ON UTILITY-BASED PRICING OF CONTINGENT CLAIMS IN INCOMPLETE MARKETS

We study the uniqueness of the marginal utility-based price of contingent claims in a semimartingale model of an incomplete financial market. In particular, we obtain that a necessary and sufficient condition for all bounded contingent claims to admit a unique marginal utility-based price is that the solution to the dual problem defines an equivalent local martingale measure.     

DYNAMIC PORTFOLIO OPTIMIZATION WITH A DEFAULTABLE SECURITY AND REGIME‐SWITCHING

We consider a portfolio optimization problem in a defaultable market with finitely‐many economical regimes, where the investor can dynamically allocate her wealth among a defaultable bond, a stock, and a money market account. The market coefficients are assumed to depend on the market regime in place, which is modeled by a finite state continuous time Markov process. By separating the utility maximization problem into a predefault and postdefault component, we deduce two coupled Hamilton–Jacobi–Bellman equations for the post‐ and predefault optimal value functions, and show a novel verification theorem for their solutions. We obtain explicit constructions of value functions and investment strategies for investors with logarithmic and Constant Relative Risk Aversion utilities, and provide a precise characterization of the directionality of the bond investment strategies in terms of corporate returns, forward rates, and expected recovery at default. We illustrate the dependence of the optimal strategies on time, losses given default, and risk aversion level of the investor through a detailed economic and numerical analysis.     

CONTINGENT CLAIMS VALUED AND HEDGED BY PRICING AND INVESTING IN A BASIS

Contingent claims with payoffs depending on finitely many asset prices are modeled as elements of a separable Hilbert space. Under fairly general conditions, including market completeness, it is shown that one may change measure to a reference measure under which asset prices are Gaussian and for which the family of Hermite polynomials serves as an orthonormal basis. Basis pricing synthesizes claim valuation and basis investment provides static hedging opportunities. For claims written as functions of a single asset price we infer from observed option prices the implicit prices of basis elements and use these to construct the implied equivalent martingale measure density with respect to the reference measure, which in this case is the Black-Scholes geometric Brownian motion model. Data on S & P 500 options from the Wall Street Journal are used to illustrate the calculations involved. On this illustrative data set the equivalent martingale measure deviates from the Black-Scholes model by relatively discounting the larger price movements with a compensating premia placed on the smaller movements.     

NO‐ARBITRAGE PRICING UNDER SYSTEMIC RISK: ACCOUNTING FOR CROSS‐OWNERSHIP

We generalize Merton’s asset valuation approach to systems of multiple financial firms where cross‐ownership of equities and liabilities is present. The liabilities, which may include debts and derivatives, can be of differing seniority. We derive equations for the prices of equities and recovery claims under no‐arbitrage. An existence result and a uniqueness result are proven. Examples and an algorithm for the simultaneous calculation of all no‐arbitrage prices are provided. A result on capital structure irrelevance for groups of firms regarding externally held claims is discussed, as well as financial leverage and systemic risk caused by cross‐ownership.     

NO‐ARBITRAGE PRICING FOR DIVIDEND‐PAYING SECURITIES IN DISCRETE‐TIME MARKETS WITH TRANSACTION COSTS

We prove a version of First Fundamental Theorem of Asset Pricing under transaction costs for discrete‐time markets with dividend‐paying securities. Specifically, we show that the no‐arbitrage condition under the efficient friction assumption is equivalent to the existence of a risk‐neutral measure. We derive dual representations for the superhedging ask and subhedging bid price processes of a contingent claim contract. Our results are illustrated with a vanilla credit default swap contract.     

ON AGENT’S AGREEMENT AND PARTIAL‐EQUILIBRIUM PRICING IN INCOMPLETE MARKETS

We consider two risk‐averse financial agents who negotiate the price of an illiquid indivisible contingent claim in an incomplete semimartingale market environment. Under the assumption that the agents are exponential utility maximizers with nontraded random endowments, we provide necessary and sufficient conditions for negotiation to be successful, i.e., for the trade to occur. We also study the asymptotic case where the size of the claim is small compared to the random endowments and we give a full characterization in this case. Finally, we study a partial‐equilibrium problem for a bundle of divisible claims and establish existence and uniqueness. A number of technical results on conditional indifference prices is provided.     

TUG‐OF‐WAR,MARKET MANIPULATION,AND OPTION PRICING

We develop an option pricing model based on a tug‐of‐war game. This two‐player zero‐sum stochastic differential game is formulated in the context of a multidimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the nonlinear and completely degenerate infinity Laplace operator.     

A MODEL‐FREE VERSION OF THE FUNDAMENTAL THEOREM OF ASSET PRICING AND THE SUPER‐REPLICATION THEOREM

We propose a Fundamental Theorem of Asset Pricing and a Super‐Replication Theorem in a model‐independent framework. We prove these theorems in the setting of finite, discrete time and a market consisting of a risky asset S as well as options written on this risky asset. As a technical condition, we assume the existence of a traded option with a superlinearly growing payoff‐function, e.g., a power option. This condition is not needed when sufficiently many vanilla options maturing at the horizon T are traded in the market.     

NO‐ARMAGEDDON MEASURE FOR ARBITRAGE‐FREE PRICING OF INDEX OPTIONS IN A CREDIT CRISIS

In this work, we consider three problems of the standard market approach to credit index options pricing: the definition of the index spread is not valid in general, the considered payoff leads to a pricing which is not always defined, and the candidate numeraire for defining a pricing measure is not strictly positive, which leads to a nonequivalent pricing measure. We give a solution to the three problems, based on modeling the flow of information through a suitable subfiltration. With this we consistently take into account the possibility of default of all names in the portfolio, that is neglected in the standard market approach. We show on market inputs that, while the pricing difference can be negligible in normal market conditions, it can become highly relevant in stressed market conditions, like the situation caused by the credit crunch.     

MODEL UNCERTAINTY AND ITS IMPACT ON THE PRICING OF DERIVATIVE INSTRUMENTS

Uncertainty on the choice of an option pricing model can lead to "model risk" in the valuation of portfolios of options. After discussing some properties which a quantitative measure of model uncertainty should verify in order to be useful and relevant in the context of risk management of derivative instruments, we introduce a quantitative framework for measuring model uncertainty in the context of derivative pricing. Two methods are proposed: the first method is based on a coherent risk measure compatible with market prices of derivatives, while the second method is based on a convex risk measure. Our measures of model risk lead to a premium for model uncertainty which is comparable to other risk measures and compatible with observations of market prices of a set of benchmark derivatives. Finally, we discuss some implications for the management of "model risk."     

THE WIENER–HOPF TECHNIQUE AND DISCRETELY MONITORED PATH‐DEPENDENT OPTION PRICING

Fusai, Abrahams, and Sgarra (2006) employed the Wiener–Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black–Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path‐dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first‐touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener–Hopf technique yields a novel formula for double‐barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Padé approximation. A Gaussian Black–Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Lévy processes generally.     

HIGH‐ORDER SHORT‐TIME EXPANSIONS FOR ATM OPTION PRICES OF EXPONENTIAL LÉVY MODELS

The short‐time asymptotic behavior of option prices for a variety of models with jumps has received much attention in recent years. In this work, a novel second‐order approximation for at‐the‐money (ATM) option prices is derived for a large class of exponential Lévy models with or without Brownian component. The results hereafter shed new light on the connection between both the volatility of the continuous component and the jump parameters and the behavior of ATM option prices near expiration. In the presence of a Brownian component, the second‐order term, in time‐t, is of the form , with d₂ only depending on Y, the degree of jump activity, on σ, the volatility of the continuous component, and on an additional parameter controlling the intensity of the “small” jumps (regardless of their signs). This extends the well‐known result that the leading first‐order term is . In contrast, under a pure‐jump model, the dependence on Y and on the separate intensities of negative and positive small jumps are already reflected in the leading term, which is of the form . The second‐order term is shown to be of the form and, therefore, its order of decay turns out to be independent of Y. The asymptotic behavior of the corresponding Black–Scholes implied volatilities is also addressed. Our method of proof is based on an integral representation of the option price involving the tail probability of the log‐return process under the share measure and a suitable change of probability measure under which the pure‐jump component of the log‐return process becomes a Y‐stable process. Our approach is sufficiently general to cover a wide class of Lévy processes, which satisfy the latter property and whose Lévy density can be closely approximated by a stable density near the origin. Our numerical results show that the first‐order term typically exhibits rather poor performance and that the second‐order term can significantly improve the approximation's accuracy, particularly in the absence of a Brownian component.     

EFFICIENT PRICING OF BARRIER OPTIONS AND CREDIT DEFAULT SWAPS IN LÉVY MODELS WITH STOCHASTIC INTEREST RATE

Recently, advantages of conformal deformations of the contours of integration in pricing formulas for European options have been demonstrated in the context of wide classes of Lévy models, the Heston model, and other affine models. Similar deformations were used in one‐factor Lévy models to price options with barrier and lookback features and credit default swaps (CDSs). In the present paper, we generalize this approach to models, where the dynamics of the assets is modeled as , where X is a Lévy process, and the interest rate is stochastic. Assuming that X and r are independent, and , the infinitesimal generator of the pricing semigroup in the model for the short rate, satisfies weak regularity conditions, which hold for popular models of the short rate, we develop a variation of the pricing procedure for Lévy models which is almost as fast as in the case of the constant interest rate. Numerical examples show that about 0.15 second suffices to calculate prices of 8 options of same maturity in a two‐factor model with the error tolerance and less; in a three‐factor model, accuracy of order 0.001–0.005 is achieved in about 0.2 second. Similar results are obtained for quanto CDS, where an additional stochastic factor is the exchange rate. We suggest a class of Lévy models with the stochastic interest rate driven by 1–3 factors, which allows for fast calculations. This class can satisfy the current regulatory requirements for banks mandating sufficiently sophisticated credit risk models.     

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.     

THE STOCHASTIC VOLATILITY MODEL OF BARNDORFF‐NIELSEN AND SHEPHARD IN COMMODITY MARKETS

We consider the non‐Gaussian stochastic volatility model of Barndorff‐Nielsen and Shephard for the exponential mean‐reversion model of Schwartz proposed for commodity spot prices. We analyze the properties of the stochastic dynamics, and show in particular that the log‐spot prices possess a stationary distribution defined as a normal variance‐mixture model. Furthermore, the stochastic volatility model allows for explicit forward prices, which may produce a hump structure inherited from the mean‐reversion of the stochastic volatility. Although the spot price dynamics has continuous paths, the forward prices will have a jump dynamics, where jumps occur according to changes in the volatility process. We compare with the popular Heston stochastic volatility dynamics, and show that the Barndorff‐Nielsen and Shephard model provides a more flexible framework in describing commodity spot prices. An empirical example on UK spot data is included.     

MEAN‐VARIANCE POLICY FOR DISCRETE‐TIME CONE‐CONSTRAINED MARKETS: TIME CONSISTENCY IN EFFICIENCY AND THE MINIMUM‐VARIANCE SIGNED SUPERMARTINGALE MEASURE

The discrete‐time mean‐variance portfolio selection formulation, which is a representative of general dynamic mean‐risk portfolio selection problems, typically does not satisfy time consistency in efficiency (TCIE), i.e., a truncated precommitted efficient policy may become inefficient for the corresponding truncated problem. In this paper, we analytically investigate the effect of portfolio constraints on the TCIE of convex cone‐constrained markets. More specifically, we derive semi‐analytical expressions for the precommitted efficient mean‐variance policy and the minimum‐variance signed supermartingale measure (VSSM) and examine their relationship. Our analysis shows that the precommitted discrete‐time efficient mean‐variance policy satisfies TCIE if and only if the conditional expectation of the density of the VSSM (with respect to the original probability measure) is nonnegative, or once the conditional expectation becomes negative, it remains at the same negative value until the terminal time. Our finding indicates that the TCIE property depends only on the basic market setting, including portfolio constraints. This motivates us to establish a general procedure for constructing TCIE dynamic portfolio selection problems by introducing suitable portfolio constraints.     

COMPLEX LOGARITHMS IN HESTON‐LIKE MODELS

The characteristic functions of many affine jump‐diffusion models, such as Heston's stochastic volatility model and all of its extensions, involve multivalued functions such as the complex logarithm. If we restrict the logarithm to its principal branch, as is done in most software packages, the characteristic function can become discontinuous, leading to completely wrong option prices if options are priced by Fourier inversion. In this paper, we prove without any restrictions that there is a formulation of the characteristic function in which the principal branch is the correct one. Because this formulation is easier to implement and numerically more stable than the so‐called rotation count algorithm of Kahl and Jäckel, we solely focus on its stability in this paper. This paper shows how complex discontinuities can be avoided in the Variance Gamma and Schöbel–Zhu models, as well as in the exact simulation algorithm of the Heston model, recently proposed by Broadie and Kaya.     

PRICING DISCRETELY MONITORED BARRIER OPTIONS AND DEFAULTABLE BONDS IN LÉVY PROCESS MODELS: A FAST HILBERT TRANSFORM APPROACH

This paper presents a novel method to price discretely monitored single- and double-barrier options in Lévy process-based models. The method involves a sequential evaluation of Hilbert transforms of the product of the Fourier transform of the value function at the previous barrier monitoring date and the characteristic function of the (Esscher transformed) Lévy process. A discrete approximation with exponentially decaying errors is developed based on the Whittaker cardinal series (Sinc expansion) in Hardy spaces of functions analytic in a strip. An efficient computational algorithm is developed based on the fast Hilbert transform that, in turn, relies on the FFT-based Toeplitz matrix–vector multiplication. Our method also provides a natural framework for credit risk applications, where the firm value follows an exponential Lévy process and default occurs at the first time the firm value is below the default barrier on one of a discrete set of monitoring dates.     

INDIFFERENCE VALUATION OF MORTGAGE‐BACKED SECURITIES IN THE PRESENCE OF PREPAYMENT RISK

We present a utility‐based methodology for the valuation and the risk management of mortgage‐backed securities subject to totally unpredictable prepayment risk. Incompleteness stems from its embedded prepayment option which affects the security's cash flow pattern. The prepayment time is constructed via deterministic or stochastic hazard rate. The relevant indifference price consists of a linear term, corresponding to the remaining outstanding balance, and a nonlinear one that incorporates the investor's risk aversion and the interest payments generated by the mortgage contract. The indifference valuation approach is also extended to the case of homogeneous mortgage pools.     

AN ANALYTICAL SOLUTION FOR THE TWO‐SIDED PARISIAN STOPPING TIME,ITS ASYMPTOTICS,AND THE PRICING OF PARISIAN OPTIONS

In this paper, we obtain a recursive formula for the density of the two‐sided Parisian stopping time. This formula does not require any numerical inversion of Laplace transforms, and is similar to the formula obtained for the one‐sided Parisian stopping time derived in Dassios and Lim. However, when we study the tails of the two distributions, we find that the two‐sided stopping time has an exponential tail, while the one‐sided stopping time has a heavier tail. We derive an asymptotic result for the tail of the two‐sided stopping time distribution and propose an alternative method of approximating the price of the two‐sided Parisian option.