Fair bilateral pricing under funding costs and exogenous collateralization

Bielecki and Rutkowski introduced and studied a generic nonlinear market model, which includes several risky assets, multiple funding accounts, and margin accounts. In this paper, we examine the pricing and hedging of contract from the perspective of both the hedger and the counterparty with arbitrary initial endowments. We derive inequalities for unilateral prices and we study the range of fair bilateral prices. We also examine the positive homogeneity and monotonicity of unilateral prices with respect to the initial endowments. Our study hinges on results from Nie and Rutkowski for backward stochastic differential equations (BSDEs) driven by continuous martingales, but we also derive the pricing partial differential equations (PDEs) for path‐independent contingent claims of a European style in a Markovian framework.     

BILATERAL COUNTERPARTY RISK UNDER FUNDING CONSTRAINTS—PART I: PRICING

This and the follow‐up paper deal with the valuation and hedging of bilateral counterparty risk on over‐the‐counter derivatives. Our study is done in a multiple‐curve setup reflecting the various funding constraints (or costs) involved, allowing one to investigate the question of interaction between bilateral counterparty risk and funding. The first task is to define a suitable notion of no arbitrage price in the presence of various funding costs. This is the object of this paper, where we develop an “additive, multiple curve” extension of the classical “multiplicative (discounted), one curve” risk‐neutral pricing approach. We derive the dynamic hedging interpretation of such an “additive risk‐neutral” price, starting by consistency with pricing by replication in the case of a complete market. This is illustrated by a completely solved example building over previous work by Burgard and Kjaer.     

Binary funding impacts in derivative valuation

We discuss the binary nature of funding impact in derivative valuation. Under some conditions, funding is either a cost or a benefit, that is, one of the lending/borrowing rates does not play a role in pricing derivatives. When derivatives are priced, considering different lending/borrowing rates leads to semilinear backward stochastic differential equations (BSDEs) and partial differential equation (PDEs), and thus it is necessary to solve the equations numerically. However, once it can be guaranteed that only one of the rates affects pricing, linear equations can be recovered, and analytical formulae can be derived. Moreover, as a by‐product, our results explain how debt value adjustment (DVA) and funding benefits are dissimilar. It is often believed that considering both DVA and funding benefits results in a double‐counting issue but it will be shown that the two components are affected by different mathematical structures of derivative transactions. We find that funding benefit is related to the decreasing property of the payoff function, but this relationship decreases as the funding choices of underlying assets are transferred to repo markets.     

BOUNDING WRONG‐WAY RISK IN CVA CALCULATION

A credit valuation adjustment (CVA) is an adjustment applied to the value of a derivative contract or a portfolio of derivatives to account for counterparty credit risk. Measuring CVA requires combining models of market and credit risk to estimate a counterparty's risk of default together with the market value of exposure to the counterparty at default. Wrong‐way risk refers to the possibility that a counterparty's likelihood of default increases with the market value of the exposure. We develop a method for bounding wrong‐way risk, holding fixed marginal models for market and credit risk and varying the dependence between them. Given simulated paths of the two models, a linear program computes the worst‐case CVA. We analyze properties of the solution and prove convergence of the estimated bound as the number of paths increases. The worst case can be overly pessimistic, so we extend the procedure by constraining the deviation of the joint model from a baseline reference model. Measuring the deviation through relative entropy leads to a tractable convex optimization problem that can be solved through the iterative proportional fitting procedure. Here, too, we prove convergence of the resulting estimate of the penalized worst‐case CVA and the joint distribution that attains it. We consider extensions with additional constraints and illustrate the method with examples.     

Arbitrage‐free XVA

We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no‐arbitrage arguments, we derive backward stochastic differential equations associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buyer's and seller's XVA, which in turn identify a no‐arbitrage interval. In the case that borrowing and lending rates coincide, we provide a fully explicit expression for the unique XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. In the general case of asymmetric funding, repo, and collateral rates, we study the semilinear partial differential equations characterizing buyer's and seller's XVA and show the existence of a unique classical solution to it. To illustrate our results, we conduct a numerical study demonstrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.     

Option pricing for the transformed‐binomial class

This article generalizes the seminal Cox‐Ross‐Rubinstein (1979) binomial option pricing model to all members of the class of transformed‐binomial pricing processes. The investigation addresses issues related with asset pricing modeling, hedging strategies, and option pricing. Formulas are derived for (a) replicating or hedging portfolios, (b) risk‐neutral transformed‐binomial probabilities, (c) limiting transformed‐normal distributions, and (d) the value of contingent claims, including limiting analytical option pricing equations. The properties of the transformed‐binomial class of asset pricing processes are also studied. The results of the article are illustrated with several examples. © 2006 Wiley Periodicals, Inc. Jrl. Fut Mark 26:759–787, 2006     

Derivative pricing model and time‐series approaches to hedging: A comparison

This research compares derivative pricing model and statistical time‐series approaches to hedging. The finance literature stresses the former approach, while the applied economics literature has focused on the latter. We compare the out‐of‐sample hedging effectiveness of the two approaches when hedging commodity price risk using futures contracts. For various methods of parameter estimation and inference, we find that the derivative pricing models cannot out‐perform a vector error‐correction model with a GARCH error structure. The derivative pricing models' unpalatable assumption of deterministically evolving futures volatility seems to impede their hedging effectiveness, even when potentially foresighted optionimplied volatility term structures are employed. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:613–641, 2005     

Hedging under counterparty credit uncertainty

This study investigates optimal production and hedging decisions for firms facing price risk that can be hedged with vulnerable contracts, i.e., exposed to nonhedgeable endogenous counterparty credit risk. When vulnerable forward contracts are the only hedging instruments available, the firm's optimal level of production is lower than without credit risk. Under plausible conditions on the stochastic dependence between the commodity price and the counterparty's assets, the firm does not sell its entire production on the vulnerable forward market. When options on forward contracts are also available, the optimal hedging strategy requires a long put position. This provides a new rationale for the hedging role of options in the over‐the‐counter markets exposed to counterparty credit risk. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28: 248–263, 2008     

Volatility options: Hedging effectiveness,pricing, and model error

Motivated by the growing literature on volatility options and their imminent introduction in major exchanges, this article addresses two issues. First, the question of whether volatility options are superior to standard options in terms of hedging volatility risk is examined. Second, the comparative pricing and hedging performance of various volatility option pricing models in the presence of model error is investigated. Monte Carlo simulations within a stochastic volatility setup are employed to address these questions. Alternative dynamic hedging schemes are compared, and various option‐pricing models are considered. It is found that volatility options are not better hedging instruments than plain‐vanilla options. Furthermore, the most naïve volatility option‐pricing model can be reliably used for pricing and hedging purposes. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:1–31, 2006     

Pricing and hedging illiquid energy derivatives: An application to the JCC index

In this paper a simple strategy for pricing and hedging a swap on the Japanese crude oil cocktail (JCC) index is discussed. The empirical performance of different econometric models is compared in terms of their computed optimal hedge ratios, using monthly data on the JCC over the period January 2000–January 2006. An explanation to how to compute a bid/ask spread and to construct the hedging position for the JCC swap contract with variable oil volume is provided. The swap pricing scheme with backtesting and rolling regression techniques is evaluated. The empirical findings show that the price‐level regression model permits one to compute more precise optimal hedge ratios relative to its competing alternatives. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:464–487, 2008     

Pricing and hedging in the freight futures market

In this article, we consider the pricing and hedging of single‐route dry bulk freight futures contracts traded on the International Maritime Exchange. Thus far, this relatively young market has received almost no academic attention. In contrast to many other commodity markets, freight services are non‐storable, making a simple cost‐of‐carry valuation impossible. We empirically compare the pricing and hedging accuracy of a variety of continuous‐time futures pricing models. Our results show that the inclusion of a second stochastic factor significantly improves the pricing and hedging accuracy. Overall, the results indicate that the Schwartz and Smith ( 2000 ) two‐factor model provides the best performance. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:440–464, 2011     

Futures hedging when the structure of the underlying asset changes: The case of the BIFFEX contract

This article is concerned with the hedging effectiveness of futures contracts whose underlying asset is an index, when the structure of this index is changing. The case of the freight futures (BIFFEX) contract is examined here. Investigation of this issue is particularly interesting as the composition of its underlying asset, the Baltic Freight Index (BFI), has been revised on a number of occasions in order to improve the hedging performance of the market; previous empirical evidence on the market indicates substantially lower variance reduction (4–19%), compared to other markets (up to 98%). The BFI is a weighted average dry‐cargo freight rate index, compiled from actual freight rates on 11 shipping routes that are dissimilar in terms of vessel sizes and transported commodities. The hedging effectiveness of the market is investigated using both constant and time‐varying hedge ratios, estimated through bivariate error correction GARCH models. Our results indicate that the effectiveness of the BIFFEX contract as a centre for risk management has strengthened over the recent years as a result of the more homogeneous composition of the index. This by itself indicates that the latest restructuring of the index, in November 1999, which is aimed at increasing its homogeneity even further, is likely to have a beneficial impact on the market. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:775–801, 2000     

Valuation and Hedging of Differential Swaps

This paper derives a general‐form formula for pricing and hedging differential swaps with the principal denominated either in a domestic, foreign, or third‐country currency. We first derive the formula for differential swaps with the principal in a domestic currency and identify an error in the formula of Wei (1994). We then show the pricing duality between differential swaps with the principal in a domestic currency and differential swaps with the principal in a foreign currency. Finally, we complete the pricing and hedging analysis on differential swaps by deriving a formula for differential swaps with the principal denominated in a third‐country currency. Simulation results indicate that constant margin rates are generally smaller than interest rate differentials and decline with the tenor of swaps. Correlation parameters associated with the exchange rate play a more important role than correlation parameters among interest rates in pricing differential swaps. © 2002 John Wiley & Sons, Inc. Jrl Fut Mark 22:73–94, 2002     

Effects of reduced government deficiency payments on post‐harvest wheat marketing strategies

Effects of reducing government deficiency payments on a wheat producer's post‐harvest marketing strategies are evaluated. The deficiency payment is predicted using an average option pricing model to properly value both intrinsic and time values of the deficiency payment. The biggest loss to producers from reducing deficiency payments is reduced revenue. The deficiency payment program was no better than hedging strategies in reducing post‐harvest risk, and when grain was sold at harvest, it even increased post‐harvest risk. Many producers will compensate for reduced deficiency payments by increasing use of futures or options contracts. For some producers, however, the optimal strategy is to sell wheat at harvest, because of high opportunity cost, storage cost, or risk aversion. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:243–263, 2000     

Hedging and value at risk

In this article, it is shown that although minimum‐variance hedging unambiguously reduces the standard deviation of portfolio returns, it can increase both left skewness and kurtosis; consequently the effectiveness of hedging in terms of value at risk (VaR) and conditional value at risk (CVaR) is uncertain. The reduction in daily standard deviation is compared with the reduction in 1‐day 99% VaR and CVaR for 20 cross‐hedged currency portfolios with the use of historical simulation. On average, minimum‐variance hedging reduces both VaR and CVaR by about 80% of the reduction in standard deviation. Also investigated, as an alternative to minimum‐variance hedging, are minimum‐VaR and minimum‐CVaR hedging strategies that minimize the historical‐simulation VaR and CVaR of the hedge portfolio, respectively. The in‐sample results suggest that in terms of VaR and CVaR reduction, minimum‐VaR and minimum‐CVaR hedging can potentially yield small but consistent improvements over minimum‐variance hedging. The out‐of‐sample results are more mixed, although there is a small improvement for minimum‐VaR hedging for the majority of the currencies considered. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:369–390, 2006     

Optimal hedging under nonlinear borrowing cost,progressive tax rates,and liquidity constraints

Empirical research using optimal hedge ratios usually suggests that producers should hedge much more than they do. In this study, a new theoretical model of hedging is derived. Optimal hedge and leverage ratios and their relationship with yield risk, price variability, basis risk, taxes, and financial risk are determined using alternative assumptions. The motivation to hedge is provided by progressive tax rates and cost of bankruptcy. An empirical example for a wheat and stocker‐steer producer is provided. Results show that there are many factors, often assumed away in the literature, that make farmers hedge little or not at all. Progressive tax rates provide an incentive for farmers to hedge in order to reduce their tax liabilities and increase their after‐tax income. Farmers will hedge when the cost of hedging is less than the benefits of hedging that come from reducing tax liabilities, liquidity costs, or bankruptcy costs. When tax‐loss carryback is allowed, hedging decreases as the amount of tax loss that can be carried back increases. Higher profitability makes benefits from futures trading negligible and hedging unattractive, since farmers move to higher income brackets with near constant marginal tax rates. Increasing basis risk or yield risk also reduce the incentive to hedge. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20: 375–396, 2000     

RISK INDIFFERENCE PRICING IN JUMP DIFFUSION MARKETS

We study the risk indifference pricing principle in incomplete markets: The (seller's)  risk indifference price        is the initial payment that makes the  risk  involved for the seller of a contract equal to the risk involved if the contract is not sold, with no initial payment. We use stochastic control theory and PDE methods to find a formula for       and similarly for      . In particular, we prove that  where    p _low   and    p _up   are the lower and upper hedging prices, respectively.     

THE COST OF ILLIQUIDITY AND ITS EFFECTS ON HEDGING

Though liquidity is commonly believed to be a major effect in financial markets, there appears to be no consensus definition of what it is or how it is to be measured. In this paper, we understand liquidity as a nonlinear transaction cost incurred as a function of rate of change of portfolio. Using this definition, we obtain the optimal hedging policy for the hedging of a call option in a Black‐Scholes model. This is a more challenging question than the more common studies of optimal strategy for liquidating an initial position, because our goal requires us to match a random final value. The solution we obtain reduces in the case of quadratic loss to the solution of three partial differential equations of Black‐Scholes type, one of them nonlinear.     

How firms should hedge: An extension

This note studies a firm's optimal hedging strategy with tailor‐made exotic derivatives under both price risk and quantity risk. It extends the analysis of Brown G. W. and Toft K.‐B. (2002) by relaxing the assumption of a bivariate normal distribution. The optimal payoff function of a derivative contract is characterized in terms of the expectation and variance of the quantity, conditional on the price. This main result is illustrated by different examples, stressing the importance of the dependence structure between price risk and quantity risk for the choice of appropriate hedging instruments. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:834–845, 2010     

FROM SMILE ASYMPTOTICS TO MARKET RISK MEASURES

The left tail of the implied volatility skew, coming from quotes on out‐of‐the‐money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations, to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear partial differential equation and provide a small time‐to‐maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parameterized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.