Incomplete Market Setting Research Articles

Pricing vulnerable options under cross-asset markov-modulated jump-diffusion dynamics

In this study, the dynamics of the underlying asset price and the counterparty's asset price are governed by a cross-asset Markov-modulated jump-diffusion model that captures the time-inhomogeneity and systematic cojumps. Additionally, the forward interest rate processes are driven by a Markov-modulated Heath–Jarrow–Morton model to depict stochastic volatilities. Under an incomplete-market setting, we apply the Markov-modulated Esscher transform technique to determine a risk-neutral martingale measure. After determining the Markov-modulated Esscher parameters, we obtain an integral expression on the prices of vulnerable European-style Black–Scholes options. The numerical illustrations indicate that the findings are consistent with Klein (1996) and contribute to the extant literature on cojumping impacts on vulnerable option prices.

Risk management under weighted limited expected loss

We present and solve an optimal asset allocation problem under a weighted limited expected loss (WLEL) constraint. This formulation encompasses the risk management problem with a limited expected loss (LEL) constraint as a specialized instance and offers a pertinent internal risk management instrument for firms. We observe that a WLEL constraint makes the optimizing investor pursue less volatile payoffs than the unconstrained Merton solution. Compared to the LEL-constrained problem with the same weighted default threshold, the WLEL optimal terminal wealth displays a less dispersed distribution with a smaller variance, suggesting a more secure risk management framework. Conducting a comprehensive equilibrium analysis in the presence of a WLEL risk manager, we validate the relatively conservative investment approach undertaken by the WLEL manager. Subsequently, we expand our findings to encompass broader incomplete market settings, wherein the uniqueness of the equivalent local martingale measure is not assured.

Pricing derivatives on foreign assets using Markov-modulated cojump-diffusion dynamics

In this study, cross-currency derivatives pricing under stochastic interest rates is analyzed when Markov-modulated cojump-diffusion dynamics with both idiosyncratic and simultaneous jumps drive the underlying foreign equity price and foreign exchange rate processes. More specifically, the noise dynamics of these two assets are captured by the correlated Wiener processes, and the two types of jump events are described as two independent compound Poisson processes with log-normal jump amplitudes. Cojumping addresses the correlation between these two asset price jumps. In an incomplete market setting, the dynamic Esscher transform approach is applied to determine a pricing kernel. According to the resulting dynamics, explicit analytical formulas for evaluating values of European foreign equity call options are provided. Numerical experiments are also described.

Equilibrium pricing of European crude oil options with stochastic behaviour and jump risks

In this paper, we investigate the pricing of European crude oil options under nonlinear dynamics with stochastic behaviour and jump risks, incorporating the features of arising convenience yield of crude oil and potential extreme fluctuation in the dynamics of crude oil prices. We present a closed-form solution to European crude oil option prices under an incomplete market setting, after deriving the pricing kernel with the equilibrium method and determining the risk neutral dynamics of crude oil prices. We extend our model to a mean-reverting stochastic volatility case, which also admits an analytical formula for the equilibrium price of European crude-oil options. Finally, our model is shown to be consistent with a number of interesting facts documented in the recent literature.

Hedging using reinforcement learning: Contextual k-armed bandit versus Q-learning

The construction of replication strategies for contingent claims in the presence of risk and market friction is a key problem of financial engineering. In real markets, continuous replication, such as in the model of Black, Scholes and Merton (BSM), is not only unrealistic but is also undesirable due to high transaction costs. A variety of methods have been proposed to balance between effective replication and losses in the incomplete market setting. With the rise of Artificial Intelligence (AI), AI-based hedgers have attracted considerable interest, where particular attention is given to Recurrent Neural Network systems and variations of the Q-learning algorithm. From a practical point of view, sufficient samples for training such an AI can only be obtained from a simulator of the market environment. Yet if an agent is trained solely on simulated data, the run-time performance will primarily reflect the accuracy of the simulation, which leads to the classical problem of model choice and calibration. In this article, the hedging problem is viewed as an instance of a risk-averse contextual k-armed bandit problem, which is motivated by the simplicity and sample-efficiency of the architecture, which allows for realistic online model updates from real-world data. We find that the k-armed bandit model naturally fits to the Profit and Loss formulation of hedging, providing for a more accurate and sample efficient approach than Q-learning and reducing to the Black-Scholes model in the absence of transaction costs and risks.

Learning equilibrium mean‐variance strategy

AbstractWe study a dynamic mean‐variance portfolio optimization problem under the reinforcement learning framework, where an entropy regularizer is introduced to induce exploration. Due to the time–inconsistency involved in a mean‐variance criterion, we aim to learn an equilibrium policy. Under an incomplete market setting, we obtain a semi‐analytical, exploratory, equilibrium mean‐variance policy that turns out to follow a Gaussian distribution. We then focus on a Gaussian mean return model and propose a reinforcement learning algorithm to find the equilibrium policy. Thanks to a thoroughly designed policy iteration procedure in our algorithm, we prove the convergence of our algorithm under mild conditions, despite that dynamic programming principle and the usual policy improvement theorem failing to hold for an equilibrium policy. Numerical experiments are given to demonstrate our algorithm. The design and implementation of our reinforcement learning algorithm apply to a general market setup.

Collective longevity swap: A novel longevity risk transfer solution and its economic pricing

To mitigate the hedger’s longevity risk exposure, this paper proposes a collective longevity swap between a reinsurer (hedge provider) and a group of hedgers (pension plans and annuity providers), and an economic framework to price longevity risk and longevity swaps. Combining the appealing features of two widely discussed longevity swaps, i.e., the indemnity swap and index-based swap, the collective longevity swap requires that reinsurer’s payments are based on the longevity risk of the aggregate portfolio of all hedgers, whereas each hedger could receive indemnity payments on their own portfolio. In a general principal-agent pricing model under the incomplete market setting, we determine the optimal risk premiums and hedge rates that maximize the reinsurer’s expected profit under a set of the hedgers’ participation constraints. We find that, in a competitive market, the proposed collective swap could lead to a higher expected profit for the reinsurer than the indemnity and the index-based swap, and simultaneously improve the benefits of the majority of hedgers without worsening the benefits of the rest. Finally, using a stochastic multi-population mortality model and real-world mortality data, a series of numerical analyses are performed to verify the theoretical findings.

Learning Equilibrium Mean-Variance Strategy

We study a dynamic mean-variance portfolio optimization problem under the reinforcement learning framework, where an entropy regularizer is introduced to induce exploration. Due to the time-inconsistency involved in a mean-variance criterion, we aim to learn an equilibrium strategy. Under an incomplete market setting, we obtain a semi-analytical, exploratory, equilibrium mean-variance strategy that turns out to follow a Gaussian distribution. We then focus on a Gaussian mean return model and propose an algorithm to find the equilibrium strategy using the reinforcement learning technique. Thanks to a thoroughly designed policy iteration procedure in our algorithm, we can prove our algorithm's convergence under mild conditions, despite that dynamic programming principle and the usual policy improvement theorem fail to hold for an equilibrium solution. Numerical experiments are given to demonstrate our algorithm.

Incidence of Capital Income Taxation in a Lifecycle Economy with Firm Heterogeneity

We study the incidence of capital income taxation in a dynamic general equilibrium model with heterogeneous firms and lifecycle households. In this incomplete market setting, marginal excess burdens of three capital taxes, namely corporate income, dividend and capital gains taxes, are vastly different due to heterogeneous responses of firms and households, and heterogeneous effects of general equilibrium adjustments. Overall, taxing capital with a corporate income tax at the firm level results in higher excess burden than taxing capital with dividend and capital gains taxes at the household level. Given the salient features of the 2013 U.S. tax system, reforms that shift tax burden from the firm to household side potentially result in aggregate efficiency gains and overall welfare improving. However, the welfare benefits of these tax reforms are quite different across households and generations over transition, depending on skill type, age-cohort and government budget balancing rule. Thus, our findings highlight the importance of accounting for firm and household heterogeneity when analyzing the aggregate and distributional effects of capital income taxation.

The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time

<p style="text-indent:20px;">Motivated by financial and empirical arguments and in order to introduce a more flexible methodology of pricing, we provide a new approach to asset pricing based on Backward Volterra equations. The approach relies on an arbitrage-free and incomplete market setting in continuous time by choosing non-unique pricing measures depending either on the time of evaluation or on the maturity of payoffs. We show that in the latter case the dynamics can be captured by a <em>time-delayed</em> backward stochastic Volterra integral equation here introduced which, to the best of our knowledge, has not yet been studied. We then prove an existence and uniqueness result for time-delayed backward stochastic Volterra integral equations. Finally, we present a Lucas-type consumption-based asset pricing model that justifies the emergence of stochastic discount factors matching the term structure of Sharpe ratios.

Pricing virtual currency-linked derivatives with time-inhomogeneity

In this study, we provide a theoretical exploration of cryptocurrency option pricing under the presence of regime-switching cryptocurrency prices with jumps and state-dependent interest rates. The properties of cryptocurrency returns are empirically investigated, and the European-style cryptocurrency options are priced when the dynamics of the cryptocurrency price and the instantaneous forward interest rate are, respectively, driven by a two-factor, state-dependent stochastic volatility model with jumps and a state-dependent Heath-Jarrow-Morton model. Under an incomplete market setting, we employ a dynamic measure change technique to determine a pricing kernel and state-dependent risk premiums and then derive the option pricing formula. Numerical illustrations verify the superior pricing performance of proposed model over competing models, and empirical results show the fitness of the proposed model.

Cojump risks and their impacts on option pricing

In this study, we investigate the no-arbitrage valuation of options on two assets whereby one asset is exchanged for another. For incorporating surprising events in the underlying asset price dynamics, we model such stochastic processes by a correlated bivariate jump-diffusion model with capturing both individual jumps and systematic cojumps. Furthermore, we employ the technique of Esscher transform to determine a pricing kernel for option valuation under an incomplete market setting. Finally, the estimated results and numerical examples are given.

A Dynamic Mean-Variance Analysis for Log Returns

We propose a dynamic portfolio choice model with the mean-variance criterion for log returns. The model yields time-consistent portfolio policies and is analytically tractable even under some incomplete market settings. The portfolio policies conform with conventional investment wisdom (e.g., richer people should invest more absolute amounts of money in risky assets; the longer the investment time horizon, the more proportional amount of money should be invested in risky assets; and for long-term investment, people should not short-sell major stock indices whose returns are higher than the risk-free rate), and the model provides a direct link with the constant relative risk aversion utility maximization in a complete market. This paper was accepted by Kay Giesecke, finance.

Variance minimizing strategies for stochastic processes with applications to tracking stock indices

AbstractThis paper extends the notion of variance optimal hedging of contingent claims under the incomplete market setting to the hedging of entire processes and applies the results to the problem of tracking stock indices. Sufficient conditions under which this is possible are given, along with the corresponding variance minimizing strategy. The performances of tracking error variance (TEV) minimizing, locally risk minimizing, and variance minimizing strategies in tracking stock indices are investigated using both simulated and historical market data. In particular, it is shown using S&amp;P500 data over the period 2000 and 2015 that the TEV of the variance minimizing strategy is statistically lower than other strategies at the 95% confidence level for 6‐month holding periods.

Volumetric Risk Hedging Strategies and Basis Risk Premium for Solar Power

This paper studies volumetric risk hedging strategies for solar power under incomplete market settings with a twofold proposal of temperature-based and solar power generation-based models for solar power derivatives and discusses the basis risk arising from solar power volumetric risk hedge with temperature. Based on an indirect modeling of solar power generation using temperature and a direct modeling of solar power generation, we design two types of call options written on the accumulated non cooling degree days (ANCDDs) and the accumulated low solar power generation days (ALSPGDs), respectively, which can hedge cool summer volumetric risk more appropriately than those on well-known accumulated cooling degree days. We offer the pricing formulas of the two options under the good-deal bounds (GDBs) framework, which can consider incompleteness of solar power derivative markets. To calculate the option prices numerically, we derive the partial differential equations for the two options using the GDBs. Empirical studies using Czech solar power generation and Prague temperature estimate the parameters of temperature-based and solar power generation-based models, respectively. We numerically calculate the call option prices on ANCDDs and ALSPGDs, respectively, as the upper and lower price boundaries using the finite difference method. Results show that the call option prices based on a solar power generation process are bigger than the call option prices based on a temperature process. This is consistent with the fact that the solar power generation approach takes into account more comprehensive risk than the temperature approach, resulting in the bigger prices for the solar power generation approach. We finally show that the basis risk premiums, i.e., solar power generation-based call option prices minus temperature-based call option prices, decrease in line with initial temperature greater than around 25 ◦C. This may be because the uncertainty in solar power generation by temperature decreases due to the cancellation between the increase in solar power generation due to the increase in solar radiation and the decrease in solar power generation due to the decrease in solar panel efficiency.

A Dynamic Mean-Variance Analysis for Log Returns

We propose a dynamic portfolio choice model with the mean-variance criterion for log-returns. The model yields time-consistent portfolio policies and is analytically tractable even under some incomplete market settings. The portfolio policies conform with conventional investment wisdom (e.g. richer people should invest more absolute amount of money in risky assets; the longer investment time horizon, the more proportional amount of money should be invested in risky assets; and for long-term investment, people should not short sell major stock indices whose returns are higher than the risk-free rate), and the model provides a direct link with the CRRA utility maximization in a complete market.

An Improvement of Gain-Loss Price Bounds on Options Based on Binomial Tree and Market-Implied Risk-Neutral Distribution

This paper investigates the approximated arbitrage bounds of option prices in an incomplete market setting and draws implications for option pricing and risk management. It gives consideration to periods of global financial crisis and European sovereign debt crisis. To this end, we employ the gain-loss ratio method combined with the market-implied risk-neutral distribution calculated by binomial tree to investigate the options price bounds. Our implied gain-loss bounds of option prices are preference-free and parametric-free to avoid the misspecification error of subjective choice on the benchmark model of gain-loss ratio, and consequently, greatly reduce model risk and market risk. The empirical results show that there are option prices breaking the gain-loss bounds, even after taking into account the market information. This means that a good risk management technique and good-deal investment opportunities exist if the implied binomial tree is used as a benchmark model in the gain-loss bounds.

Integrability of Demand in Incomplete Markets: Kreps-Porteus-Selden Preferences

We provide necessary and sufficient conditions such that consumption and asset demands in an incomplete market setting can be rationalized by Kreps-Porteus-Selden preferences and provide a means for recovering the underlying unique representations of risk and time preferences. The incompleteness of asset markets introduces two serious problems in attempting to use the classic Slutsky symmetry and negative semideniteness properties employed in certainty demand analysis. First, contingent claim prices are not unique and second, they do not vary independently. Non-uniqueness is a key obstacle to rationalizing conditional asset demand by a representation of risk preferences. Non-independence precludes proving the existences of an overall Kreps-Porteus-Selden representation dened over consumption and contingent claims and hence the existence of a representation of time preferences. Mechanisms are provided for overcoming both obstacles.

Temporal and Spatial Turnpike-Type Results Under Forward Time-Monotone Performance Criteria

We present turnpike-type results for the risk tolerance function in an incomplete market setting under time-monotone forward performance criteria. We show that, contrary to the classical case, the temporal and spatial limits do not coincide. We also show that they depend directly on the left- and right-end of the support of an underlying measure, which is used to construct the forward performance criterion. We provide examples with discrete and continuous measures, and discuss the asymptotic behavior of the risk tolerance for each case.

Variance Minimizing Strategies for Stochastic Processes with Applications to Tracking Stock Indices

This paper extends the notion of variance optimal hedging of contingent claims under the incomplete market setting to the hedging of entire processes, and applies the results to the problem of tracking stock indices. Sufficient conditions under which this is possible are given, along with the corresponding variance optimal strategy in feedback form as given in Schweizer (1996) and Pham, Rheinlander, and Schweizer (1998) for contingent claims. The performances of tracking error variance minimizing, locally risk minimizing, and variance minimizing strategies in tracking stock indices are investigated using both simulated and historical market data.