derive fully analytical optimal hedges and demonstrate that they can easily be computed in various stochastic environments. Our dynamic hedges preserve the simple structure of complete-market perfect hedges and are in terms of generalized Greeks, familiar in risk management applications, as well as retaining the intuitive features of their static counterparts. We obtain our time-consistent hedges by dynamic programming, while the extant literature characterizes either static or myopic hedges, or dynamic ones that minimize the variance criterion at an initial date and from which the hedger may deviate unless she can pre-commit to follow them. We apply our results to the discrete hedging problem of derivatives when trading occurs infrequently. We determine the corresponding optimal hedge and replicating portfolio value, and show that they have structure similar to their complete-market counterparts and reduce to generalized Black-Scholes expressions when specialized to the Black-Scholes setting. We also generalize our results to richer settings to study dynamic hedging with Poisson jumps, stochastic correlation and portfolio management with benchmarking.

2008 M&amp;SOM Best Paper Award

It is a pleasure to announce that the 2008 Manufacturing & Service Operations Management Best Paper is “Hedging Inventory Risk Through Market Instruments,” by Vishal Gaur and Sridhar Seshadri (2005). This annual award is given to a paper, published in one of the prior three volumes of M&SOM, deemed by the M&SOM editorial board to be most deserving for its contribution to the theory and practice of operations management. For their accomplishment, Professors Gaur and Seshadri will share $2,000, which was contributed by the Manufacturing and Service Operations Management Society of INFORMS. This paper asks an intriguing question—can a firm use financial hedging strategies to reduce the risk it faces from its inventory investment? The operations management literature is full of papers that offer advice on making optimal inventory decisions, but that literature had not considered how inventory decisions should change when there exists the possibility of creating a financial hedge. Of course, for a hedging strategy to be effective a firm needs to be able to trade a financial asset that is sufficiently correlated with the firm’s inventory risk. Instead of merely assuming the existence of such an asset, Gaur and Seshadri (2005) use actual data to nicely demonstrate that such a financial instrument may indeed exist for some firms: a regression of same-store sales growth among 60 large U.S. general merchandisers with annual returns to the S&P 500 index reveals an R2 = 0 81. This is indeed an eye-opening motivation and a very nice example of data-driven research. With the plausibility of a financial hedge established, the authors begin with a simple single-period inventory problem and an ideal underlying financial asset that is perfectly correlated with demand. A perfect hedge is constructed so that the firm earns the expected value of its inventory investment without any variance in returns, upfront investment, or a need to dynamically revise the hedge over time. Furthermore, the optimal inventory policy remains unaffected by the presence of the hedge. Hence, there is no need to coordinate the firm’s financial strategy with its inventory management. However, the authors next analyze minimum variance hedges under a more realistic situation: an underlying asset that is only partially correlated with demand. In this setting they provide conditions under which a riskaverse decision maker invests in inventory, rather than just the financial asset. Now, the decision maker purchases more inventory with hedging than without, demonstrating the link between the financial and operational strategies. To complete the loop, the authors finish with an application of their results to a numerical example based on sales data for computer game CDs sold at a consumer electronics retailing chain. The strength of this paper is not that it ends a line of research but rather that it starts a vein of research with significant promise. Others will develop more sophisticated hedges, consider more elaborate inventory models and apply these ideas to other operational issues (sourcing strategies, investments in production flexibility, etc.), but the basic idea to combine financial hedging with operational decisions has been established. As such, the paper need not make a contribution by providing new analytical methods. Instead, its contribution comes from combining two sets of well-established tools, just like an artist who brings two familiar media together to create something entirely new and exciting. Finally, the paper’s blend of analytics and data is undeniably powerful: not only will these results influence academics, these

On the Pricing of Options Written on the Last Exit Time

We establish formulae of the pricing of options which are written on the last exit time with finite maturity. Our main theorem gives a description of a perfect hedging strategy for the Black-Scholes model.

Hedging Strategies and Minimal Variance Portfolios for European and Exotic Options in a Levy Market

This paper presents hedging strategies for European and exotic options in a Levy market. By applying Taylor’s theorem, dynamic hedging portfolios are constructed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk-free bank account, the underlying asset, and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results.

Analysis of Participating Life Insurance Contracts: A Unification Approach

AbstractFair pricing of embedded options in life insurance contracts is usually conducted by using risk‐neutral valuation. This pricing framework assumes a perfect hedging strategy, which insurance companies can hardly pursue in practice. In this article, we extend the risk‐neutral valuation concept with a risk measurement approach. We accomplish this by first calibrating contract parameters that lead to the same market value using risk‐neutral valuation. We then measure the resulting risk assuming that insurers do not follow perfect hedging strategies. As the relevant risk measure, we use lower partial moments, comparing shortfall probability, expected shortfall, and downside variance. We show that even when contracts have the same market value, the insurance company's risk can vary widely, a finding that allows us to identify key risk drivers for participating life insurance contracts.

Hedging with risk for game options in discrete time

We study the problems of efficient hedging of game (Israeli) options when the initial capital in the portfolio is less than the fair option price. In this case a perfect hedging is impossible and one can only try to minimise the risk (which can be defined in different ways) of having not enough funds in the portfolio to pay the required amount at the excercise time. We solve the minimization problems and find via dynamical programming appropriate efficient hedging strategies for discrete time game options in multinomial markets. The approach and some of the results are new also for standard American options.

Reply to “A comment on 'A hedging deficiency in eurodollar futures'”

AbstractKawaller's argument that a perfect hedge is possible with eurodollar futures is limited to the notion of a perfect accounting hedge, whereby the time value of money is ignored. In addition, his attempt to show the importance of sizing the hedge merely introduces a change to the problem, leading to a difference in our results that he erroneously believes is a mistake on my part. It is easy to show that his example is driven by an irrelevant distraction. I correct his example and show that had he worked the same problem I worked, he would have worked it the same way I did. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:195–201, 2007

A comment on “A hedging deficiency in eurodollar futures”

AbstractProfessor Chance's analysis shows that hedge results from eurodollar futures are imperfect; and he credits the futures contract design as being the source of the error. This comment argues that the unanticipated outcomes that Professor Chance evidences stem not from the design of the contract, but rather from improperly sizing hedge transactions. If appropriately sized hedges are used, perfect hedge outcomes in fact, will follow. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:187–193, 2007

Table of Contents

Abstract Features LNG Supply Surge to Affect Prices in North America? How? Benjamin Schlesinger Renewables Are REC Markets a WRECk Waiting to Happen? Jurgen Weiss Alternate Energy Is Gasification of Solid Waste the Perfect Hedge? William H. Davis Producers Natural Gas Storage—Another Boom to Squander? Kenneth L. Beckman and Peggy L. Determeyer Columns Pipelines &amp; Rates Historical Pipeline Allowed Rates of Return Have Supported Infrastructure Health Richard G. Smead Federal Oversight and Energy Issues Electric Transmission Problems Affect US Competitiveness James J. Hoecker Workforce Issues Knowledge‐Retention Programs Can Keep Critical Skills from Vanishing John Juliano

Risk measures for derivatives with Markov-modulated pure jump processes

We consider a regime-switching HJB approach to evaluate risk measures for derivative securities when the price process of the underlying risky asset is governed by the exponential of a pure jump process with drift and a Markov switching compensator. The pure jump process is flexible enough to incorporate both the infinite, (small), jump activity and the finite, (large), jump activity. The drift and the compensator of the pure jump process switch over time according to the state of a continuous-time hidden Markov chain representing the state of an economy. The market described by our model is incomplete. Hence, there is more than one pricing kernel and there is no perfect hedging strategy for a derivative security. We derive the regime-switching HJB equations for coherent risk measures for the unhedged position of derivative securities, including standard European options and barrier options. For measuring risk inherent in the unhedged option position, we first need to mark the position into the market by valuing the option. We employ a well-known tool in actuarial science, namely, the Esscher transform to select a pricing kernel for valuation of an option and to generate a family of real-world probabilities for risk measurement. We also derive the regime-switching HJB-variational inequalities for coherent risk measures for American-style options.

Common Structural Time Series Components in Inflation and Residential Property Prices

Executive Summary. This paper uses the multivariate common components model to investigate the inflation-hedging characteristics of residential property in five markets: Hong Kong, Tokyo, Singapore, Taipei and London. The results show that Singapore residential property offers perfect hedges against both short-term and permanent inflations, whereas Taipei residential property is the most effective in hedging long-term inflation. For investors who invest in residential properties in Asia and the United Kingdom, with the expectation that their investment portfolio will provide natural hedges against inflation, the results imply that they will have to adopt different timing strategies to minimize their exposure to different inflation risks in individual markets.

Risk measure pricing and hedging in incomplete markets

This article attempts to extend the complete market option pricing theory to incomplete markets. Instead of eliminating the risk by a perfect hedging portfolio, partial hedging will be adopted and some residual risk at expiration will be tolerated. The risk measure (or risk indifference) prices charged for buying or selling an option are associated to the capital required for dynamic hedging so that the risk exposure will not increase. The associated optimal hedging portfolio is decided by minimizing a convex measure of risk. I will give the definition of risk-efficient options and confirm that options evaluated by risk measure pricing rules are indeed risk-efficient. Relationships to utility indifference pricing and pricing by valuation and stress measures will be discussed. Examples using the shortfall risk measure and average VaR will be shown.

Valuation and Hedging of Limited Price Indexed Liabilities

ABSTRACTThis paper considers the market or economic valuation and the hedging of Limited Price Indexed (LPI) liabilities. This involves finding optimal static and dynamic hedging strategies which minimise the riskiness of the investment portfolio relative to the liability.In this paper we do not aim to find the perfect hedge in a perfect world. Instead, it is assumed that optimisation is restricted to three commonly used asset classes in pension funds: cash; long-term (or irredeemable) fixed-interest bonds; and long-dated index-linked bonds. The economic value of the liability is then defined as the value of the best matching portfolio using a mean/variance type of loss function. Specifically, we adopt the risk minimising approach of Föllmer &amp; Sondermann (1986) and Schweizer &amp; Föllmer (1988). Even with such a simple loss function, establishing the theoretically optimal solution can be difficult. We propose that a practical solution close to the theoretical optimum can be found using two approximations. First, we approximate the ‘true’ stochastic economic model by a vector autoregressive model of order one. Second, we use a sequence of linearisations to approximate non-linear by straightforward quadratic minimisation problems.The proposed approach is illustrated with various numerical examples, and we compare the results of the approximately optimal hedging strategy with static strategies.

Acquiring foreign equity assets without currency risk

This paper introduces a practical method of acquiring foreign equity assets without currency risk. We design a self-financing portfolio, such as the matured value of the portfolio is our goal. Currency risk is reflected in an adjustment term when the price process of foreign equity becomes a martingale after discounting under the domestic martingale measure. The perfect hedging of currency risk is obtained by taking into consideration the forward prices of foreign equity and exchange rate simultaneously.

Modern Financial Engineering and Discounts: The Collar Message

An abundance of literature addresses the appropriate marketability discount to be applied to a block of publicly traded securities that cannot be immediately liquidated. Typically, the discounts are based upon inferences drawn from measuring discounts indirectly. Here, we take a different approach. We ask the specific question: What is the direct cost of providing full marketability to a less than liquid financial asset? Our findings indicate that marketability discounts for restricted stock and stock option grants, as well as large blocks of publicly traded securities that can be converted to publicly traded securities at some future date, can be determined with reasonable accuracy. This direct approach is superior to that of relying on observed discounts of restricted stock prices in private placements relative to their publicly traded counterparts. Liquidity or marketability is comprised of two components. The elimination of price risk is the first component. Price risk represents the risk that one may not receive the current market value of the underlying freely traded asset due to price fluctuations in the marketplace. The second component of marketability is the ability to convert an asset to cash in a timely fashion. Restricted stock units received as part of a compensation package may not convert to a freely traded stock for periods ranging from a few months to several years. The cost of restoring liquidity to an illiquid asset represents the appropriate marketability discount. We show that an at-the-money equity collar, engineered by buying and selling options on the underlying publicly traded stock creates a perfect hedge that removes all price risk at a point in time. In addition, borrowing against this hedged position will provide funds and restore full liquidity. In the absence of transaction costs, the return on the collar portfolio will be equal to the risk free rate. In practice this return is reduced by transactions costs and interest costs that are incurred. Therefore, the net transaction cost and net interest cost in creating the equity collar represents an appropriate marketability discount. This direct cost approach is the best tool available for calculating the size of the marketability discount. We demonstrate the application of this strategy to blockage discounts and discounts applied to restricted stock units. This approach cannot be directly applied to situations where an equity collar cannot be engineered. However, even where a perfect hedge cannot be directly created we can develop a sense of the appropriate marketability discount that should be applied in a given situation. We begin by looking at previous work dealing with the measurement of liquidity discounts for blockage and restricted stock. Next we apply the collar strategy to determine the appropriate discount to both restricted stock and stock subject to blockage discounts. Blockage Discount Literature Moore (1992) presents a checklist of 22 factors to consider when determining the appropriate blockage discount. The weighting of the factors is based upon the judgment of the valuation professional. Several of the factors he suggests are useful for determining whether a blockage discount is warranted. For example, The share size of the block – especially as compared to the number of shares outstanding of that

Moderne Finanzinstrumente im Rahmen des Katastrophen-Risk-Managements — Basisrisiko versus Ausfallrisiko

A major problem for insuring catastrophic risk is that, as a disaster causes damages to many insureds at the same time, such insurance and in particular reinsurance contracts are often subject to considerable default risk. On the other hand, the securitization of insurance risk, for example via a catastrophe bond, can be designed to completely avoid default risk. Typically, however, the payout from an insurance-linked security is tied to some stochastic variable, an index, which is correlated, but not identical, with the insured’s actual losses. Therefore, such an instrument will usually not provide a perfect hedge. There will be some mismatch, the so-called basis risk. This paper investigates how the trade off between default respectively credit risk and basis risk affects optimal risk management solutions, when (re)insurance and index-linked risk securitization are used simultaneously. In particular, the impact of credit risk and risk securitization on the optimal reinsurance contract is analyzed.

Option Pricing and Hedging on Correlated Stocks

We develop a theory for option pricing with perfect hedging in an inefficient market model where the underlying price variations are autocorrelated over a time. This is accomplished by assuming that the underlying noise in the system is derived by an Ornstein-Uhlenbeck, rather than from a Wiener process. After obtaining an effective one-dimensional market model, we achieve a closed expression for the European call price within the Black-Scholes framework and find that our price is always lower than the Black-Scholes price. We obtain the same price if we start from a modified portfolio although now we get a different hedging strategy than that of Black-Scholes. We compare these strategies and study the sensitivity of the call price to several parameters where the correlation effects are also observed.