We find optimal trading policies for long-term investors with constant relative risk aversion and constant investment opportunities, which include one safe asset, liquid risky assets, and an illiquid risky asset trading with proportional costs. Access to liquid assets creates a diversification motive, which reduces illiquid trading, and a hedging motive, which both reduces illiquid trading and increases liquid trading. A further tempering effect depresses the liquid asset's weight when the illiquid asset's weight is close to ideal, to keep it near that level by reducing its volatility. Multiple liquid assets lead to portfolio separation in four funds: the safe asset, the myopic portfolio, the illiquid asset, and its hedging portfolio.

]]>The aim of this paper is to put forward a new family of risk measures that could guide investment decisions of private companies. But at the difference of the classical approach of Artzner, Delbaen, Eber, and Heath and the subsequent extensions of this model, our risk measures are built to reflect the risk perception of shareholders rather than regulators. Instead of an axiomatic approach, we derive risk measures from the optimal policies of a shareholder value-maximizing company. We study these optimal policies and the related risk measures that we call shareholder risk measures. We emphasize the fact that due to the specific corporate environment, in particular the limited shareholders' liability and the possibility to pay out dividends from cash reserves, these risk measures are not convex. Also, they depend on the specific economic situation of the firm, in particular its current cash level, and thus they are not translation invariant. This paper bridges the gap between two important branches of mathematical finance: risk measures and optimal dividends.

]]>We study utility indifference prices and optimal purchasing quantities for a nontraded contingent claim in an incomplete semimartingale market with vanishing hedging errors. We make connections with the theory of large deviations. We concentrate on sequences of semicomplete markets where in the *n*th market, the claim admits the decomposition . Here, is replicable by trading in the underlying assets , but is independent of . Under broad conditions, we may assume that vanishes in accordance with a large deviations principle (LDP) as *n* grows. In this setting, for an exponential investor, we identify the limit of the average indifference price , for units of , as . We show that if , the limiting price typically differs from the price obtained by assuming bounded positions , and the difference is explicitly identifiable using large deviations theory. Furthermore, we show that optimal purchase quantities occur at the large deviations scaling, and hence large positions arise endogenously in this setting.

In a new scheme for hedge fund managerial compensation known as the first-loss scheme, a fund manager uses her investment in the fund to cover any fund losses first; by contrast, in the traditional scheme currently used in most US funds, the manager does not cover investors' losses in the fund. We propose a framework based on cumulative prospect theory to compute and compare the trading strategies, fund risk, and managers' and investors' utilities in these two schemes analytically. The model is calibrated to the historical attrition rates of US hedge funds. We find that with reasonable parameter values, both fund managers' and investors' utilities can be improved and fund risk can be reduced simultaneously by replacing the traditional scheme (with 10% internal capital and 20% performance fee) with a first-loss scheme (with 10% first-loss capital and 30% performance fee). When the performance fee in the first-loss scheme is 40% (a current market practice), however, such substitution renders investors worse off.

]]>We study the cost of shocks, that is, jump risk, with respect to reserve management when the reserve process is formulated as a drift-switching jump diffusion with a reflecting barrier at 0. Inspired by the Brownian drift switching model, our model results in a more realistic dynamic behavior of international reserves than the buffer stock model. The new model can capture both the jump behavior in reserve dynamics and the leptokurtic feature of the increment distribution which has a higher peak and two asymmetric heavier tails than the normal distribution. Through the selection of an initial distribution that reflects certain steady state behaviors, the reserve process becomes a regenerative process. This selection enables us to derive a closed-form expression for the total expected discounted cost of managing reserves, thus helping us to numerically find management strategies that minimize costs. The numerical results show that shocks at the reserve level have a significant effect on reserve management strategies and that model misspecification can result in nonnegligible additional costs.

]]>Motivated by the European sovereign debt crisis, we propose a hybrid sovereign default model that combines an accessible part taking into account the evolution of the sovereign solvency and the impact of critical political events, and a totally inaccessible part for the idiosyncratic credit risk. We obtain closed-form formulas for the probability that the default occurs at critical political dates in a Markovian setting. Moreover, we introduce a generalized density framework for the hybrid default time and deduce the compensator process of default. Finally, we apply the hybrid model and the generalized density to the valuation of sovereign bonds and explain the significant jumps in long-term government bond yields during the sovereign crisis.

]]>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.

]]>In this paper, we study the aggregate risk of inhomogeneous risks with dependence uncertainty, evaluated by a generic risk measure. We say that a pair of risk measures is asymptotically equivalent if the ratio of the worst-case values of the two risk measures is almost one for the sum of a large number of risks with unknown dependence structure. The study of asymptotic equivalence is particularly important for a pair of a noncoherent risk measure and a coherent risk measure, as the worst-case value of a noncoherent risk measure under dependence uncertainty is typically difficult to obtain. The main contribution of this paper is to establish general asymptotic equivalence results for the classes of distortion risk measures and convex risk measures under different mild conditions. The results implicitly suggest that it is only reasonable to implement a coherent risk measure for the aggregation of a large number of risks with uncertainty in the dependence structure, a relevant situation for risk management practice.

]]>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.

]]>The two main approaches in credit risk are the structural approach pioneered by Merton and the reduced-form framework proposed by Jarrow and Turnbull and by Artzner and Delbaen. The goal of this paper is to provide a unified view on both approaches. This is achieved by studying reduced-form approaches under weak assumptions. In particular, we do not assume the global existence of a default intensity and allow default at fixed or predictable times, such as coupon payment dates, with positive probability. In this generalized framework, we study dynamic term structures prone to default risk following the forward-rate approach proposed by Heath, Jarrow, and Morton. It turns out that previously considered models lead to arbitrage possibilities when default can happen at a predictable time. A suitable generalization of the forward-rate approach contains an additional stochastic integral with atoms at predictable times and necessary and sufficient conditions for an appropriate no-arbitrage condition are given. For efficient implementations, we develop a new class of affine models that do not satisfy the standard assumption of stochastic continuity. The chosen approach is intimately related to the theory of enlargement of filtrations, for which we provide an example by means of filtering theory where the Azéma supermartingale contains upward and downward jumps, both at predictable and totally inaccessible stopping times.

]]>We consider a general local-stochastic volatility model and an investor with exponential utility. For a European-style contingent claim, whose payoff may depend on either a traded or nontraded asset, we derive an explicit approximation for both the buyer's and seller's indifference prices. For European calls on a traded asset, we translate indifference prices into an explicit approximation of the buyer's and seller's implied volatility surfaces. For European claims on a nontraded asset, we establish rigorous error bounds for the indifference price approximation. Finally, we implement our indifference price and implied volatility approximations in two examples.

]]>We consider an investor who has access both to a traditional venue and a dark pool for liquidating a position in a single asset. While trade execution is certain on the traditional exchange, she faces linear price impact costs. On the other hand, dark pool orders suffer from adverse selection and trade execution is uncertain. Adverse selection decreases order sizes in the dark pool while it speeds up trading at the exchange. For small orders, it is optimal to avoid the dark pool completely. Adverse selection can prevent profitable round-trip trading strategies that otherwise would arise if permanent price impact were included in the model.

]]>The well-known theorem of Dybvig, Ingersoll, and Ross shows that the long zero-coupon rate can never fall. This result, which, although undoubtedly correct, has been regarded by many as surprising, stems from the implicit assumption that the long-term discount function has an exponential tail. We revisit the problem in the setting of modern interest rate theory, and show that if the long “simple” interest rate (or Libor rate) is finite, then this rate (unlike the zero-coupon rate) acts viably as a state variable, the value of which can fluctuate randomly in line with other economic indicators. New interest rate models are constructed, under this hypothesis and certain generalizations thereof, that illustrate explicitly the good asymptotic behavior of the resulting discount bond systems. The conditions necessary for the existence of such “hyperbolic” and “generalized hyperbolic” long rates are those of so-called social discounting, which allow for long-term cash flows to be treated as broadly “just as important” as those of the short or medium term. As a consequence, we are able to provide a consistent arbitrage-free valuation framework for the cost-benefit analysis and risk management of long-term social projects, such as those associated with sustainable energy, resource conservation, and climate change.

]]>We analyze the convergence of the Longstaff–Schwartz algorithm relying on only a single set of independent Monte Carlo sample paths that is repeatedly reused for all exercise time-steps. We prove new estimates on the stochastic component of the error of this algorithm whenever the approximation architecture is any uniformly bounded set of *L*^{2} functions of finite Vapnik–Chervonenkis dimension (VC-dimension), but in particular need not necessarily be either convex or closed. We also establish new overall error estimates, incorporating bounds on the approximation error as well, for certain nonlinear, nonconvex sets of neural networks.

We provide an extension of the explicit solution of a mixed optimal stopping–optimal stochastic control problem introduced by Henderson and Hobson. The problem examines whether the optimal investment problem on a local martingale financial market is affected by the optimal liquidation of an independent indivisible asset. The indivisible asset process is defined by a homogeneous scalar stochastic differential equation, and the investor's preferences are defined by a general expected utility function. The value function is obtained in explicit form, and we prove the existence of an optimal stopping–investment strategy characterized as the limit of an explicit maximizing strategy. Our approach is based on the standard dynamic programming approach.

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

We derive the joint density of a Skew Brownian motion, its last visit to the origin, its local and occupation times. The result enables us to obtain explicit analytical formulas for pricing European options under both a two-valued local volatility model and a displaced diffusion model with constrained volatility.

]]>We study the problem of expected utility maximization in a large market, i.e., a market with countably many traded assets. Assuming that agents have von Neumann–Morgenstern preferences with stochastic utility function and that consumption occurs according to a stochastic clock, we obtain the “usual” conclusions of the utility maximization theory. We also give a characterization of the value function in a large market in terms of a sequence of value functions in finite-dimensional models.

]]>The growth of the exchange-traded fund (ETF) industry has given rise to the trading of options written on ETFs and their leveraged counterparts (LETFs). We study the relationship between the ETF and LETF implied volatility surfaces when the underlying ETF is modeled by a general class of local-stochastic volatility models. A closed-form approximation for prices is derived for European-style options whose payoffs depend on the terminal value of the ETF and/or LETF. Rigorous error bounds for this pricing approximation are established. A closed-form approximation for implied volatilities is also derived. We also discuss a scaling procedure for comparing implied volatilities across leverage ratios. The implied volatility expansions and scalings are tested in three settings: Heston, limited constant elasticity of variance (CEV), and limited SABR; the last two are regularized versions of the well-known CEV and SABR models.

]]>We introduce a new stochastic volatility model that includes, as special instances, the Heston (1993) and the 3/2 model of Heston (1997) and Platen (1997). Our model exhibits important features: first, instantaneous volatility can be uniformly bounded away from zero, and second, our model is mathematically and computationally tractable, thereby enabling an efficient pricing procedure. This called for using the Lie symmetries theory for partial differential equations; doing so allowed us to extend known results on Bessel processes. Finally, we provide an exact simulation scheme for the model, which is useful for numerical applications.

]]>We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family of possible physical measures. A robust notion of no-arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot be superhedged for free by using simple trading strategies. Our first main result is a version of the fundamental theorem of asset pricing: holds if and only if every admits a martingale measure that is equivalent up to a certain lifetime. The second main result provides the existence of optimal superhedging strategies for general contingent claims and a representation of the superhedging price in terms of martingale measures.

]]>We study the Merton portfolio optimization problem in the presence of stochastic volatility using asymptotic approximations when the volatility process is characterized by its timescales of fluctuation. This approach is tractable because it treats the incomplete markets problem as a perturbation around the complete market constant volatility problem for the value function, which is well understood. When volatility is fast mean-reverting, this is a singular perturbation problem for a nonlinear Hamilton–Jacobi–Bellman partial differential equation, while when volatility is slowly varying, it is a regular perturbation. These analyses can be combined for multifactor multiscale stochastic volatility models. The asymptotics shares remarkable similarities with the linear option pricing problem, which follows from some new properties of the Merton risk tolerance function. We give examples in the family of mixture of power utilities and also use our asymptotic analysis to suggest a “practical” strategy that does not require tracking the fast-moving volatility. In this paper, we present formal derivations of asymptotic approximations, and we provide a convergence proof in the case of power utility and single-factor stochastic volatility. We assess our approximation in a particular case where there is an explicit solution.

]]>We consider the fundamental theorem of asset pricing (FTAP) and the hedging prices of options under nondominated model uncertainty and portfolio constraints in discrete time. We first show that no arbitrage holds if and only if there exists some family of probability measures such that any admissible portfolio value process is a local super-martingale under these measures. We also get the nondominated optional decomposition with constraints. From this decomposition, we obtain the duality of the super-hedging prices of European options, as well as the sub- and super-hedging prices of American options. Finally, we get the FTAP and the duality of super-hedging prices in a market where stocks are traded dynamically and options are traded statically.

]]>We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. We also establish rigorous error estimates for these quantities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under four different model dynamics: constant elasticity of variance local volatility, Heston stochastic volatility, three-halves stochastic volatility, and SABR local-stochastic volatility.

]]>We investigate the general structure of optimal investment and consumption with small proportional transaction costs. For a safe asset and a risky asset with general continuous dynamics, traded with random and time-varying but small transaction costs, we derive simple formal asymptotics for the optimal policy and welfare. These reveal the roles of the investors' preferences as well as the market and cost dynamics, and also lead to a fully dynamic model for the implied trading volume. In frictionless models that can be solved in closed form, explicit formulas for the leading-order corrections due to small transaction costs are obtained.

]]>Approximations to utility indifference prices are provided for a contingent claim in the large position size limit. Results are valid for general utility functions on the real line and semi-martingale models. It is shown that as the position size approaches infinity, the utility function's decay rate for large negative wealths is the primary driver of prices. For utilities with exponential decay, one may price like an exponential investor. For utilities with a power decay, one may price like a power investor after a suitable adjustment to the rate at which the position size becomes large. In a sizable class of diffusion models, limiting indifference prices are explicitly computed for an exponential investor. Furthermore, the large claim limit arises endogenously as the hedging error for the claim vanishes.

]]>We consider *n* risk-averse agents who compete for liquidity in an Almgren–Chriss market impact model. Mathematically, this situation can be described by a Nash equilibrium for a certain linear quadratic differential game with state constraints. The state constraints enter the problem as terminal boundary conditions for finite and infinite time horizons. We prove existence and uniqueness of Nash equilibria and give closed-form solutions in some special cases. We also analyze qualitative properties of the equilibrium strategies and provide corresponding financial interpretations.

This paper considers the pricing and hedging of a call option when liquidity matters, that is, either for a large nominal or for an illiquid underlying asset. In practice, as opposed to the classical assumptions of a price-taking agent in a frictionless market, traders cannot be perfectly hedged because of execution costs and market impact. They indeed face a trade-off between hedging errors and costs that can be solved by using stochastic optimal control. Our modeling framework, which is inspired by the recent literature on optimal execution, makes it possible to account for both execution costs and the lasting market impact of trades. Prices are obtained through the indifference pricing approach. Numerical examples are provided, along with comparisons to standard methods.

]]>We study a problem of optimal investment/consumption over an infinite horizon in a market with two possibly correlated assets: one liquid and one illiquid. The liquid asset is observed and can be traded continuously, while the illiquid one can be traded only at discrete random times, corresponding to the jumps of a Poisson process with intensity λ, is observed at the trading dates, and is partially observed between two different trading dates. The problem is a nonstandard mixed discrete/continuous optimal control problem, which we solve by a dynamic programming approach. When the utility has a general form, we prove that the value function is the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation and characterize the optimal allocation in the illiquid asset. In the case of power utility, we establish the regularity of the value function needed to prove the verification theorem, providing the complete theoretical solution of the problem. This enables us to perform numerical simulations, so as to analyze the impact of time illiquidity and how this impact is affected by the degree of observation.

]]>In a financial market with a continuous price process and proportional transaction costs, we investigate the problem of utility maximization of terminal wealth. We give sufficient conditions for the existence of a shadow price process, i.e., a least favorable frictionless market leading to the same optimal strategy and utility as in the original market under transaction costs. The crucial ingredients are the continuity of the price process and the hypothesis of “no unbounded profit with bounded risk.” A counterexample reveals that these hypotheses cannot be relaxed.

]]>This paper studies the problem of option replication in general stochastic volatility markets with transaction costs, using a new specification for the volatility adjustment in Leland's algorithm. We prove several limit theorems for the normalized replication error of Leland's strategy, as well as that of the strategy suggested by Lépinette. The asymptotic results obtained not only generalize the existing results, but also enable us to fix the underhedging property pointed out by Kabanov and Safarian. We also discuss possible methods to improve the convergence rate and to reduce the option price inclusive of transaction costs.

]]>We generalize the primal–dual methodology, which is popular in the pricing of early-exercise options, to a backward dynamic programming equation associated with time discretization schemes of (reflected) backward stochastic differential equations (BSDEs). Taking as an input some approximate solution of the backward dynamic program, which was precomputed, e.g., by least-squares Monte Carlo, this methodology enables us to construct a confidence interval for the unknown true solution of the time-discretized (reflected) BSDE at time 0. We numerically demonstrate the practical applicability of our method in two 5-dimensional nonlinear pricing problems where tight price bounds were previously unavailable.

]]>An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth. In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market, cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to optimal execution problems and to other market frictions such as proportional and fixed transaction costs.

]]>In a companion paper, we studied a control problem related to swing option pricing in a general non-Markovian setting. The main result there shows that the value process of this control problem can uniquely be characterized in terms of a first-order backward stochastic partial differential equation (BSPDE) and a pathwise differential inclusion. In this paper, we additionally assume that the cash flow process of the swing option is left-continuous in expectation. Under this assumption, we show that the value process is continuously differentiable in the space variable that represents the volume in which the holder of the option can still exercise until maturity. This gives rise to an existence and uniqueness result for the corresponding BSPDE in a classical sense. We also explicitly represent the space derivative of the value process in terms of a nonstandard optimal stopping problem over a subset of predictable stopping times. This representation can be applied to derive a dual minimization problem in terms of martingales.

]]>We derive the process followed by trading volume, in a market with finite depth and constant investment opportunities, where a large investor, with a long horizon and constant relative risk aversion, trades a safe and a risky asset. Trading volume approximately follows a Gaussian, mean-reverting diffusion, and increases with depth, volatility, and risk aversion. Unlike the frictionless theory, finite depth excludes leverage and short sales because such positions may not be solvent even with continuous trading.

]]>This paper provides a coherent method for scenario aggregation addressing model uncertainty. It is based on divergence minimization from a reference probability measure subject to scenario constraints. An example from regulatory practice motivates the definition of five fundamental criteria that serve as a basis for our method. Standard risk measures, such as value-at-risk and expected shortfall, are shown to be robust with respect to minimum divergence scenario aggregation. Various examples illustrate the tractability of our method.

]]>In this paper, we study the pricing and hedging of typical life insurance liabilities for an insurance portfolio with dependent mortality risk by means of the well-known risk-minimization approach. As the insurance portfolio consists of individuals of different age cohorts in order to capture the cross-generational dependency structure of the portfolio, we introduce affine models for the mortality intensities based on Gaussian random fields that deliver analytically tractable results. We also provide specific examples consistent with historical mortality data and correlation structures. Main novelties of this work are the explicit computations of risk-minimizing strategies for life insurance liabilities written on an insurance portfolio composed of primary financial assets (a risky asset and a money market account) and a family of longevity bonds, and the simultaneous consideration of different age cohorts.

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

]]>This paper is concerned with the axiomatic foundation and explicit construction of a general class of optimality criteria that can be used for investment problems with multiple time horizons, or when the time horizon is not known in advance. Both the investment criterion and the optimal strategy are characterized by the Hamilton–Jacobi–Bellman equation on a semi-infinite time interval. In the case where this equation can be linearized, the problem reduces to a time-reversed parabolic equation, which cannot be analyzed via the standard methods of partial differential equations. Under the additional uniform ellipticity condition, we make use of the available description of all minimal solutions to such equations, along with some basic facts from potential theory and convex analysis, to obtain an explicit integral representation of all positive solutions. These results allow us to construct a large family of the aforementioned optimality criteria, including some closed-form examples in relevant financial models.

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

]]>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.

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.

]]>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. (2012): Risk aversion asymptotics for power utility maximization. *Probab. Theory & Relat. Fields* 152, 703–749], which establishes the convergence for a sequence of power utilities.

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.

]]>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.

]]>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.

]]>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).

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.

]]>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.

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