Continuous-time Financial Market Research Articles

No-arbitrage conditions and pricing from discrete-time to continuous-time strategies

In this paper, a general framework is developed for continuous-time financial market models defined from simple strategies through conditional topologies that avoid stochastic calculus and do not necessitate semimartingale models. We then compare the usual no-arbitrage conditions of the literature, e.g. the usual no-arbitrage conditions NFL, NFLVR and NUPBR and the recent AIP condition. With appropriate pseudo-distance topologies, we show that they hold in continuous time if and only if they hold in discrete time. Moreover, the super-hedging prices in continuous time coincide with the discrete-time super-hedging prices, even without any no-arbitrage condition.

Short Communication: A Primer on Perpetuals

We consider a continuous-time financial market with no arbitrage and no transactions costs. In this setting, we introduce two types of perpetual contracts, one in which the payoff to the long side is a fixed function of the underlyers and the long side pays a funding rate to the short side, the other in which the payoff to the long side is a fixed function of the underlyers times a discount factor that changes over time but no funding payments are required. Assuming asset prices are continuous and strictly positive, we derive model-free expressions for the funding rate and discount rate of these perpetual contracts as well as replication strategies for the short side. When asset prices can jump, we derive expressions for the funding and discount rates, which are semirobust in the sense that they do not depend on the dynamics of the volatility process of the underlying risky assets, but do depend on the intensity of jumps under the market’s pricing measure. When asset prices can jump and the volatility process is independent of the underlying risky assets, we derive an explicit replication strategy for the short side of a perpetual contract. Throughout the paper, we illustrate through examples how specific perpetual contracts relate to traditional financial instruments such as variance swaps and leveraged exchange traded funds.

Dynamic Mean-Downside Risk Portfolio Selection Problem with Stochastic Interest Rate in Continuous-Time

Even though a consensus has been reached a long time ago that the interest rate is a stochastic process, most of the existing works on dynamic mean-downside risk portfolio selection still focus on a deterministic interest rate. This work studies a dynamic mean-downside risk portfolio selection problem with a stochastic interest rate in a continuous-time financial market. Specifically, we choose the lower-partial moments(LPM), value-at-risk (VaR), and conditional value-at-risk(CVaR) to model our downside risk measures in the Vasicek interest rate model. By using the martingale method and the inverse Fourier Transformation, we successfully derive the semi-analytical optimal portfolio policies and the optimal wealth processes for the mean-downside risk measures with a stochastic interest rate. The results suggest that the higher the degree of negative correlation, the better the performance of the portfolio valued by mean-downside risk framework when the other market opportunity set is the same.

Scalarized utility-based multi-asset risk measures

Financial institutions have to satisfy capital adequacy tests required, e.g., by the Basel Accords for banks or by Solvency II for insurers. If the financial situation of an institution is tight, then it can happen that no reallocation of the initial endowment would pass the capital adequacy test. The classical portfolio optimization approach breaks down and a capital increase is needed. We introduce the scalarized utility-based multi-asset (SUBMA) risk measure which optimizes the hedging costs and the expected utility of an agent simultaneously subject to the capital adequacy test. We find that the SUBMA risk measure is coherent if the utility function has constant relative risk aversion and the capital adequacy test leads to a coherent acceptance set. In a one-period financial market model we present a sufficient condition for the SUBMA risk measure to be finite-valued and continuous. Under further assumptions on the utility function we obtain existence and uniqueness results for the optimal hedging strategies. Finally, we calculate the SUBMA risk measure in a continuous-time financial market model for two benchmark capital adequacy tests.

BEHAVIORAL PORTFOLIO CHOICE UNDER HYPERBOLIC ABSOLUTE RISK AVERSION

This paper studies the optimal investment problem for a behavioral investor with probability distortion functions and an S-shaped utility function whose utility on gains satisfies the Inada condition at infinity, albeit not necessarily at zero, in a complete continuous-time financial market model. In particular, a piecewise utility function with hyperbolic absolute risk aversion (HARA) is applied. The considered behavioral framework, cumulative prospect theory (CPT), was originally introduced by [A. Tversky & D. Kahneman (1992) Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty 5 (4), 297–323]. The utility model allows for increasing, constant or decreasing relative risk aversion. The continuous-time portfolio selection problem under the S-shaped HARA utility function in combination with probability distortion functions on gains and losses is solved theoretically for the first time, the optimal terminal wealth and its replicating wealth process and investment strategy are stated. In addition, conditions on the utility and the probability distortion functions for well-posedness and closed-form solutions are provided. A specific probability distortion function family is presented which fulfills all those requirements. This generalizes the work by [H. Jin & X. Y. Zhou (2008) Behavioral portfolio selection in continuous time, Mathematical Finance 18 (3), 385–426]. Finally, a numerical case study is carried out to illustrate the impact of the utility function and the probability distortion functions.

Behavioral Portfolio Choice under Hyperbolic Absolute Risk Aversion

This paper studies the optimal investment problem for a behavioral investor with probability distortion functions and an S-shaped utility function whose utility on gains satisfies the Inada condition at infinity, albeit not necessarily at zero, in a complete continuous-time financial market model. In particular, a piecewise utility function with hyperbolic absolute risk aversion (HARA) is applied. The considered behavioral framework, Cumulative Prospect Theory (CPT), was originally introduced by Tversky and Kahneman (1992). The utility model allows for increasing, constant or decreasing relative risk aversion. The continuous-time portfolio selection problem under the S-shaped HARA utility function in combination with probability distortion functions on gains and losses is solved theoretically for the first time, the optimal terminal wealth and its replicating wealth process and investment strategy are stated. In addition, conditions on the utility and the probability distortion functions for well-posedness and closed-form solutions are provided. A specific probability distortion function family is presented which fulfills all those requirements. This generalizes the work by Jin and Zhou (2008). Finally, a numerical case study is carried out to illustrate the impact of the utility function and the probability distortion functions.

Super-replication with fixed transaction costs

We study super-replication of contingent claims in markets with fixed transaction costs. This can be viewed as a stochastic impulse control problem with a terminal state constraint. The first result in this paper reveals that in reasonable continuous time financial market models the super-replication price is prohibitively costly and leads to trivial buy-and-hold strategies. Our second result derives nontrivial scaling limits of super-replication prices for binomial models with small fixed costs.

Consumption and investment with interest rate risk

This paper investigates the optimal investment and consumption problem in a continuous-time financial market for agents who maximize expected power utility from consumption, facing partially hedgeable interest rate risk. With no analytic solution available for the optimal strategy, we find closed-form approximate strategies by solving the same optimization problem in two fictitious complete markets. The approximate strategies help verify the existence and the optimality of the solution to the original optimization problem and provide bounds of the optimal consumption strategy and the approximation error, both in closed form. As the interest rate increases, if the investor's risk aversion is greater than one, the wealth effect dominates the substitution effect, and consumption increases. If the risk aversion is less than one, then the substitution effect dominates, and the investor consumes less.

General Equilibrium Under Convex Portfolio Constraints and Heterogeneous Risk Preferences

This paper characterizes the equilibrium in a continuous time financial market populated by heterogeneous agents who differ in their rate of relative risk aversion and face convex portfolio constraints. The model is studied in an application to margin constraints and found to match real world observations about financial variables and leverage cycles. It is shown how margin constraints increase the market price of risk and decrease the interest rate by forcing more risk averse agents to hold more risky assets, producing a higher equity risk premium. In addition, heterogeneity and margin constraints are shown to produce both pro- and counter-cyclical leverage cycles. Beyond two types, it is shown how constraints can cascade and how leverage can exhibit highly non-linear dynamics. Finally, empirical results are given, documenting a novel stylized fact which is predicted by the model, namely that the leverage cycle is both pro- and counter-cyclical.

On utility maximization without passing by the dual problem

We treat utility maximization from terminal wealth for an agent with utility function who dynamically invests in a continuous-time financial market and receives a possibly unbounded random endowment. We prove the existence of an optimal investment without introducing the associated dual problem. We rely on a recent result of Orlicz space theory, due to Delbaen and Owari which leads to a simple and transparent proof. Our results apply to non-smooth utilities and even strict concavity can be relaxed. We can handle certain random endowments with non-hedgeable risks, complementing earlier papers. Constraints on the terminal wealth can also be incorporated. As examples, we treat frictionless markets with finitely many assets and large financial markets.

Approximation for portfolio optimization in a financial market with shot-noise jumps

For an investor in a continuous-time financial market the portfolio optimization problem of maximizing expected utility of terminal wealth is considered. Stock prices are driven by a Brownian motion and a shot-noise process. The latter leads to jumps in the stock prices whose influence decays exponentially with time. We analyze this model using a stochastic control approach based on the Hamilton–Jacobi–Bellman (HJB) equation. Special cases are discussed motivating that an explicit solution is difficult. Two approximations are derived. Firstly, an approximation in the HJB equation using a Taylor expansion. Secondly, a Gaussian approximation of the shot noise process which leads to an explicit solution for the trading strategy. These approximations are compared in a simulation study with different strategies showing that for a wide range of model parameters the derived approximate strategies have a good performance.

Semi-static completeness and robust pricing by informed investors

We consider a continuous-time financial market that consists of securities available for dynamic trading, and securities only available for static trading. We work in a robust framework where a set of nondominated models is given. The concept of semi-static completeness is introduced: it corresponds to having exact replication by means of semi-static strategies. We show that semi-static completeness is equivalent to an extremality property, and give a characterization of the induced filtration structure. Furthermore, we consider investors with additional information and, for specific types of extra information, we characterize the models that are semi-statically complete for the informed investors. Finally, we provide some examples where robust pricing for informed and uninformed agents can be done over semi-statically complete models.

Stochastic maximum principle for partial information optimal investment and dividend problem of an insurer

We study an optimal investment and dividend problem of an insurer, where the aggregate insurance claims process is modeled by a pure jump Levy process. We allow the management of the dividend payment policy and the investment of surplus in a continuous-time financial market, which is composed of a risk free asset and a risky asset. The information available to the insurer is partial information. We generalize this problem as a partial information regular-singular stochastic control problem, where the control variable consists of regular control and singular control. Then maximum principles are established to give sufficient and necessary optimality conditions for the solutions of the regular-singular control problem. Finally we apply the maximum principles to solve the investment and dividend problem of an insurer.

Convex Duality with Transaction Costs

Convex duality for two different super-replication problems in a continuous time financial market with proportional transaction cost is proved. In this market, static hedging in a finite number of options, in addition to usual dynamic hedging with the underlying stock, are allowed. The first one of the problems considered is the model-independent hedging that requires the super-replication to hold for every continuous path. In the second one the market model is given through a probability measure ℙ and the inequalities are understood the probability measure almost surely. The main result, using the convex duality, proves that the two super-replication problems have the same value provided that the probability measure satisfies the conditional full support property. Hence, the transaction costs prevents one from using the structure of a specific model to reduce the super-replication cost.

Consumption with Unhedgeable Interest Rate Risk

This paper investigates the optimal investment and consumption problem in a continuous-time financial market for investors with power utility on consumption, who face partially hedgeable interest rate risk. With no analytical solution to the optimal strategies, closed-form approximate strategies are derived by solving the same optimization problem in two fictitious complete markets. The approximate solution helps verify the existence and the optimality of the solution to the original optimization problem and provides bounds of the optimal consumption strategy and the approximation error, both in closed form. As the interest rate increases, if the investor's risk aversion is greater than one, the wealth effect dominates the substitution effect, and consumption increases. If risk aversion is less than one, then the substitution effect dominates, and the investor consumes less.

When Prospect Theory Preference Meets Mean-Reverting Asset Returns: A Dynamic Asset Allocation Model

We examine how the evidence of mean-reversion in stock returns affects dynamic trading behavior for investors with prospect-theory preferences. Particular attention is paid to the trading incentives created by the interaction between prospect-theory preferences and mean-reverting return dynamics. Under general assumptions for the continuous-time financial market, we develop the semi-analytical portfolio policy by inverse Fourier Transformation method. By the revealed policy, we find that a small degree of mean reversion can be sufficient to reverse the direction of the investors' trading patterns. Further simulation results demonstrate that the combination of prospect theory and mean reversion can generate the disposition effect close to the data at the reasonable values of the underlying parameters. The results suggest that trading behavior patterns can be seriously misleading if the prospect theory allocation framework ignores time-variation in expected returns such as mean reversion.

Convex Duality with Transaction Costs

Convex duality for two two different super-replication problems in a continuous time financial market with proportional transaction cost is proved. In this market, static hedging in a finite number of options, in addition to usual dynamic hedging with the underlying stock, are allowed. The first one of the problems considered is the model-independent hedging that requires the super-replication to hold for every continuous path. In the second one the market model is given through a probability measure P and the inequalities are understood P almost surely. The main result, using the convex duality, proves that the two super-replication problems have the same value provided that P satisfies the conditional full support property. Hence, the transaction costs prevents one from using the structure of a specific model to reduce the super-replication cost.

Robust No Arbitrage of the Second Kind with a Continuum of Assets and Proportional Transaction Costs

Motivated by bond markets with a continuum of possible maturities, we study a robust version of the absence of arbitrage opportunity of the second kind introduced by Rasonyi in a continuous-time and infinite-dimensional financial market model with proportional transaction costs.

Contingent claim pricing through a continuous time variational bargaining scheme

We consider a variational problem modelling the evolution with time of two probability measures representing the subjective beliefs of a couple of agents engaged in a continuous-time bargaining pricing scheme with the goal of finding a unique price for a contingent claim in a continuous-time financial market. This optimization problem is coupled with two finite dimensional portfolio optimization problems, one for each agent involved in the bargaining scheme. Under mild conditions, we prove that the optimization problem under consideration here admits a unique solution, yielding a unique price for the contingent claim.

Optimal Diversification in the Presence of Parameter Uncertainty for a Risk Averse Investor

We consider an investor who faces parameter uncertainty in a continuoustime financial market. We model the investor's preference by a power utility function leading to constant relative risk aversion. We show that the loss in expected utility is large when using a simple plug-in strategy for unknown parameters. We also provide theoretical results that show the trade-off between holding a well-diversified portfolio and a portfolio that is robust against estimation errors. To reduce the effect of estimation, we constrain the weights of the risky assets with an L1-norm leading to a sparse portfolio. We provide analytical results that show how the sparsity of the constrained portfolio depends on the coefficient of relative risk aversion. Based on a simulation study, we demonstrate the existence and the uniqueness of an optimal bound on the L1-norm for each level of relative risk aversion.