This paper develops the procedure of multivariate subordination for a collection of independent Markov processes with killing. Starting from *d* independent Markov processes with killing and an independent *d*-dimensional time change , we construct a new process by time, changing each of the Markov processes with a coordinate . When is a *d*-dimensional Lévy subordinator, the time changed process is a time-homogeneous Markov process with state-dependent jumps and killing in the product of the state spaces of . The dependence among jumps of its components is governed by the *d*-dimensional Lévy measure of the subordinator. When is a *d*-dimensional additive subordinator, *Y* is a time-inhomogeneous Markov process. When with forming a multivariate Markov process, is a Markov process, where each plays a role of stochastic volatility of . This construction provides a rich modeling architecture for building multivariate models in finance with time- and state-dependent jumps, stochastic volatility, and killing (default). The semigroup theory provides powerful analytical and computational tools for securities pricing in this framework. To illustrate, the paper considers applications to multiname unified credit-equity models and correlated commodity models.

In this paper we ask whether, given a stock market and an illiquid derivative, there exists arbitrage-free prices at which a utility-maximizing agent would always want to buy the derivative, irrespectively of his own initial endowment of derivatives and cash. We prove that this is false for any given investor if one considers *all* initial endowments with finite utility, and that it can instead be true if one restricts to the endowments in the interior. We show, however, how the endowments on the boundary can give rise to very odd phenomena; for example, an investor with such an endowment would choose not to trade in the derivative even at prices arbitrarily close to some arbitrage price.

This paper deals with multidimensional dynamic risk measures induced by conditional *g*-expectations. A notion of multidimensional *g*-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem, and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, and translatability properties of multidimensional conditional *g*-expectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic *g*-risk measure is nonincreasingly convex if and only if the generator *g* satisfies a quasi-monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex *g*-risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacting subsidiaries; optimal risk sharing for -tolerant *g*-risk measures, and risk contribution for coherent *g*-risk measures are investigated. Insurance *g*-risk measure and other ways to induce *g*-risk measures are also studied at the end of the paper.

This paper discusses the problem of hedging not perfectly replicable contingent claims using the numéraire portfolio. The proposed concept of benchmarked risk minimization leads beyond the classical no-arbitrage paradigm. It provides in incomplete markets a generalization of the pricing under classical risk minimization, pioneered by Föllmer, Sondermann, and Schweizer. The latter relies on a quadratic criterion, requests square integrability of claims and gains processes, and relies on the existence of an equivalent risk-neutral probability measure. Benchmarked risk minimization avoids these restrictive assumptions and provides symmetry with respect to all primary securities. It employs the real-world probability measure and the numéraire portfolio to identify the minimal possible price for a contingent claim. Furthermore, the resulting benchmarked (i.e., numéraire portfolio denominated) profit and loss is only driven by uncertainty that is orthogonal to benchmarked-traded uncertainty, and forms a local martingale that starts at zero. Consequently, sufficiently different benchmarked profits and losses, when pooled, become asymptotically negligible through diversification. This property makes benchmarked risk minimization the least expensive method for pricing and hedging diversified pools of not fully replicable benchmarked contingent claims. In addition, when hedging it incorporates evolving information about nonhedgeable uncertainty, which is ignored under classical risk minimization.

We provide conditions on a one-period-two-date pure exchange economy with rank-dependent utility agents under which Arrow–Debreu equilibria exist. When such an equilibrium exists, we show that the state-price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.

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 *d*_{2} 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.

We give a general formulation of the utility maximization problem under nondominated model uncertainty in discrete time and show that an optimal portfolio exists for any utility function that is bounded from above. In the unbounded case, integrability conditions are needed as nonexistence may arise even if the value function is finite.

Classical put–call symmetry relates the price of puts and calls under a suitable dual market transform. One well-known application is the semistatic hedging of path-dependent barrier options with European options. This, however, in its classical form requires the price process to observe rather stringent and unrealistic symmetry properties. In this paper, we develop a general self-duality theorem to develop valuation schemes for barrier options in stochastic volatility models with correlation.

We study an optimal control problem related to swing option pricing in a general non-Markovian setting in continuous time. As a main result we uniquely characterize the value process in terms of a first-order nonlinear backward stochastic partial differential equation and a differential inclusion. Based on this result we also determine the set of optimal controls and derive a dual minimization problem.

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.

Hedge fund managers receive a large fraction of their funds' profits, paid when funds exceed their high-water marks. We study the incentives of such performance fees. A manager with long-horizon, constant investment opportunities and relative risk aversion, chooses a constant Merton portfolio. However, the effective risk aversion shrinks toward one in proportion to performance fees. Risk shifting implications are ambiguous and depend on the manager's own risk aversion. Managers with equal investment opportunities but different performance fees and risk aversions may coexist in a competitive equilibrium. The resulting leverage increases with performance fees—a prediction that we confirm empirically.

It is well known from the work of Schönbucher that the marginal laws of a loss process can be matched by a unit increasing time inhomogeneous Markov process, whose deterministic jump intensity is called local intensity. The stochastic local intensity (SLI) models such as the one proposed by Arnsdorf and Halperin allow to get a stochastic jump intensity while keeping the same marginal laws. These models involve a nonlinear stochastic differential equation (SDE) with jumps. The first contribution of this paper is to prove the existence and uniqueness of such processes. This is made by means of an interacting particle system, whose convergence rate toward the nonlinear SDE is analyzed. Second, this approach provides a powerful way to compute pathwise expectations with the SLI model: we show that the computational cost is roughly the same as a crude Monte Carlo algorithm for standard SDEs.

We consider the pricing of American put options in a model-independent setting: that is, we do not assume that asset prices behave according to a given model, but aim to draw conclusions that hold in any model. We incorporate market information by supposing that the prices of European options are known.

In this setting, we are able to provide conditions on the American put prices which are necessary for the absence of arbitrage. Moreover, if we further assume that there are finitely many European and American options traded, then we are able to show that these conditions are also sufficient. To show sufficiency, we construct a model under which both American and European options are correctly priced at all strikes simultaneously. In particular, we need to carefully consider the optimal stopping strategy in the construction of our process.

For longer horizons, assuming no dividend distributions, models for discounted stock prices in balanced markets are formulated as conditional expectations of nontrivial terminal random variables defined at infinity. Observing that extant models fail to have this property, new models are proposed. The new concept of a balanced market proposed here permits a distinction between such markets and unduly optimistic or pessimistic ones. A tractable example is developed and termed the discounted variance gamma model. Calibrations to market data provide empirical support. Additionally, procedures are presented for the valuation of path dependent stochastic perpetuities. Evidence is provided for long dated equity linked claims paying coupon for time spent by equity above a lower barrier, being underpriced by extant models relative to the new discounted ones. Given the popularity of such claims, the resulting mispricing could possibly take some corrections. Furthermore for these new discounted models, implied volatility curves do not flatten out at the larger maturities.

We consider a sequence of financial markets that converges weakly in a suitable sense and maximize a behavioral preference functional in each market. For expected *concave* utilities, it is well known that the maximal expected utilities and the corresponding final positions converge to the corresponding quantities in the limit model. We prove similar results for nonconcave utilities and distorted expectations as employed in behavioral finance, and we illustrate by a counterexample that these results require a stronger notion of convergence of the underlying models compared to the *concave* utility maximization. We use the results to analyze the stability of behavioral portfolio selection problems and to provide numerically tractable methods to solve such problems in complete continuous-time models.

We propose to interpret distribution model risk as sensitivity of expected loss to changes in the risk factor distribution, and to measure the distribution model risk of a portfolio by the maximum expected loss over a set of plausible distributions defined in terms of some divergence from an estimated distribution. The divergence may be relative entropy or another *f*-divergence or Bregman distance. We use the theory of minimizing convex integral functionals under moment constraints to give formulae for the calculation of distribution model risk and to explicitly determine the worst case distribution from the set of plausible distributions. We also evaluate related risk measures describing divergence preferences.

It is said that risky asset *h* *acceptance dominates* risky asset *k* if any decision maker who rejects the investment in *h* also rejects the investment in *k*. While in general acceptance dominance is a partial order, we show that it becomes a complete order if only infinitely short investment time horizons are considered. Two indices that induce different variants of this order are proposed, absolute acceptance dominance and relative acceptance dominance, and their properties are studied. We then show that many indices of riskiness that are compatible with the acceptance dominance order coincide with our indices in the continuous-time setup.

We derive rigorous asymptotic results for the magnitude of contagion in a large counterparty network and give an analytical expression for the asymptotic fraction of defaults, in terms of network characteristics. Our results extend previous studies on contagion in random graphs to inhomogeneous-directed graphs with a given degree sequence and arbitrary distribution of weights. We introduce a criterion for the resilience of a large financial network to the insolvency of a small group of financial institutions and quantify how contagion amplifies small shocks to the network. Our results emphasize the role played by “contagious links” and show that institutions which contribute most to network instability have both large connectivity and a large fraction of contagious links. The asymptotic results show good agreement with simulations for networks with realistic sizes.

]]>We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short-end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model-independent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on short-maturity option prices.

]]>The objective of this paper is to study the mean–variance portfolio optimization in continuous time. Since this problem is time inconsistent we attack it by placing the problem within a game theoretic framework and look for subgame perfect Nash equilibrium strategies. This particular problem has already been studied in Basak and Chabakauri where the authors assumed a constant risk aversion parameter. This assumption leads to an equilibrium control where the dollar amount invested in the risky asset is independent of current wealth, and we argue that this result is unrealistic from an economic point of view. In order to have a more realistic model we instead study the case when the risk aversion depends dynamically on current wealth. This is a substantially more complicated problem than the one with constant risk aversion but, using the general theory of time-inconsistent control developed in Björk and Murgoci, we provide a fairly detailed analysis on the general case. In particular, when the risk aversion is inversely proportional to wealth, we provide an analytical solution where the equilibrium dollar amount invested in the risky asset is proportional to current wealth. The equilibrium for this model thus appears more reasonable than the one for the model with constant risk aversion.

]]>We consider evaluation methods for payoffs with an inherent financial risk as encountered for instance for portfolios held by pension funds and insurance companies. Pricing such payoffs in a way consistent to market prices typically involves combining actuarial techniques with methods from mathematical finance. We propose to extend standard actuarial principles by a new market-consistent evaluation procedure which we call “two-step market evaluation.” This procedure preserves the structure of standard evaluation techniques and has many other appealing properties. We give a complete axiomatic characterization for two-step market evaluations. We show further that in a dynamic setting with continuous stock prices every evaluation which is time-consistent and market-consistent is a two-step market evaluation. We also give characterization results and examples in terms of *g*-expectations in a Brownian-Poisson setting.

Dynamic capital structure models with roll-over debt rely on widely accepted arguments that have never been formalized. This paper clarifies the literature and provides a rigorous formulation of the equity holders’ decision problem within a game theory framework. We spell out the linkage between default policies in a rational expectations equilibrium and optimal stopping theory. We prove that there exists a unique equilibrium in constant barrier strategies, which coincides with that derived in the literature. Furthermore, that equilibrium is the unique equilibrium when the firm loses all its value at default time. Whether the result holds when there is a recovery at default remains a conjecture.

]]>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.

]]>We develop an arbitrage-free valuation framework for bilateral counterparty risk, where collateral is included with possible rehypothecation. We show that the adjustment is given by the sum of two option payoff terms, where each term depends on the netted exposure, i.e., the difference between the on-default exposure and the predefault collateral account. We then specialize our analysis to credit default swaps (CDS) as underlying portfolios, and construct a numerical scheme to evaluate the adjustment under a doubly stochastic default framework. In particular, we show that for CDS contracts a perfect collateralization cannot be achieved, even under continuous collateralization, if the reference entity’s and counterparty’s default times are dependent. The impact of rehypothecation, collateral margining frequency, and default correlation-induced contagion is illustrated with numerical examples.

]]>We prove that in a discrete-time market model the lower arbitrage bound of an American contingent claim is itself an arbitrage-free price if and only if it corresponds to the price of the claim optimally exercised under some equivalent martingale measure.

]]>We introduce a new approach for the numerical pricing of American options. The main idea is to choose a finite number of suitable excessive functions (randomly) and to find the smallest majorant of the gain function in the span of these functions. The resulting problem is a linear semi-infinite programming problem, that can be solved using standard algorithms. This leads to good upper bounds for the original problem. For our algorithms no discretization of space and time and no simulation is necessary. Furthermore it is applicable even for high-dimensional problems. The algorithm provides an approximation of the value not only for one starting point, but for the complete value function on the continuation set, so that the optimal exercise region and, for example, the Greeks can be calculated. We apply the algorithm to (one- and) multidimensional diffusions and show it to be fast and accurate.

]]>In this paper, having been inspired by the work of Kunita and Seko, we study the pricing of δ-penalty game call options on a stock with a dividend payment. For the perpetual case, our result reveals that the optimal stopping region for the option seller depends crucially on the dividend rate *d*. More precisely, we show that when the penalty δ is small, there are two critical dividends 0 < *d*_{1} < *d*_{2} < ∞ such that the optimal stopping region for the option seller takes one of the following forms: (1) an interval if *d* < *d*_{1}; (2) a singleton if *d*∈ [*d*_{1}, *d*_{2}]; or (3) an empty set if *d* > *d*_{2}. When *d*∈ [*d*_{1}, *d*_{2}], the value function is not continuously differentiable at the optimal stopping boundary for the option seller, therefore our result in the perpetual case cannot be established by the free boundary approach with smooth-fit conditions imposed on both free boundaries. For the finite time horizon case, the dependence of the optimal stopping region for the option seller on the time to maturity is exhibited; more precisely, when both δ and *d* are small, we show that there are two critical times 0 < *T*_{1} < *T*_{2} < *T*, such that the optimal stopping region for the option seller takes one of the following forms: (1) an interval if *t* < *T*_{1}; (2) a singleton if *t*∈ [*T*_{1}, *T*_{2}]; or (3) an empty set if *t* > *T*_{2}. In summary, for both the perpetual and the finite horizon cases, we characterize in terms of model parameters how the optimal stopping region for the option seller shrinks when the dividend rate *d* increases and the time to maturity decreases; these results complete the original work of Emmerling for the perpetual case and Kunita and Seko for the finite maturity case. In addition, for the finite time horizon case, we also extend the probabilistic method for the establishment of existence and regularity results of the classical American option pricing problem to the game option setting. Finally, we characterize the pair of optimal stopping boundaries for both the seller and the buyer as the unique pair of solutions to a couple of integral equations and provide numerical illustrations.