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.

]]>The classic approach to modeling financial markets consists of four steps. First, one fixes a currency unit. Second, one describes in that unit the evolution of financial assets by a stochastic process. Third, one chooses in that unit a numéraire, usually the price process of a positive asset. Fourth, one divides the original price process by the numéraire and considers the class of admissible strategies for trading. This approach has one fundamental drawback: Almost all concepts, definitions, and results, including no-arbitrage conditions like NA, NFLVR, and NUPBR depend *by their very definition*, at least formally, on initial choices of a currency unit and a numéraire. In this paper, we develop a new framework for modeling financial markets, *which is not based on ex-ante choices of a currency unit and a numéraire*. In particular, we introduce a “numéraire-independent” notion of no-arbitrage and derive its dual characterization. This yields a numéraire-independent version of the fundamental theorem of asset pricing (FTAP). We also explain how the classic approach and other recent approaches to modeling financial markets and studying no-arbitrage can be embedded in our framework.

In this paper, we examine irreversible investment decisions in duopoly games with a variable economic climate. Integrating timing flexibility, competition, and changes in the economic environment in the form of a cash flow process with regime switching, the problem is formulated as a stopping-time game under Stackelberg leader-follower competition, in which both players determine their respective optimal market entry time. By extending the variational inequality approach, we solve for the free boundaries and obtain optimal investment strategies for each player. Despite the lack of regularity in the leader's obstacle and the cash flow regime uncertainty, the regime-dependent optimal policies for both the leader and the follower are obtained. In addition, we perform comprehensive numerical experiments to demonstrate the properties of solutions and to gain insights into the implications of regime switching.

]]>We consider an optimal investment problem with intermediate consumption and random endowment, in an incomplete semimartingale model of the financial market. We establish the key assertions of the utility maximization theory, assuming that both primal and dual value functions are finite in the interiors of their domains and that the random endowment at maturity can be dominated by the terminal value of a self-financing wealth process. In order to facilitate the verification of these conditions, we present alternative, but equivalent conditions, under which the conclusions of the theory hold.

]]>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.

]]>When the planning horizon is long, and the safe asset grows indefinitely, isoelastic portfolios are nearly optimal for investors who are close to isoelastic for high wealth, and not too risk averse for low wealth. We prove this result in a general arbitrage-free, frictionless, semimartingale model. As a consequence, optimal portfolios are robust to the perturbations in preferences induced by common option compensation schemes, and such incentives are weaker when their horizon is longer. Robust option incentives are possible, but require several, arbitrarily large exercise prices, and are not always convex.

]]>In some options markets (e.g., commodities), options are listed with only a single maturity for each underlying. In others (e.g., equities, currencies), options are listed with multiple maturities. In this paper, we analyze a special class of pure jump Markov martingale models and provide an algorithm for calibrating such models to match the market prices of European options with multiple strikes and maturities. This algorithm matches option prices exactly and only requires solving several one-dimensional root-search problems and applying elementary functions. We show how to construct a time-homogeneous process which meets a single smile, and a piecewise time-homogeneous process which can meet multiple smiles.

]]>Lions and Musiela give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert and Mijatović and Urusov give necessary and sufficient conditions in the case of perfect correlation (). For financial applications, such as checking the martingale property of the stock price process in correlated stochastic volatility models, we extend their work to the arbitrary correlation case (). We give a complete classification of the convergence properties of both perpetual and capped integral functionals of time-homogeneous diffusions and generalize results in Mijatović and Urusov with direct proofs avoiding the use of *separating times* (concept introduced by Cherny and Urusov and extensively used in the proofs of Mijatović and Urusov).

It is well known that, under a continuity assumption on the price of a stock *S*, the realized variance of *S* for maturity *T* can be replicated by a portfolio of calls and puts maturing at *T*. This paper assumes that call prices on *S* maturing at *T* are known for all strikes but makes no continuity assumptions on *S*. We derive semiexplicit expressions for the supremum lower bound on the hedged payoff, at maturity *T*, of a long position in the realized variance of *S*. Equivalently, is the supremum strike *K* such that an investor with a long position in a variance swap with strike *K* can ensure a nonnegative payoff at *T*. We study examples with constant implied volatilities and with a volatility skew. In our examples, is close to the fair variance strike obtained under the continuity assumption.

We develop two novel approaches to solving for the Laplace transform of a time-changed stochastic process. We discard the standard assumption that the background process () is Lévy. Maintaining the assumption that the business clock () and the background process are independent, we develop two different series solutions for the Laplace transform of the time-changed process . In fact, our methods apply not only to Laplace transforms, but more generically to expectations of smooth functions of random time. We apply the methods to introduce stochastic time change to the standard class of default intensity models of credit risk, and show that stochastic time-change has a very large effect on the pricing of deep out-of-the-money options on credit default swaps.

]]>Since risky positions in multivariate portfolios can be offset by various choices of capital requirements that depend on the exchange rules and related transaction costs, it is natural to assume that the risk measures of random vectors are set-valued. Furthermore, it is reasonable to include the exchange rules in the argument of the risk measure and so consider risk measures of set-valued portfolios. This situation includes the classical Kabanov's transaction costs model, where the set-valued portfolio is given by the sum of a random vector and an exchange cone, but also a number of further cases of additional liquidity constraints. We suggest a definition of the risk measure based on calling a set-valued portfolio acceptable if it possesses a selection with all individually acceptable marginals. The obtained selection risk measure is coherent (or convex), law invariant, and has values being upper convex closed sets. We describe the dual representation of the selection risk measure and suggest efficient ways of approximating it from below and from above. In the case of Kabanov's exchange cone model, it is shown how the selection risk measure relates to the set-valued risk measures considered by Kulikov (, *Theory Probab. Appl*. 52, 614–635), Hamel and Heyde (, *SIAM J. Financ. Math*. 1, 66–95), and Hamel, Heyde, and Rudloff (, *Math. Financ. Econ*. 5, 1–28).

This paper studies the price-setting problem of market makers under risk neutrality and perfect competition in continuous time. The classic approach of Glosten–Milgrom is followed. Bid and ask prices are defined as conditional expectations of a true value of the asset given the market makers' partial information that includes the customers' trading decisions. The true value is modeled as a Markov process that can be observed by the customers with some noise at Poisson times. A mathematically rigorous analysis of the price-setting problem is carried out, solving a filtering problem with endogenous filtration that depends on the bid and ask price processes quoted by the market maker. The existence and uniqueness of the bid and ask price processes is shown under some conditions.

]]>In this paper, we propose a sensitivity-based analysis to study the nonlinear behavior under nonexpected utility with probability distortions (or “distorted utility” for short). We first discover the “monolinearity” of distorted utility, which means that after properly changing the underlying probability measure, distorted utility becomes locally linear in probabilities, and the derivative of distorted utility is simply an expectation of the sample path derivative under the new measure. From the monolinearity property, simulation algorithms for estimating the derivative of distorted utility can be developed, leading to gradient-based search algorithms for the optimum of distorted utility. We then apply the sensitivity-based approach to the portfolio selection problem under distorted utility with complete and incomplete markets. For the complete markets case, the first-order condition is derived and optimal wealth deduced. For the incomplete markets case, a dual characterization of optimal policies is provided; a solvable incomplete market example with unhedgeable interest rate risk is also presented. We expect this sensitivity-based approach to be generally applicable to optimization problems involving probability distortions.

]]>The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile-based risk measures such as value at risk are elicitable. In this paper, the existing result of the nonelicitability of expected shortfall is extended to all law-invariant spectral risk measures unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law-invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles.

]]>We propose a fast and accurate numerical method for pricing European swaptions in multifactor Gaussian term structure models. Our method can be used to accelerate the calibration of such models to the volatility surface. The pricing of an interest rate option in such a model involves evaluating a multidimensional integral of the payoff of the claim on a domain where the payoff is positive. In our method, we approximate the exercise boundary of the state space by a hyperplane tangent to the maximum probability point on the boundary and simplify the multidimensional integration into an analytical form. The maximum probability point can be determined using the gradient descent method. We demonstrate that our method is superior to previous methods by comparing the results to the price obtained by numerical integration.

]]>To assure price admissibility—that all bond prices, yields, and forward rates remain positive—we show how to control the state variables within the class of arbitrage-free linear price function models for the evolution of interest rate yield curves over time. Price admissibility is necessary to preclude cash-and-carry arbitrage, a market imperfection that can happen even with a risk-neutral diffusion process and positive bond prices. We assure price admissibility by (i) defining the state variables to be scaled partial sums of weighted coefficients of the exponential terms in the bond pricing function, (ii) identifying a simplex within which these state variables remain price admissible, and (iii) choosing a general functional form for the diffusion that selectively diminishes near the simplex boundary. By assuring that prices, yields, and forward rates remain positive with tractable diffusions for the physical and risk-neutral measures, an obstacle is removed from the wider acceptance of interest rate methods that are linear in prices.

]]>Many investment models in discrete or continuous-time settings boil down to maximizing an objective of the quantile function of the decision variable. This quantile optimization problem is known as the quantile formulation of the original investment problem. Under certain monotonicity assumptions, several schemes to solve such quantile optimization problems have been proposed in the literature. In this paper, we propose a change-of-variable and relaxation method to solve the quantile optimization problems without using the calculus of variations or making any monotonicity assumptions. The method is demonstrated through a portfolio choice problem under rank-dependent utility theory (RDUT). We show that this problem is equivalent to a classical Merton's portfolio choice problem under expected utility theory with the same utility function but a different pricing kernel explicitly determined by the given pricing kernel and probability weighting function. With this result, the feasibility, well-posedness, attainability, and uniqueness issues for the portfolio choice problem under RDUT are solved. It is also shown that solving functional optimization problems may reduce to solving probabilistic optimization problems. The method is applicable to general models with law-invariant preference measures including portfolio choice models under cumulative prospect theory (CPT) or RDUT, Yaari's dual model, Lopes' SP/A model, and optimal stopping models under CPT or RDUT.

]]>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.

]]>We propose a tractable framework for quantifying the impact of loss-triggered fire sales on portfolio risk, in a multi-asset setting. We derive analytical expressions for the impact of fire sales on the realized volatility and correlations of asset returns in a fire sales scenario and show that our results provide a quantitative explanation for the spikes in volatility and correlations observed during such deleveraging episodes. These results are then used to develop an econometric framework for the forensic analysis of fire sales episodes, using observations of market prices. We give conditions for the identifiability of model parameters from time series of asset prices, propose a statistical test for the presence of fire sales, and an estimator for the magnitude of fire sales in each asset class. Pathwise consistency and large sample properties of the estimator are studied in the high-frequency asymptotic regime. We illustrate our methodology by applying it to the forensic analysis of two recent deleveraging episodes: the Quant Crash of August 2007 and the Great Deleveraging following the default of Lehman Brothers in Fall 2008.

]]>This paper studies stability of the exponential utility maximization when there are small variations on agent's utility function. Two settings are considered. First, in a general semimartingale model where random endowments are present, a sequence of utilities defined on converges to the exponential utility. Under a uniform condition on their marginal utilities, convergence of value functions, optimal payoffs, and optimal investment strategies are obtained, their rate of convergence is also determined. Stability of utility-based pricing is studied as an application. Second, a sequence of utilities defined on converges to the exponential utility after shifting and scaling. Their associated optimal strategies, after appropriate scaling, converge to the optimal strategy for the exponential hedging problem. This complements Theorem 3.2 in [Nutz, M. (): Risk aversion asymptotics for power utility maximization. *Probab. Theory & Relat. Fields* 152, 703–749], which establishes the convergence for a sequence of power utilities.

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.

This paper discusses the gambling contest introduced in Seel and Strack (, Gambling in Contests, *Journal of Economic Theory*, 148(5), 2033–2048) and considers the impact of adding a penalty associated with failure to follow a winning strategy. The Seel and Strack model consists of *n*-agents each of whom privately observes a transient diffusion process and chooses when to stop it. The player with the highest stopped value wins the contest, and each player's objective is to maximize her probability of winning the contest. We give a new derivation of the results of Seel and Strack based on a Lagrangian approach. Moreover, we consider an extension of the problem to a behavioral finance context in the sense of regret theory. In particular, an agent is penalized when her chosen strategy does not win the contest, but there existed an alternative strategy that would have resulted in victory.

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 classical literature on optimal liquidation, rooted in Almgren–Chriss models, tackles the optimal liquidation problem using a trade-off between market impact and price risk. It answers the general question of optimal scheduling but the very question of the actual way to proceed with liquidation is rarely dealt with. Our model, which incorporates both price risk and nonexecution risk, is an attempt to tackle this question using limit orders. The very general framework we propose to model liquidation with limit orders generalizes existing ones in two ways. We consider a risk-averse agent, whereas the model of Bayraktar and Ludkovski only tackles the case of a risk-neutral one. We consider very general functional forms for the execution process intensity, whereas Guéant, Lehalle and Fernandez-Tapia are restricted to exponential intensity. Eventually, we link the execution cost function of Almgren–Chriss models to the intensity function in our model, providing then a way to see Almgren–Chriss models as a limit of ours.

In this paper, we investigate a method based on risk minimization to hedge observable but nontradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the conditional value-at-risk (CVaR) using three main tools: a stochastic approximation algorithm, optimal quantization, and variance reduction techniques (importance sampling and linear control variable), as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till maturity. We devise a stochastic approximation algorithm and study its a.s. convergence and weak convergence rate. Then, we extend our approach to the dynamic case under the assumption that the process modeling the nontradable source of risk and financial assets prices is Markovian. Finally, we illustrate our approach by considering several portfolios in connection with energy markets.

]]>We consider the optimal liquidation of a position of stock (long or short) where trading has a temporary market impact on the price. The aim is to minimize both the mean and variance of the order slippage with respect to a benchmark given by the market volume-weighted average price (VWAP). In this setting, we introduce a new model for the relative volume curve which allows simultaneously for accurate data fit, economic justification, and mathematical tractability. Tackling the resulting optimization problem using a stochastic control approach, we derive and solve the corresponding Hamilton–Jacobi–Bellman equation to give an explicit characterization of the optimal trading rate and liquidation trajectory.

]]>We study a class of optimization problems involving linked recursive preferences in a continuous-time Brownian setting. Such links can arise when preferences depend directly on the level or volatility of wealth, in principal–agent (optimal compensation) problems with moral hazard, and when the impact of social influences on preferences is modeled via utility (and utility diffusion) externalities. We characterize the necessary first-order conditions, which are also sufficient under additional conditions ensuring concavity. We also examine applications to optimal consumption and portfolio choice, and applications to Pareto optimal allocations.

]]>For an investor with constant absolute risk aversion and a long horizon, who trades in a market with constant investment opportunities and small proportional transaction costs, we obtain explicitly the optimal investment policy, its implied welfare, liquidity premium, and trading volume. We identify these quantities as the limits of their isoelastic counterparts for high levels of risk aversion. The results are robust with respect to finite horizons, and extend to multiple uncorrelated risky assets. In this setting, we study a Stackelberg equilibrium, led by a risk-neutral, monopolistic market maker who sets the spread as to maximize profits. The resulting endogenous spread depends on investment opportunities only, and is of the order of a few percentage points for realistic parameter values.

]]>Execution traders know that market impact greatly depends on whether their orders lean with or against the market. We introduce the OEH model, which incorporates this fact when determining the optimal trading horizon for an order, an input required by many sophisticated execution strategies. This model exploits the trader's private information about her trade's side and size, and how it will shift the prevailing order flow. From a theoretical perspective, OEH explains why market participants may rationally “dump” their orders in an increasingly illiquid market. We argue that trade side and order imbalance are key variables needed for modeling market impact functions, and their dismissal may be the reason behind the apparent disagreement in the literature regarding the functional form of the market impact function. We show that in terms of its information ratio OEH performs better than participation rate schemes and VWAP strategies. Our backtests suggest that OEH contributes substantial “execution alpha” for a wide variety of futures contracts. An implementation of OEH is provided in Python language.

]]>We propose a rank-dependent portfolio choice model in continuous time that captures the role in decision making of three emotions: hope, fear, and aspirations. Hope and fear are modeled through an inverse-S shaped probability weighting function and aspirations through a probabilistic constraint. By employing the recently developed approach of quantile formulation, we solve the portfolio choice problem both thoroughly and analytically. These solutions motivate us to introduce a fear index, a hope index, and a lottery-likeness index to quantify the impacts of three emotions, respectively, on investment behavior. We find that a sufficiently high level of fear endogenously necessitates portfolio insurance. On the other hand, hope is reflected in the agent's perspective on good states of the world: a higher level of hope causes the agent to include more scenarios under the notion of good states and leads to greater payoffs in sufficiently good states. Finally, an exceedingly high level of aspirations results in the construction of a lottery-type payoff, indicating that the agent needs to enter into a pure gamble in order to achieve his goal. We also conduct numerical experiments to demonstrate our findings.

]]>We propose a framework to study optimal trading policies in a one-tick pro rata limit order book, as typically arises in short-term interest rate futures contracts. The high-frequency trader chooses to post either market orders or limit orders, which are represented, respectively, by impulse controls and regular controls. We discuss the consequences of the two main features of this microstructure: first, the limit orders are only partially executed, and therefore she has no control on the executed quantity. Second, the high-frequency trader faces the overtrading risk, which is the risk of large variations in her inventory. The consequences of this risk are investigated in the context of optimal liquidation. The optimal trading problem is studied by stochastic control and dynamic programming methods, and we provide the associated numerical resolution procedure and prove its convergence. We propose dimension reduction techniques in several cases of practical interest. We also detail a high-frequency trading strategy in the case where a (predictive) directional information on the price is available. Each of the resulting strategies is illustrated by numerical tests.

]]>Motivated by analytical valuation of timer options (an important innovation in realized variance-based derivatives), we explore their novel mathematical connection with stochastic volatility and Bessel processes (with constant drift). Under the Heston (1993) stochastic volatility model, we formulate the problem through a first-passage time problem on realized variance, and generalize the standard risk-neutral valuation theory for fixed maturity options to a case involving random maturity. By time change and the general theory of Markov diffusions, we characterize the joint distribution of the first-passage time of the realized variance and the corresponding variance using Bessel processes with drift. Thus, explicit formulas for a useful joint density related to Bessel processes are derived via Laplace transform inversion. Based on these theoretical findings, we obtain a Black–Scholes–Merton-type formula for pricing timer options, and thus extend the analytical tractability of the Heston model. Several issues regarding the numerical implementation are briefly discussed.

]]>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.

]]>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.

The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is nondominated. We propose studying this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power, and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally, several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models.

]]>Considering a positive portfolio diffusion *X* with negative drift, we investigate optimal stopping problems of the form

where *f* is a nonincreasing function, *τ* is the next random time where the portfolio *X* crosses zero and *θ* is any stopping time smaller than *τ*. Hereby, our motivation is the obtention of an optimal selling strategy minimizing the relative distance between the liquidation value of the portfolio and its highest possible value before it reaches zero. This paper unifies optimal selling rules observed for the quadratic absolute distance criteria in this stationary framework with bang–bang type ones observed for monetary invariant criteria but in finite horizon. More precisely, we provide a verification result for the general stopping problem of interest and derive the exact solution for two classical criteria *f* of the literature. For the power utility criterion with , instantaneous selling is always optimal, which is consistent with previous observations for the Black-Scholes model in finite observation. On the contrary, for a relative quadratic error criterion, , selling is optimal as soon as the process *X* crosses a specified function *φ* of its running maximum . These results reinforce the idea that optimal stopping problems of similar type lead easily to selling rules of very different nature. Nevertheless, our numerical experiments suggest that the practical optimal selling rule for the relative quadratic error criterion is in fact very close to immediate selling.

We consider a continuous-time framework featuring a central bank, private agents, and a financial market. The central bank's objective is to maximize a functional, which measures the classical trade-off between output and inflation over time plus income from the sales of inflation-indexed bonds minus payments for the liabilities that the inflation-indexed bonds produce at maturity. Private agents are assumed to have adaptive expectations. The financial market is modeled in continuous-time Black–Scholes–Merton style and financial agents are averse against inflation risk, attaching an inflation risk premium to nominal bonds. Following this route, we explain demand for inflation-indexed securities on the financial market from a no-arbitrage assumption and derive pricing formulas for inflation-linked bonds and calls, which lead to a supply-demand equilibrium. Furthermore, we study the consequences that the sales of inflation-indexed securities have on the observed inflation rate and price level. Similar to the study of Walsh, we find that the inflationary bias is significantly reduced, and hence that markets for inflation-indexed bonds provide a mechanism to reduce inflationary bias and increase central bank's credibility.

]]>An investor with constant absolute risk aversion trades a risky asset with general Itô-dynamics, in the presence of small proportional transaction costs. In this setting, we formally derive a leading-order optimal trading policy and the associated welfare, expressed in terms of the local dynamics of the frictionless optimizer. By applying these results in the presence of a random endowment, we obtain asymptotic formulas for utility indifference prices and hedging strategies in the presence of small transaction costs.

We consider the problem of optimal investment when agents take into account their relative performance by comparison to their peers. Given *N* interacting agents, we consider the following optimization problem for agent *i*, :

where is the utility function of agent *i*, his portfolio, his wealth, the average wealth of his peers, and is the parameter of relative interest for agent *i*. Together with some mild technical conditions, we assume that the portfolio of each agent *i* is restricted in some subset . We show existence and uniqueness of a Nash equilibrium in the following situations:

- -unconstrained agents,
- -constrained agents with exponential utilities and Black–Scholes financial market.

We also investigate the limit when the number of agents *N* goes to infinity. Finally, when the constraints sets are vector spaces, we study the impact of the s on the risk of the market.

We present a general equilibrium model of a moral-hazard economy with many firms and financial markets, where stocks and bonds are traded. Contrary to the principal-agent literature, we argue that optimal contracting in an infinite economy is not about a tradeoff between risk sharing and incentives, but it is all about incentives. Even when the economy is finite, optimal contracts do not depend on principals’ risk aversion, but on market prices of risks. We also show that optimal contracting does not require relative performance evaluation, that the second best risk-free interest rate is lower than that of the first best, and that the second-best equity premium can be higher or lower than that of the first best. Moral hazard can contribute to the resolution of the risk-free rate puzzle. Its potential to explain the equity premium puzzle is examined.

In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cash flows which are subject to volume constraints modeled by integer-valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers (2012), Bender (2011a), Bender (2011b), Aleksandrov and Hambly (2010), and Meinshausen and Hambly (2004) on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cash flow structures than the additive structure in the above references. For example, some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for prices of multiple exercise options and illustrate it with a numerical study on the pricing of a swing option in an electricity market.

]]>We investigate correlations of asset returns in stress scenarios where a common risk factor is truncated. Our analysis is performed in the class of normal variance mixture (NVM) models, which encompasses many distributions commonly used in financial modeling. For the special cases of jointly normally and *t*-distributed asset returns we derive closed formulas for the correlation under stress. For the NVM distribution, we calculate the asymptotic limit of the correlation under stress, which depends on whether the variables are in the maximum domain of attraction of the Fréchet or Gumbel distribution. It turns out that correlations in heavy-tailed NVM models are less sensitive to stress than in medium- or light-tailed models. Our analysis sheds light on the suitability of this model class to serve as a quantitative framework for stress testing, and as such provides valuable information for risk and capital management in financial institutions, where NVM models are frequently used for assessing capital adequacy. We also demonstrate how our results can be applied for more prudent stress testing.

We consider an optimal insurance design problem for an individual whose preferences are dictated by the rank-dependent expected utility (RDEU) theory with a concave utility function and an inverse-S shaped probability distortion function. This type of RDEU is known to describe human behavior better than the classical expected utility. By applying the technique of quantile formulation, we solve the problem explicitly. We show that the optimal contract not only insures large losses above a deductible but also insures small losses fully. This is consistent, for instance, with the demand for warranties. Finally, we compare our results, analytically and numerically, both to those in the expected utility framework and to cases in which the distortion function is convex or concave.

]]>We consider a framework for solving optimal liquidation problems in limit order books. In particular, order arrivals are modeled as a point process whose intensity depends on the liquidation price. We set up a stochastic control problem in which the goal is to maximize the expected revenue from liquidating the entire position held. We solve this optimal liquidation problem for power-law and exponential-decay order book models explicitly and discuss several extensions. We also consider the continuous selling (or fluid) limit when the trading units are ever smaller and the intensity is ever larger. This limit provides an analytical approximation to the value function and the optimal solution. Using techniques from viscosity solutions we show that the discrete state problem and its optimal solution converge to the corresponding quantities in the continuous selling limit uniformly on compacts.

]]>In financial markets, liquidity is not constant over time but exhibits strong seasonal patterns. In this paper, we consider a limit order book model that allows for time-dependent, deterministic depth and resilience of the book and determine optimal portfolio liquidation strategies. In a first model variant, we propose a trading-dependent spread that increases when market orders are matched against the order book. In this model, no price manipulation occurs and the optimal strategy is of the wait region/buy region type often encountered in singular control problems. In a second model, we assume that there is no spread in the order book. Under this assumption, we find that price manipulation can occur, depending on the model parameters. Even in the absence of classical price manipulation, there may be transaction triggered price manipulation. In specific cases, we can state the optimal strategy in closed form.

]]>In a limit order book model with exponential resilience, general shape function, and an unaffected stock price following the Bachelier model, we consider the problem of optimal liquidation for an investor with constant absolute risk aversion. We show that the problem can be reduced to a two-dimensional deterministic problem which involves no buy orders. We derive an explicit expression for the value function and the optimal liquidation strategy. The analysis is complicated by the fact that the intervention boundary, which determines the optimal liquidation strategy, is discontinuous if there are levels in the limit order book with relatively little market depth. Despite this complication, the equation for the intervention boundary is fairly simple. We show that the optimal liquidation strategy possesses the natural properties one would expect, and provide an explicit example for the case where the limit order book has a constant shape function.

]]>Our main goal is to re-examine and extend certain results from the papers by Galluccio et al. and Pietersz and van Regenmortel. We establish several results providing alternate necessary and sufficient conditions for admissibility of a family of forward swaps, that is, the property that it is supported by a (positive) family of bonds associated with the underlying tenor structure. We also derive the generic expression for the joint dynamics of a family of forward swap rates under a single probability measure and we show that these dynamics are uniquely determined by a selection of volatility processes with respect to the set of driving martingales.

]]>Several approximations have been proposed in the literature for the pricing of European-style swaptions under multifactor term structure models. However, none of them provides an estimate for the inherent approximation error. Until now, only the Edgeworth expansion technique of Collin-Dufresne and Goldstein is able to characterize the order of the approximation error. Under a multifactor HJM Gaussian framework, this paper proposes a new approximation for European-style swaptions, which is able to set bounds on the magnitude of the approximation error and is based on the *conditioning approach* initiated by Curran and Rogers and Shi. All the proposed pricing bounds will arise as a simple by-product of the Nielsen and Sandmann setup, and will be shown to provide a better accuracy–efficiency trade-off than all the approximations already proposed in the literature.

This paper shows that Singleton and Umantsev's method for swaption pricing in affine models can be simplified and extended to other models. Two alternative methods for approximating the option exercise boundary are introduced: one based on the multivariate Taylor series expansion, and the other based on duration-matched zero-coupon bond approximation. Applied to affine models and quadratic-Gaussian models, these methods are found to give accurate swaption prices.

]]>We develop a theory of robust pricing and hedging of a weighted variance swap given market prices for a finite number of co-maturing put options. We assume the put option prices do not admit arbitrage and deduce no-arbitrage bounds on the weighted variance swap along with super- and sub-replicating strategies that enforce them. We find that market quotes for variance swaps are surprisingly close to the model-free lower bounds we determine. We solve the problem by transforming it into an analogous question for a European option with a convex payoff. The lower bound becomes a problem in semi-infinite linear programming which we solve in detail. The upper bound is explicit. We work in a model-independent and probability-free setup. In particular, we use and extend Föllmer's pathwise stochastic calculus. Appropriate notions of arbitrage and admissibility are introduced. This allows us to establish the usual hedging relation between the variance swap and the “log contract” and similar connections for weighted variance swaps. Our results take the form of a FTAP: we show that the absence of (weak) arbitrage is equivalent to the existence of a classical model which reproduces the observed prices via risk-neutral expectations of discounted payoffs.

]]>Most of the existing pricing models of variance derivative products assume continuous sampling of the realized variance processes, though actual contractual specifications compute the realized variance based on sampling at discrete times. We present a general analytic approach for pricing discretely sampled generalized variance swaps under the stochastic volatility models with simultaneous jumps in the asset price and variance processes. The resulting pricing formula of the gamma swap is in closed form while those of the corridor variance swaps and conditional variance swaps take the form of one-dimensional Fourier integrals. We also verify through analytic calculations the convergence of the asymptotic limit of the pricing formulas of the discretely sampled generalized variance swaps under vanishing sampling interval to the analytic pricing formulas of the continuously sampled counterparts. The proposed methodology can be applied to any affine model and other higher moments swaps as well. We examine the exposure to convexity (volatility of variance) and skew (correlation between the equity returns and variance process) of these discretely sampled generalized variance swaps. We explore the impact on the fair strike prices of these exotic variance swaps with respect to different sets of parameter values, like varying sampling frequencies, jump intensity, and width of the monitoring corridor.

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