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.     

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.     

OPTIMAL INVESTMENT IN CREDIT DERIVATIVES PORTFOLIO UNDER CONTAGION RISK

We consider the optimal portfolio problem of a power investor who wishes to allocate her wealth between several credit default swaps (CDSs) and a money market account. We model contagion risk among the reference entities in the portfolio using a reduced‐form Markovian model with interacting default intensities. Using the dynamic programming principle, we establish a lattice dependence structure between the Hamilton‐Jacobi‐Bellman equations associated with the default states of the portfolio. We show existence and uniqueness of a classical solution to each equation and characterize them in terms of solutions to inhomogeneous Bernoulli type ordinary differential equations. We provide a precise characterization for the directionality of the CDS investment strategy and perform a numerical analysis to assess the impact of default contagion. We find that the increased intensity triggered by default of a very risky entity strongly impacts size and directionality of the investor strategy. Such findings outline the key role played by default contagion when investing in portfolios subject to multiple sources of default risk.     

Network valuation in financial systems

We introduce a general model for the balance‐sheet consistent valuation of interbank claims within an interconnected financial system. Our model represents an extension of clearing models of interdependent liabilities to account for the presence of uncertainty on banks' external assets. At the same time, it also provides a natural extension of classic structural credit risk models to the case of an interconnected system. We characterize the existence and uniqueness of a valuation that maximizes individual and total equity values for all banks. We apply our model to the assessment of systemic risk and in particular for the case of stress testing. Further, we provide a fixed‐point algorithm to carry out the network valuation and the conditions for its convergence.     

Does the More Risk‐Averse Investor hold a Less Risky Portfolio?*

We study the suitability of using absolute risk aversion as a measure of willingness to take risk in the Arrow–Debreu portfolio framework. We define a global measure of risk for the Arrow–Debreu portfolio, which is measured by the sensitivity of an individual's Arrow–Debreu portfolio payoff to the change in the market return. We call this measure ‘conservatism’ and show that the concept of ‘more conservative’ is stronger than that of ‘more risk‐averse.’ A higher absolute risk aversion is only necessary but not sufficient to induce a less risky Arrow–Debreu portfolio. Our results not only challenge the well‐accepted notion that a more risk‐averse investor holds a less risky portfolio, but also suggest a stronger measure – conservatism – for evaluating the riskiness of portfolio.     

CHOQUET PRICING FOR FINANCIAL MARKETS WITH FRICTIONS1

In markets where dealers play a central role, bid-ask spreads inhibit asset valuation as defined by the formation cost of a replicating portfolio. We introduce a nonlinear valuation formula similar to the usual expectation with respect to the risk-adjusted probability measure. This formula expresses the asset's selling and buying prices set by dealers as the Choquet integrals of their random payoffs We investigate several price puzzles: the violation of the put-call parity and the fact that the components of a security can sell at a premium to the underlying security (primes and scores).     

THE NUMÉRAIRE PROPERTY AND LONG‐TERM GROWTH OPTIMALITY FOR DRAWDOWN‐CONSTRAINED INVESTMENTS

We consider the portfolio choice problem for a long‐run investor in a general continuous semimartingale model. We combine the decision criterion of pathwise growth optimality with a flexible specification of attitude toward risk, encoded by a linear drawdown constraint imposed on admissible wealth processes. We define the constrained numéraire property through the notion of expected relative return and prove that drawdown‐constrained numéraire portfolio exists and is unique, but may depend on the investment horizon. However, when sampled at the times of its maximum and asymptotically as the time‐horizon becomes distant, the drawdown‐constrained numéraire portfolio is given explicitly through a model‐independent transformation of the unconstrained numéraire portfolio. The asymptotically growth‐optimal strategy is obtained as limit of numéraire strategies on finite horizons.     

PORTFOLIO OPTIMIZATION WITH JUMPS AND UNOBSERVABLE INTENSITY PROCESS

We consider a financial market with one bond and one stock. The dynamics of the stock price process allow jumps which occur according to a Markov-modulated Poisson process. We assume that there is an investor who is only able to observe the stock price process and not the driving Markov chain. The investor's aim is to maximize the expected utility of terminal wealth. Using a classical result from filter theory it is possible to reduce this problem with partial observation to one with complete observation. With the help of a generalized Hamilton–Jacobi–Bellman equation where we replace the derivative by Clarke's generalized gradient, we identify an optimal portfolio strategy. Finally, we discuss some special cases of this model and prove several properties of the optimal portfolio strategy. In particular, we derive bounds and discuss the influence of uncertainty on the optimal portfolio strategy.     

TRANSFORM ANALYSIS FOR POINT PROCESSES AND APPLICATIONS IN CREDIT RISK

This paper develops a formula for a transform of a vector point process with totally inaccessible arrivals. The transform is expressed in terms of a Laplace transform under an equivalent probability measure of the point process compensator. The Laplace transform of the compensator can be calculated explicitly for a wide range of model specifications, because it is analogous to the value of a simple security. The transform formula extends the computational tractability offered by extant security pricing models to a point process and its applications, which include valuation and risk management problems arising in single‐name and portfolio credit risk.     

ROBUST UTILITY MAXIMIZATION WITH LÉVY PROCESSES

We study a robust portfolio optimization problem under model uncertainty for an investor with logarithmic or power utility. The uncertainty is specified by a set of possible Lévy triplets, that is, possible instantaneous drift, volatility, and jump characteristics of the price process. We show that an optimal investment strategy exists and compute it in semi‐closed form. Moreover, we provide a saddle point analysis describing a worst‐case model.     

TRACTABLE ROBUST EXPECTED UTILITY AND RISK MODELS FOR PORTFOLIO OPTIMIZATION

Expected utility models in portfolio optimization are based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance, and support information. No additional restrictions on the type of distribution such as normality is made. The investor’s utility is modeled as a piecewise‐linear concave function. We derive exact and approximate optimal trading strategies for a robust (maximin) expected utility model, where the investor maximizes his worst‐case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Extensions of the model to capture asymmetry using partitioned statistics information and box‐type uncertainty in the mean and covariance matrix are provided. Using the optimized certainty equivalent framework, we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst‐case risk under distributional ambiguity. New closed‐form results for the worst‐case optimized certainty equivalent risk measures and optimal portfolios are provided for two‐ and three‐piece utility functions. For more complicated utility functions, computational experiments indicate that such robust approaches can provide good trading strategies in financial markets.     

Credit portfolio selection with decaying contagion intensities

We develop a fixed‐income portfolio framework capturing the exponential decay of contagious intensities between successive default events. We show that the value function of the control problem is the classical solution to a recursive system of second‐order uniformly parabolic Hamilton–Jacobi–Bellman partial differential equations. We analyze the interplay between risk premia, decay of default intensities, and their volatilities. Our comparative statics analysis finds that the investor chooses to go long only if he is capturing enough risk premia. If the default intensities deteriorate faster, the investor increases the size of his position if he goes short, or reduces the size of his position if he goes long.     

PORTFOLIO LIQUIDATION IN DARK POOLS IN CONTINUOUS TIME

We consider an illiquid financial market where a risk averse investor has to liquidate a portfolio within a finite time horizon [0, T] and can trade continuously at a traditional exchange (the “primary venue”) and in a dark pool. At the primary venue, trading yields a linear price impact. In the dark pool, no price impact costs arise but order execution is uncertain, modeled by a multidimensional Poisson process. We characterize the costs of trading by a linear‐quadratic functional which incorporates both the price impact costs of trading at the primary exchange and the market risk of the position. The solution of the cost minimization problem is characterized by a matrix differential equation with singular boundary condition; by means of stochastic control theory, we provide a verification argument. If a single‐asset position is to be liquidated, the investor slowly trades out of her position at the primary venue, with the remainder being placed in the dark pool at any point in time. For multi‐asset liquidations this is generally not the case; for example, it can be optimal to oversize orders in the dark pool in order to turn a poorly balanced portfolio into a portfolio bearing less risk.     

Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix

This paper studies a robust continuous‐time Markowitz portfolio selection problem where the model uncertainty affects the covariance matrix of multiple risky assets. This problem is formulated into a min–max mean‐variance problem over a set of nondominated probability measures that is solved by a McKean–Vlasov dynamic programming approach, which allows us to characterize the solution in terms of a Bellman–Isaacs equation in the Wasserstein space of probability measures. We provide explicit solutions for the optimal robust portfolio strategies and illustrate our results in the case of uncertain volatilities and ambiguous correlation between two risky assets. We then derive the robust efficient frontier in closed form, and obtain a lower bound for the Sharpe ratio of any robust efficient portfolio strategy. Finally, we compare the performance of Sharpe ratios for a robust investor and for an investor with a misspecified model.     

Developing the selection and valuation capabilities through learning: The case of corporate venture capital

The objective of this paper is to examine the impacts of experience intensity, experience diversity and acquisitive experience on the development of selection and valuation capabilities that help the parent (investor) company generate higher short-term financial returns and improve long-term strategic performance. Based on our analysis of 2110 cases of CVC investments in the VenureXpert data base, we find that industry diversity of a CVC program's experience is positively related to its selection of portfolio companies with relatively high financial potential. The CVC program's experience intensity, stage diversity of its experience, and syndication improve its selection of portfolio companies with greater strategic potential. In addition, stage diversity may enhance valuation capability. We also find that experience accumulation is more effective when a CVC program invests in a portfolio company in the later stage rather than in the early stage.     

Principal Component Value at Risk

Value at risk (VaR) is an industrial standard for monitoring financial risk in an investment portfolio. It measures potential losses within a given confidence interval. The implementation, calculation, and interpretation of VaR contains a wealth of mathematical issues that are not fully understood. In this paper we present a methodology for an approximation to value at risk that is based on the principal components of a sensitivity‐adjusted covariance matrix. The result is an explicit expression in terms of portfolio deltas, gammas, and the variance/covariance matrix. It can be viewed as a nonlinear extension of the linear model given by the delta‐normal VaR or RiskMetrics (J.P. Morgan, 1996).     

DERIVATIVE ASSET PRICING WITH TRANSACTION COSTS1

In the modern theory of finance, the valuation of derivative assets is commonly based on a replication argument. When there are transaction costs, this argument is no longer valid. In this paper, we try to address the general problem of finding the optimal portfolio among those which dominate a given derivative asset at maturity. We derive an interval for its price. the upper bound is the minimum amount one has to invest initially in order to obtain proceeds at least as valuable as the derivative asset. the lower bound is the maximum amount one can borrow initially against the proceeds of the derivative asset. We show that, in some instances, this interval may be strictly bounded above by the price of the replicating strategy. Prima facie, the cost of a dominating strategy should appear to be higher than that of the replicating one. But because trading is costly, it may pay to weigh the benefits of replication against those of potential savings on transaction costs.     

Vulnerable options,risky corporate bond,and credit spread

In the current literature, the focus of credit‐risk analysis has been either on the valuation of risky corporate bond and credit spread or on the valuation of vulnerable options, but never both in the same context. There are two main concerns with existing studies. First, corporate bonds and credit spreads are generally analyzed in a context where corporate debt is the only liability of the firm and a firm’s value follows a continuous stochastic process. This setup implies a zero short‐term spread, which is strongly rejected by empirical observations. The failure of generating non‐zero short‐term credit spreads may be attributed to the simplified assumption on corporate liabilities. Because a corporation generally has more than one type of liability, modeling multiple liabilities may help to incorporate discontinuity in a firm’s value and thereby lead to realistic credit term structures. Second, vulnerable options are generally valued under the assumption that a firm can fully pay off the option if the firm’s value is above the default barrier at the option’s maturity. Such an assumption is not realistic because a corporation can find itself in a solvent position at option’s maturity but with assets insufficient to pay off the option. The main contribution of this study is to address these concerns. The proposed framework extends the existing equity‐bond capital structure to an equity‐bond‐derivative setting and encompasses many existing models as special cases. The firm under study has two types of liabilities: a corporate bond and a short position in a call option. The risky corporate bond, credit spreads, and vulnerable options are analyzed and compared with their counterparts from previous models. Numerical results show that adding a derivative type of liability can lead to positive short‐term credit spreads and various shapes of credit‐spread term structures that were not possible in previous models. In addition, we found that vulnerable options need not always be worth less than their default‐free counterparts. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:301–327, 2001     

INCORPORATING RISK AND AMBIGUITY AVERSION INTO A HYBRID MODEL OF DEFAULT

It is well known that purely structural models of default cannot explain short‐term credit spreads, while purely intensity‐based models lead to completely unpredictable default events. Here we introduce a hybrid model of default, in which a firm enters a “distressed” state once its nontradable credit worthiness index hits a critical level. The distressed firm then defaults upon the next arrival of a Poisson process. To value defaultable bonds and credit default swaps (CDSs), we introduce the concept of robust indifference pricing. This paradigm incorporates both risk aversion and model uncertainty. In robust indifference pricing, the optimization problem is modified to include optimizing over a set of candidate measures, in addition to optimizing over trading strategies, subject to a measure dependent penalty. Using our model and valuation framework, we derive analytical solutions for bond yields and CDS spreads, and find that while ambiguity aversion plays a similar role to risk aversion, it also has distinct effects. In particular, ambiguity aversion allows for significant short‐term spreads.     

Valuation Bounds on Barrier Options Under Model Uncertainty

This article investigates valuation bounds on barrier options under model uncertainty. This investigation enriches the literature on the model‐free valuation of these exotic options. It is found that with weak assumptions on underlying price processes, tight valuation bounds on barrier options can be sought from a set of European options. As a result, the numerical routine developed in this article can be reviewed as a new method for the evaluation of barrier options, which is independent of model assumptions. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:199–234, 2013