Perfect Hedging Research Articles

Financial Hedging and Optimal Procurement Policies Under Correlated Price and Demand

We consider a firm that procures an input commodity to produce an output commodity to sell to the end retailer. Retailer’s demand for the output commodity is negatively correlated with the price of the output commodity. The firm can sell the output commodity to the retailer either through a spot or a forward contract. Input and output commodity prices are also correlated and follow a joint stochastic price process. The firm maximizes the value of its stakeholders by jointly determining the optimal procurement policy of the input commodity, and the financial hedging policy for the sales of the output commodity. We show that partial hedging dominates both perfect hedging and no-hedging when input price, output price, and demand are correlated, and we characterize this optimal hedging policy. We identify sufficient conditions under which myopic policy is optimal for the price taking firm. We also show that expected base-stock policy is optimal in the presence of yield uncertainty. Our analysis illustrates that hedging is most beneficial when output price volatility is high and input price volatility is low. Our model is tested on the futures price data between 4/1/2005 to 12/31/2011 from Chicago Board of Trade (CBOT) for corn and ethanol.

Pricing and Hedging in Stochastic Volatility Regime Switching Models

We consider general regime switching stochastic volatility models where both the asset and the volatility dynamics depend on the values of a Markov jump process. Due to the stochastic volatility and the Markov regime switching, this financial market is thus incomplete and perfect pricing and hedging of options are not possible. Thus, we are interested in finding formulae to solve the problem of pricing and hedging options in this framework. For this, we use the local risk minimization approach to obtain pricing and hedging formulae based on solving a system of partial differential equations. Then we get also formulae to price volatility and variance swap options on these general regime switching stochastic volatility models.

Pricing and Hedging Contingent Claims Using Variance and Higher-Order Moment Futures

This paper suggests perfect hedging strategies of contingent claims under stochastic volatility and/or random jumps of the underlying asset price. This is done by enlarging the market with appropriate swap contracts whose payoffs depend on higher-order sample moments of the underlying asset price process. It also derives a model-free relation between these higher-order moment swaps and the value of a composite portfolio of European options, which can be employed to perfectly hedge variance swaps. Based on the theoretical results of the paper, and on options and variance swaps rates written on the S&P 500 index, the paper provides clear cut evidence that, first, random jumps are priced in the market and, second, hedging strategies for European options employing variance and higher-order moment swaps considerably improves upon the performance of traditional delta hedging strategies. These results indicate the kind of new financial instruments that can be introduced to perfectly hedge the market.

International trade and hedging under joint price and exchange rate uncertainty

This paper examines the behavior of a competitive exporting firm under joint price and exchange rate uncertainty. We show that the firm's optimal production and hedging decisions depend crucially on the degree of forward market incompleteness, and on the correlation structure of the price and exchange rate risk. The separation theorem holds if there are complete forward markets for hedging purposes. Should the forward markets be incomplete, the firm employs operational hedging by adjusting its output so as to better cope with the residual risk that is unhedgeable by simply trading the existing forward contracts. In the case that the price risk cannot be directly managed by financial hedging, we construct a reasonable example in which the firm optimally produces more, not less, than the benchmark output level under perfect hedging. Our results thus offer new insights into the interaction between financial and operational hedging in the context of multiple sources of uncertainty.

Quantile Hedging for Guaranteed Minimum Death Benefits with Regime Switching

The efficient hedging minimizes the average of the shortfall risk weighted by a loss function, where the hedging efficiency refers to the effectiveness of a hedge to accomplish the desired goal of risk management. Quantile hedging refers to the percentage of the hedge that can cover the contingent claim, which plays a key role for contingent claims in incomplete markets when perfect hedging is not possible. As observed in [8], the concept of quantile hedging can be considered as a dynamic version of the familiar value at risk concept (VaR). Treating regime switching diffusion models, this article focuses on guaranteed minimum death benefits (GMDBs), which are present in many variableannuity contracts, and act as a form of portfolio insurance. The GMDBs cannot be perfectly hedged due to the mortality component and incompleteness resulting from the regime switching, and as a result, quantile hedges are developed. Numerical examples are also presented to illustrate our results.

Optimal hedging of variance derivatives

We examine the optimal hedging of derivatives written on realised variance, focussing principally on variance swaps (VS) (but, en route, also considering skewness swaps), when the underlying stock price has discontinuous sample paths, i.e. jumps. In general, with jumps in the underlying, the market is incomplete and perfect hedging is not possible. We derive easily implementable formulae which give optimal (or nearly optimal) hedges for VS under very general dynamics for the underlying stock which allow for multiple jump processes and stochastic volatility. We illustrate how, for parameters which are realistic for options on the S&P 500 and Nikkei-225 stock indices, our methodology gives significantly better hedges than the standard log-contract replication approach of Neuberger and Dupire which assumes continuous sample paths. Our analysis seeks to emphasise practical implications for financial institutions trading variance derivatives.

Understanding ETNs on VIX Futures

This paper aims to improve transparency in the market for direct, leveraged and inverse exchange-traded notes (ETNs) on VIX futures. The first VIX futures ETNs were issued in 2009. Now there are about 30 of them, with a market cap of about $3 billion and trading volume on some of these products can reach $5 billion per day. Yet volatility trading is highly complex and regulators are rightly concerned that many market participants lack sufficient understanding of the risks they are taking. We recommend that exchanges, market-makers, issuers and potential investors, as well as regulators, read this paper to improve their understanding of these ETNs.We provide a detailed explanation of the roll yield and convexity effects that drive the returns on VIX futures ETNs, and we track their volatility and assess their performance over an eight-year period starting in March 2004, by replicating their values using daily close VIX futures prices. We explain how ETN issuers can construct almost perfect hedges of their suite of ETNs and control their issue (most ETNs are callable) to make very significant profits under all bootstrapped scenarios. However, market knowledge has precipitated front-running of the issuer’s hedging activities, making profits more difficult to control. Moreover, for hedging the ETNs such large positions must be taken on VIX futures that the ETN market now leads the VIX futures that they are supposed to track. The result has been an evident increase in the volatility of VIX futures since 2009. If this increase in statistical volatility induces an increase in VIX futures implied volatility, a knock-on effect would be higher prices of VIX options whilst S&P options are unaffected.A previous discussion paper, Alexander and Korovilas (2012), provided incontrovertible evidence that single positions on direct VIX futures ETNs of any maturity – including mid-term and longer-term trackers – could only provide a diversification/hedge of equity exposure during the first few months of a great crisis of similar magnitude to the banking collapse in late 2008. By contrast, the present discussion paper shows that some highly attractive long-term investment vehicles can be simply constructed by holding certain portfolios of VIX futures ETNs. In particular, we introduce a new class of 'roll-yield arbitrage' ETN portfolios which we call ETN2 (because they allocate between direct and inverse VIX futures tracker ETNs) and ETN3 portfolios (that allocate between static and dynamic ETN2). These portfolios have positive exposure to mid-term direct-tracker ETNs and (typically) negative exposure to short-term direct-tracker ETNs (equivalently, positive exposure to short-term inverse-tracker ETNs). Their unique risk and return characteristics make them highly attractive long-term investments, as well as superb diversifiers of stocks, bonds and commodities.

Ambiguous Volatility and Asset Pricing in Continuous Time

This paper formulates a model of utility for a continuous time frame-work that captures the decision-maker’s concern with ambiguity about both volatility and drift. Corresponding extensions of some basic results in asset pricing theory are presented. First, we derive arbitrage-free pricing rules based on hedging arguments. Ambiguous volatility implies market incompleteness that rules out perfect hedging. Consequently, hedging arguments determine prices only up to intervals. However, sharper predictions can be obtained by assuming preference maximization and equilibrium. Thus we apply the model of utility to a representative agent endowment economy to study equilibrium asset returns. A version of the C-CAPM is derived and the effects of ambiguous volatility are described.

Estimate for the discrete time hedging error of the American option on a dividend-paying stock

This work is devoted to the discrete time hedging of the American option on a dividendpaying stock with a convex payoff, the particular case of which is American call option. Perfect hedging requires continuous trading in time and knowledge of the partial derivative of the value function of the American option in the underlying asset. Neither one can trade continuously in time nor the closed-form expression of the value function of the American option is known. Several approximation methods have been developed for the calculation of this unknown value function. We justify in this work that having at hand any such nonnegative uniform approximation, it is possible to construct a discrete time hedging strategy the value process of which uniformly approximates the value process of the corresponding continuous time perfect hedging portfolio. Mathematics subject classification (2010): 60J75, 60G40, 60J60.

Arbitrage‐Free Conditions and Hedging Strategies for Markets withPenalty Costs on Short Positions

We consider a discrete‐time financial model in a general sample space with penalty costs on short positions. We consider a friction market closely related to the standard one except that withdrawals from the portfolio value proportional to short positions are made. We provide necessary and sufficient conditions for the nonexistence of arbitrages in this situation and for a self‐financing strategy to replicate a contingent claim. For the finite‐sample space case, this result leads to an explicit and constructive procedure for obtaining perfect hedging strategies.

Partial Hedging Using Malliavin Calculus

Under the constraint that the initial capital is not enough for a perfect hedge, the problem of deriving an optimal partial hedging portfolio so as to minimize the shortfall risk is worked out by solving two connected subproblems sequentially. One subproblem is to find the optimal terminal wealth that minimizes the shortfall risk. The shortfall risk is quantified by a general convex risk measure to accommodate different levels of risk tolerance. A convex duality approach is used to obtain an explicit formula for the optimal terminal wealth. The second subproblem is to derive the explicit expression for the admissible replicating portfolio that generates the optimal terminal wealth. We show by examples that to solve the second subproblem, the Malliavin calculus approach outperforms the traditional delta-hedging approach even for the simplest claim. Explicit worked-out examples include a European call option and a standard lookback put option.

Large traders and illiquid options: Hedging vs. manipulation

In this article, we study the effects on derivative pricing arising from price impacts by large traders. When a large trader issues a derivative and (partially) hedges his risk by trading in the underlying, he influences both his hedge portfolio and the derivative's payoff. In a Black–Scholes model with a price impact on the drift, we analyze the resulting trade-off by explicitly solving the utility maximization problem of a large investor endowed with an illiquid contingent claim. We find several interesting phenomena which cannot occur in frictionless markets. First, the indifference price is a convex function of the contingent claim – and not concave as in frictionless markets – implying that for any claim the buyer's indifference price is larger than the seller's indifference price. Second, the seller's indifference prices of large positions in derivatives are smaller than the Black–Scholes replication costs. Therefore, a large trader might have an incentive to issue options if they are traded at Black–Scholes prices. Furthermore, he hedges option positions only partly if he has a negative price impact and thus exploits his ability to manipulate the option's payoff. For a positive price impact he overhedges the option position leading to an extra profit from the stock position exceeding a perfect hedge. Finally, we also study a model where the large shareholder has a price impact on both drift and volatility.

DISCRETE TIME HEDGING OF THE AMERICAN OPTION

In a complete financial market we consider the discrete time hedging of the American option with a convex payoff. It is well known that for the perfect hedging the writer of the option must trade continuously in time, which is impossible in practice. In reality, the writer hedges only at some discrete time instants. The perfect hedging requires the knowledge of the partial derivative of the value function of the American option in the underlying asset, the explicit form of which is unknown in most cases of practical importance. Several approximation methods have been developed for the calculation of the value function of the American option. We claim in this paper that having at hand any uniform approximation of the American option value function at equidistant discrete rebalancing times it is possible to construct a discrete time hedging portfolio, the value process of which uniformly approximates the value process of the continuous time perfect delta-hedging portfolio. We are able to estimate the corresponding discrete time hedging error that leads to a complete justification of our hedging method for nonincreasing convex payoff functions including the important case of the American put. This method is essentially based on a new type square integral estimate for the derivative of an arbitrary convex function recently found by Shashiashvili.

Operational Flexibility and Financial Hedging: Complements or Substitutes?

We consider a firm that invests in capacity under demand uncertainty and thus faces two related but distinct types of risk: mismatch between capacity and demand and profit variability. Whereas mismatch risk can be mitigated with greater operational flexibility, profit variability can be reduced through financial hedging. We show that the relationship between these two risk mitigating strategies depends on the type of flexibility: Product flexibility and financial hedging tend to be complements (substitutes)—i.e., product flexibility tends to increase (decrease) the value of financial hedging, and, vice versa, financial hedging tends to increase (decrease) the value of product flexibility—when product demands are positively (negatively) correlated. In contrast to product flexibility, postponement flexibility is a substitute to financial hedging as intuitively expected. Although our analytical results assume perfect flexibility and perfect hedging and rely on a linear approximation of the value of hedging, we validate their robustness in an extensive numerical study.

A further inquiry into FTR properties

William Hogan introduced financial transmission rights as a tool to hedge the locational risk inherent in locational marginal prices. FTRs are claimed to serve four main purposes: (1) provide a hedge for nodal price differences, (2) provide revenue sufficiency for contracts for differences, (3) distribute the merchandizing surplus an RTO accrues in market operations, and (4) provide a price signal for transmission and generation developers. This paper examines the hedging and redistributional properties of FTRs. It argues that FTR allocation has important distributional impacts and related implications for retail rates. This observation adds an additional explanation for rate increases in light of decreased production costs due to restructuring. This paper also shows that RTO practices have important implications for the hedging characteristics of FTRs. It further shows, via counterexample, that, even in theory, FTRs may not serve as a perfect hedge against congestion charges. The paper concludes with a series of recommendations for FTR allocation and the functions that FTRs should serve.

Option Derivatives in Electricity Hedging

Despite the high volatility of electricity prices, there is still little demand for electricity power options, and the liquidity on the power exchanges of these power derivatives is quite low. One of the reasons is the uncertainty about how to evaluate these electricity options and about finding the right fair value of this product. Hedging of electricity is associated mainly with products such as futures and forwards. However, due to new trends in electricity trading and hedging, it is also useful to think more about options and the principles for working with them in hedging various portfolio positions and counterparties. We can quite often encounter a situation when we need to have a perfect hedge for our customer’s (end user consuming electricity) portfolio, or we have to evaluate the volumetric risk (inability of a customer to predict consumption, which is very similar to selling options. Now comes the moment to compare the effects of using options or futures to hedge these open positions. From a practical viewpoint, the Black-Scholes prices appear to be the best available and the simplest method for evaluating option premiums, but there are some limitations that we have to consider.

Maximizing the Probability of a Perfect Hedge in the Case of Stochastic Interest Rate

SYNOPTIC ABSTRACTIn the financial market including n risky assets with prices following the full-observed geometric-Brownian Motion model and a bank account with stochastic interest rate, we study the problem of maximizing the probability of an investor's wealth at terminal time T meeting or exceeding the value of a contingent claim C which expires at T. In this incomplete market, using an adapted duality methodology familiar in the utility maximization literatures, we characterize the optimal probability and the corresponding portfolio strategies, in the most general framework of interest rate models and contingent claims. Then we calculate in detail the case of Vasicek model for interest rate and some specific contingent claim, via the classical theorem of Time-Changing for Martingales and a generalized Cameron-Martin formula developed in the literatures.

No arbitrage pricing of non-marketed claims in multi-period markets

In this paper, we modify the arbitrage-free interval of prices for a non-marketed contingent claim in the finite event-tree model of financial markets, according to the perfect hedging approach being well-known for the two-period model. We prove the existence of solution to the corresponding seller's and buyer's price problem for such a claim under no-arbitrage prices for the marketed contracts and we show that each of these problems can be solved by decomposing it into a finite number of one-period linear programming problems solved backwards. Finally, we indicate that the set of the no-arbitrage prices for a non-marketed contingent claim is the interval of the real numbers whose supremum and infimum is the seller's and the buyer's price of the claim, respectively. The determination of the set of no-arbitrage prices for a non-marketed contingent claim is related to the utility pricing of such a claim.

Large Traders and Illiquid Options: Hedging vs. Manipulation

In this article, we study the effects on derivative pricing arising from price impacts by large traders. When a large trader issues a derivative and (partially) hedges his risk by trading in the underlying, he influences both his hedge portfolio and the derivative's payoff. In a Black-Scholes model with a price impact on the drift, we analyze the resulting trade-off by explicitly solving the utility maximization problem of a large investor endowed with an illiquid contingent claim. We find several interesting phenomena which cannot occur in frictionless markets. Firstly, the indifference price is a convex function of the contingent claim -- and not concave as in frictionless markets -- implying that for any claim the buyer's indifference price is larger than the seller's indifference price. Secondly, the seller's indifference prices of large positions in derivatives are smaller than the Black-Scholes replication costs. Therefore, a large trader might have an incentive to issue options if they are traded at Black-Scholes prices. Furthermore, he hedges option positions only partly if he has a negative price impact and thus exploits his ability to manipulate the option's payoff. For a positive price impact he overhedges the option position leading to an extra profit from the stock position exceeding a perfect hedge. Finally, we also study a model where the large shareholder has a price impact on both drift and volatility.

Credit default swaps: the quest of the hedge

AbstractIt appears from discussions held with practitioners on forums and conferences that hedging products with CDSs is often considered to be an easy task. For instance, a bond‐CDS position is frequently seen to be a perfect hedge as is the index CDS and its corresponding single‐name counterparts. On the other hand, theoreticians claim that a perfect hedge is impossible to achieve with credit derivatives: currently it seems there is no model from which a perfect hedge of credit derivatives can be set up. So, who's right? This paper aims to show that, surprisingly, both are right, in some sense. The misunderstanding comes from a misspecification of the “hedge” term, together with a confusion between hedging and arbitraging. We focus on Credit Default Swaps and explain why, among other things, the “Bond‐CDS on the Bond” position does not result in a perfect hedge (except in a very specific case, which is not market standard). By contrast, partial hedge (that is, hedging some parts of the risk) can be obtained and, of course, arbitrage opportunities could be found. We then explain, adopting a philosophical point of view, why it is so problematic to set up a credit portfolio being perfectly hedged (even in a first‐order approximation sense). It appears that the nature of the derivative's underlying prevents us reaching this with the currently available financial instruments. Copyright © 2009 Wilmott Magazine Ltd