Perfect Hedging Research Articles

Maximizing the probability of a perfect hedge

In the framework of continuous-time, Itô processes models for financial markets, we study the problem of maximizing the probability of an agent's wealth at time T being no less than the value C of a contingent claim with expiration time T. The solution to the problem has been known in the context of complete markets and recently also for incomplete markets; we rederive the complete markets solution using a powerful and simple duality method, developed in utility maximization literature. We then show how to modify this approach to solve the problem in a market with partial information, the one in which we have only a prior distribution on the vector of return rates of the risky assets. Finally, the same problem is solved in markets in which the wealth process of the agent has a nonlinear drift. These include the case of different borrowing and lending rates, as well as "large investor" models. We also provide a number of explicitly solved examples.

Was Going “Short-Against-The-Box” Really a Perfect Hedge?

The author first introduces the strategy called “short-against-the-box,” as one which was frequently used, prior to the implementation of the Taxpayer Relief Act of 1997, by investors who held concentrated equity positions, as a means of hedging and monetizing that position in a tax-advantage manner. Though these transactions were often marketed and described as “perfect hedges” with minor, if any, cost to the investor, the author questions that view. He points to the fact that investor9s heirs lose their ability to receive a step-up in the basis upon transference of the shares at the death of the investor and that the investor faced an increasing cost to maintain the position if the stock that was hedged appreciated in price. He then shows why an investor who entered into a short-against-the-box prior to June 1997 on a stock that has continued to appreciate may well be better off unwinding it re-hedging using a different hedging strategy – a variable pre-paid forward. He points to three potential benefits: regain the step-up the basis upon transference to heirs; sell a certain number of shares at the current market price tax-free; and regain the ability to participate in upward price movement on some of the underlying stock.

Exports and Indirect Hedging of Foreign Currency Risk

We present a model of a risk-averse competitive exporting firm under exchange rate risk. Direct hedging instruments are not available. However, there are domestic assets whose prices are correlated to the foreign currency. We consider a market for futures contracts in these domestic assets and investigate the firm's indirect hedging and export policy. It is shown that the availability of many financial instruments correlated with foreign exchange may, under some circumstances, provide the same results as a perfect hedge. JEL Classification Numbers: F21, F31.

Spreading currency forwards: why and how?

In this paper we have adopted a continuous time framework in which a hedger has a currency risk-sensitive non-traded cash position. The hedger has the opportunity to hedge against risk by using two currency forward contracts that differ only in their maturities. The hedger is able to reach a perfect hedge of his non-traded position by using only two currency forward contracts with different maturities even though four sources of uncertainty drive the international economy. This perfect hedge is reached by spreading forward contracts, where the hedger is short in the nearby contract and long in the deferred contract. This strategy is easy to implement since no parameter has to be estimated empirically when the hedger is a price taker. Finally, the analysis has been extended to currency futures contracts. We show in explicit terms how the marking-to-market of futures positions affects the spreading strategy.

Methods of Partial Hedging

In this article we survey methods of dealing with the following problem: A financial agent is trying to hedge a claim C, without having enough initial capital to perform a perfect (super) replication. In particular, we describe results for minimizing the expected loss of hedging the claim C both in complete and incomplete continuous-time financial market models, and for maximizing the probability of perfect hedge in complete markets and markets with partial information. In these cases, the optimal strategy is in the form of a binary option on C, depending on the Radon-Nikodym derivative of the equivalent martingale measure which is optimal for a corresponding dual problem. We also present results on dynamic measures for the risk associated with the liability C, defined as the supremum over different scenarios of the minimal expected loss of hedging C.

On Minimizing Risk in Incomplete Markets Option Pricing Models

I study the Bouchaud–Sornette, Schweizer and Schäl way of pricing options, presenting the methodology in accordance with Bouchaud–Sornette. The definitions of the wealth balance and risk from trading in options and stocks are presented. The problem of finding a risk minimizing strategy in an incomplete market model where a perfect hedge is not possible is analyzed. Using this strategy according to the approach of Bouchaud and Sornette the option is priced by a fair game condition. In this article I establish the equivalence between global and local risk minimization and prove an option price conjecture of Wolczyńska. I also investigate optimality for a stock portfolio with extra profit.

Derivative Asset Pricing with Transaction Costs: An Extension

This paper examines the optimal perfect hedging (super-replication) of an option by a cash-plus-riskless asset portfolio within the context of the binomial model. The cases discussed here were not covered by the earlier studies of Boyle and Vorst (1992) and Bensaid, Lesne, Pages and Scheinkman (1992). It is argued that these cases are empirically important, and that there is some indication that they are encountered very often in practice in the Swiss options market. A new algorithm is developed to compute the option price lower bound (bid price) for such cases. It is then shown that, for most such cases, the portfolio hedging the short call when replication is not optimal coincides with the Merton (1973) lower bound.

Optimal price policy and the futures markets

Because of the short-term nature of existing commodity futures contracts, optimally hedged producers remain subject to intertemporal income uncertainty, but price stabilization may be detrimental because it negates the benefits of intertemporal production flexibility. Multiperiod futures would be preferred to price stabilization, although they would not provide perfect hedging opportunities, thus leaving scope for government intervention. The optimal price policy requires the support price to be positively correlated to the futures price that prevails when production decisions are made.

Cross-Hedging of Exchange-Rate Risk

For currencies with highly developed forward markets a well-known separation theorem holds which implies that international firms fully hedge the exchange rate risk if the forward markets are unbiased. In this paper we present a model of a risk-averse firm when perfect hedging instruments are not available. Instead the firm can cross-hedge the exchange-rate risk by using the forward markets of a third country's currency. We demonstrate that the unbiasedness of all forward markets does not imply full hedge, although the firm has the option to hedge all the risks.

Imperfect Hedging and Export Production

Firms engaged in international operations are highly interested in developing ways to protect themselves from exchange rate risk. The incentive for risk management comes from the enormous volatility of the floating foreign exchange rates.' Our study shows that an exporting firm can benefit from hedging exchange rate risks even when no perfect hedge is possible. Since in reality, not every currency is traded in a futures market [7, Chap. 15], the exporting firm uses futures contracts with other underlying assets whose spot prices are highly correlated with the foreign exchange spot rate. In the real world hedging must often be accomplished by using futures contracts on different deliverable instruments. Such hedging may result in imperfect hedging as shown by Anderson and Danthine [1], Eaker and Grant [11], Dellas and Zilberfarb [9], Broll, Wahl and Zilcha [6]. It has been shown in recent publications [12; 8; 15; 13; 23; 16; 2; 14; 5; 18; 22] that an international firm facing exchange rate risk can eliminate this risk altogether if it can use a currency forward market, another financial asset or a portfolio of assets which is perfectly correlated to the exchange rate. In the absence of such markets, the firm can reduce its income risk by engaging in a hedging activity of assets correlated to the foreign exchange. Recent studies of firm behavior under exchange rate uncertainty examine the influence of futures markets on the export and hedging decision. These papers derive two major theorems: One is the separation which states that, when futures markets exist, the firm's export production decision is determined solely by technology and input-output prices, including the futures prices. This result holds if the gain from the futures contract is perfectly correlated with export revenue. The other theorem is the full hedging which asserts that with unbiased futures markets, the firm completely avoids exchange rate risk by entering into optimum futures contracts.

Mean-Variance Hedging of Options on Stocks with Markov Volatilities

We consider the problem of hedging an European call option for a diffusion model where drift and volatility are functions of a Markov jump process. The market is thus incomplete implying that perfect hedging is not possible. To derive a hedging strategy, we follow the approach based on the idea of hedging under a mean-variance criterion as suggested by Follmer, Sondermann, and Schweizer. This also leads to a generalization of the Black–Scholes formula for the corresponding option price which, for the simplest case when the jump process has only two states, is given by an explicit expression involving the distribution of the integrated telegraph signal (known also as the Kac process). In the Appendix we derive this distribution by simple considerations based on properties of the order statistics.

A note on experience rating, reinsurance and premium principles

The connection between experience rating, reinsurance covers and premium calculation principles is analyzed. Under natural and order-preserving properties for the premium calculation principle and a ‘perfect hedge’ condition for the direct insurer, an adequate premium principle is derived. It turns out to be an extension of the Dutch premium principle. Actuarial applications to the design of company internal reinsurance covers and to the construction of appropriate bonus strategies are presented.

Optimal hedging with currency forwards, calls, and calls on forwards for the competitive exporting firm facing exchange rate uncertainty

We extend the well-known full hedge theorems of the hedging literature to random profits that are nonlinear in the random exchange rate. This arises when production flexibility is added to the standard model of the risk-averse exporting firm, where all production decisions have to be made before the exchange rate is known. Hence, hedging with currency derivatives that provide a linear payoff in the exchange rate can no longer provide a perfect hedge. Therefore forward selling is replaced by writing a certain call portfolio. Adding delayed revenue to the model induces the firm to sell calls on forwards. Because our generalized full hedge proposition is proved for random profits that might as well decrease in the exchange rate, the result is applicable to certain types of importing firms, too. — Given the absence of speculative motives on the part of the firm, it turns out that long-term investments in capital goods are chosen in risk-neutral manner.

Size Effects in the Market Valuation of Fundamental Factors

relative contributions to risk from foreign exchange and total interest rate volatilities and their correlation are examined in Dym, 'Measuring the Risk of Foreign Bonds, op. cit. Here we concentrate on attributing risk to the global and local decomposition of total interest rate movements. 12. Omitting the final six months of data from the sample largely preserves these nsk rankings. See footnote 7. 13. England had not yet agreed to join the EMS during the period under study. These results, as well as other conclusions of the study, are not inconsistent with the interest rate parity theorem. The latter relates interest rate differentials between pairs of countries to a one-point market estimate of expected exchange rate movements, as opposed to historical volatilities around a mean value, which are treated here. 14. Sellingforward theforeign exchange value of all the future cash flows of a foreign bond on their respective payment dates effectively transforms the bond into a US.dollar asset-that is, one sensitive to the US. interest rate environment. Except for the rare case of a temporary yield advantage, this defeats the purpose offoreign bond investing. Instead, short-datedforwards can be used to bedge the foreign exchange risk, while retaining the foreign interest rate exposure of the bond. In Measuring the Risk of Foreign Bonds, op. cit., I describe this strategy and estimate the remaining risk due to the lack of a perfect hedge.

A Unified Approach to Hedging Interest Rate Risk with Financial Futures*

ABSTRACTThere is abundant literature on the use of financial futures to reduce interest rate risk. While many applications have been developed and evaluated in the literature, little has been done to provide a simple, mathematical model unifying the disparate types of hedges. The purpose of this paper is to provide such a unifying framework. Under idealized conditions, an equation is developed giving a perfect hedge solution for arbitrary choice of planning horizon, existing or planned cash market position, and asset/liability mix. The paper is pedagogic in nature.

Inflation and Real Interest Rates on Assets with Different Risk Characteristics

Several recent studies find that ex ante real returns for short-term U.S. Treasury securities are negatively correlated both with inflation and with nominal interest rates. This paper examines whether these findings extend to the short-term holding return on publicly and privately issued securities of longer maturity, are robust with respect to the choice of price index, and are stable over time. Our results show that before 1979 a negative relationship of ex ante real returns with inflation and nominal interest rates does appear for the longer maturity assets. In fact, the relationship grows stronger with increases in maturity length. This suggests that although short-term U.S. Treasury bills were, of all the assets we study, the best hedge against expected inflation, none of the assets were a perfect hedge. We find a statistically significant change in the stochastic process of bond returns in 1979, with nominal interest rates and ex ante real holding returns being positively correlated in this latter period. This is not true for stocks, however. While the above results are robust to the choice of price index, we show that estimating the level of ex ante real returns depends crucially on the price index chosen.

The Arbitrage Pricing Model and Returns on Assets Under Uncertain Inflation*

THE IMPACT OF INFLATION on equilibrium returns in capital markets has been the subject of several articles in recent years. Chen and Boness [1], Friend, Landskroner and Losq [4], Long [7] and Roll [13] have all constructed general equilibrium models which incorporate the impact of uncertain inflation on returns.' All of these authors have derived models in a mean-variance world utilizing methodologies similar to that first utilized by Mossin [9] or Merton [8]. The basic underlying tenet of mean-variance analysis is that for any given level of expected return investors wish to minimize the variance of return. In mean-variance equilibrium models the only risk that cannot be diversified away is the covariance of an asset's return with the market return. Assets which have the same covariance with the market have the same equilibrium required return. Thus in a mean-variance world if each member of the following pairs of assets had the same covariance with the market each member would have the same expected return: (a) a stock which was a perfect hedge against inflation versus one which was not, (b) a stock which was a perfect hedge against oil price changes versus one which was not, (c) a security that hedged against interest rate changes versus one which did not. In this article we break out of the limitations of the mean variance assumptions by utilizing the Arbitrage Pricing Theory (APT) of Ross [15], [16], [17]. This

Optimal Liquidation of Assets in the Presence of Personal Taxes: Implications for Asset Pricing

WITH A TAX on realized capital gains, investors realize capital losses and defer capital gains. However, if individuals realize capital gains to consume or for other reasons (such as a cash stock merger) they may defer capital gains tax payments by hedging. The hedge produces a capital loss in one tax year and an equal capital gain in the next tax year. If the hedge is operational, capital gains tax is deferred until death and the capital gains tax is avoided entirely.' With this idealized hedge, the effective marginal tax rate on both short-term and long-term capital gains and losses is zero. Also, investors are indifferent to changes in the actual capital gains rate. Call options and commodity futures contracts are used to construct these hedges. Without transactions costs (no bid-ask spreads and brokerage commission fees),, perfect hedges can indeed be constructed and used to defer capital gains tax, despite IRS regulations designed to discourage their use. Transactions costs may dissipate the hedge's tax benefits, and therefore hedges reduce, but do not eliminate, the capital gains tax's effect on consumption and investment. The loophole in the tax code which enables tax deferral, at least in the absence of transactions costs, arises from the institutional structure of both the options exchange and the commodity futures exchange. A contract writer is not matched with a particular contract buyer. For example, if a call price falls, the call buyer can sell the call, realize a capital loss, and immediately buy a new call. The call writer is not forced to realize a corresponding capital gain. There is a net tax gain at the government's expense. Note that the net tax gain would be zero if the buyer and writer were matched. A tax gain for the buyer would be canceled by a tax loss for the writer. With a blunted but not eliminated capital gains tax, the capital gains tax's effect on optimal investor behavior is quite complex. Under the simplifying assumption that short-term and long-term rates on capital gains and losses are equal, the optimal liquidation policy is to realize capital losses and defer capital gains. With this lock-in effect, it is important to study welfare effects of given

A theoretical and empirical investigation of the dual purpose funds

Using the option pricing methods developed by Black and Scholes as a general technique for contingent claims analysis, this paper examines a class of mutual funds known as Dual Purpose Funds. By constructing a simplified model for these funds under the ‘perfect hedge’ conditions of Black and Scholes, it is demonstrated that the asset value of the fund will always exceed the market value and that it is not inconsistent with market equilibrium or efficiency for the capital shares to sell at a discount. The simple model predicts price fluctuations in the seven dual funds studied quite well; however, there is a persistent downward bias in the predicted price level. Finally, refinements to the model are examined to determine the nature of the misspecification causing this bias.