Incomplete Markets Research Articles

Intermittency and Electricity Retailing: An Incomplete Market Approach

Intermittency and Electricity Retailing: An Incomplete Market Approach

The Generalized Demand Model for a Risky Asset in the Presence of a Background Risk: Market Incompleteness Premium and the Solution

The Generalized Demand Model for a Risky Asset in the Presence of a Background Risk: Market Incompleteness Premium and the Solution

Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets

Abstract The pricing and hedging of financial derivatives have become one of the hot research issues in mathematical finance today. In the case of non-risk neutrality, this article uses the martingale method and probability measurement method to study the pricing method and hedging strategy of financial derivatives. This paper also further studies the hedging strategy of financial derivatives in the incomplete market based on the BSM model and converts the solution of this problem into solving a vector on the Hilbert space to its closure. The problem of space projection is to use projection theory to decompose financial derivatives under a given martingale measure. In the imperfect market, the vertical projection theory is used to obtain the approximate pricing method and hedging strategy of financial derivatives in which the underlying asset follows the martingale process; the projection theory is further expanded, and the pricing problem of financial derivatives under the mixed-asset portfolio is obtained. Approximate pricing of financial derivatives; in the discrete state, the hedging investment strategy of financial derivatives H in the imperfect market is found through the method of variance approximation.

Capital and Labor Income Pareto Exponents Across Time and Space

We estimate capital and labor income Pareto exponents across 475 country‐year observations that span 52 countries over half a century (1967–2018). We document two stylized facts: (i) capital income is more unequally distributed than labor income in the tail; namely, the capital exponent (1–3, median 1.46) is smaller than labor (2–5, median 3.35), and (ii) capital and labor exponents are nearly uncorrelated. To explain these findings, we build an incomplete market model with job ladders and capital income risk that gives rise to a capital income Pareto exponent smaller than but nearly unrelated to the labor exponent. Our results suggest the importance of distinguishing income and wealth inequality.

Analysis of Option-Like Fund Performance Fees in Asset Management via Monte Carlo Actuarial Distortion Pricing

A detailed analysis of management and performance fees for asset managers and investment funds is undertaken. Whilst, fund fees are considered as a cost of capital for investors, the structuring of such fee mechanisms in a fund can also influence a fund manager’s decisions and investment strategy, thereby also influencing the investment performance of the investors funds. The study undertaken will allow for an assessment of the effect of fee structures and the potential for asymmetric incentives to arise that may promote adverse risk taking behaviours by the fund manager, to the detriment of the investor or retiree who places a portion of their retirement savings into such a managed fund with such fee structures. As such, understanding the mechanism of fee charging as well as pricing the fees correctly is vital. An exploration of the application of actuarial distortion pricing methods for complete and incomplete market valuation is performed on a variety of path dependent option-like performance fee structures for various funds in the European and American markets. Furthermore, several scenario analysis and sensitivity studies are undertaken. The class of Net Asset Value (NAV) models adopted are Levy processes and the pricing is performed via Monte Carlo techniques.

Optimal exercise of American puts with transaction costs under utility maximization

American option pricing plays an essential role in quantitative finance and has been extensively studied in the past. However, how transaction costs affect the American option price, particularly the most important feature of American options, the optimal exercise price, is much less investigated. It is primarily because such a study must be conducted under an incomplete market, which presents additional difficulties on top of an already difficult nonlinear mathematical problem. This paper attempts to provide a supplement study in this area by analyzing the optimal exercise price of an American option in addition to the option price itself in the presence of transaction costs through a utility-based approach. With a computationally efficient numerical scheme, we are able to demonstrate clearly how the optimal exercise price should be calculated and consequently how the option prices for the buyer and writer as well as the early exercise decision are affected by the inclusion of transaction cost.

Optimal decision of dynamic wealth allocation with life insurance for mitigating health risk under market incompleteness

This research proposes a dynamic control modeling of income allocation between life insurance purchase and consumption subject to market incompleteness. We adopt a no-good-deal specification of the multiplicity of stochastic discount factors, and treat death as a stopping time on the individual’s health status. Health status non-tradability renders death risk non-hedgeable and the life insurance market incomplete. The no-good-deal condition provides useful guidance for individuals to choose the insurance-income ratio inside good-deal bounds. The magnitude of the influence of market incompleteness on life insurance demand (measured as the width of good-deal bounds) displays life-cycle patterns that converge over time and are highly sensitive to health status characteristics.

Equity home bias and consumption-real exchange rate puzzles: A joint solution

In a standard dynamic stochastic general equilibrium model with a complete asset market, home agents should hold a foreign equity biased portfolio to hedge the non-traded labor income risk, which contradicts home equity biased portfolios observed worldwide. As the labor income share increases, the degree of home bias should decrease because there is more incentive to hold foreign equity. In the data, there is not any evidence that the labor income share and the degree of home bias are negatively correlated. The standard model also predicts that the consumption differential-real exchange rate correlation is positive, while it is negative in the data. I show that a combination of market incompleteness, non-tradable goods, and labor supply can explain the three features above. My model can generate a large equity home bias, despite the strong positive correlation of non-traded human capital return with domestic equity return. The home bias is not sensitive to the labor income share. The consumption differential-real exchange rate unconditional correlation generated by my model simulation is zero.

American options in a non-linear incomplete market model with default

We study the superhedging problem for American options with completely irregular payoffs in a non-linear and incomplete market model with default. We give a dual representation of the seller’s (superhedging) price in terms of the value of a non-linear mixed control/stopping problem, involving a suitable set of equivalent probability measures. We characterize the seller’s price process as the minimal supersolution of two types of reflected BSDEs: a constrained one and an optional one. Under some regularity assumptions on the pay-off, we show a duality result for the buyer’s price in terms of the value of a non-linear control/stopping game problem.

Pricing analysis of wind power derivatives for renewable energy risk management

The objective of this study is to analyse the theoretical pricing of wind power derivatives, which is important for renewable energy risk management but has a problem in the pricing due to the illiquidity of the assets and to show the application of the theory to the practical implementation of the pricing. We make three contributions to the literature. First, to the best of our knowledge, we are the first to conduct a detailed econometric analysis of the wind power futures underlying, i.e., the electricity production based on windmills, resulting in strong support of seasonality and mean reversion in the logit-transformed wind power load factors. Second, after proposing a new model of wind power load factors based on the econometric findings, we analyse the theoretical prices of wind power futures and call option contracts to which the good-deal bounds pricing within an illiquid market situation is applied as well as we show the application of the theory to the practical pricing with the illiquidity. Third, our empirical pricing analysis shows that theoretical wind power futures prices derived using seasonal modelling more accurately reflect reality than those derived without seasonality compared to market observations, resulting in the importance of seasonality modelling in theoretical wind power derivatives pricing. In particular, considering that the upper and lower price boundaries represent the selling and the buying prices in the incomplete market, respectively, we show that the pricing of the short position is more affected by the seasonality than the pricing of the long position. Finally, we illustrate and discuss the practical applications of the results obtained in our study.

Pricing of Index Options in Incomplete Markets

We characterize a set of risk-neutral measures associated with a comprehensive class of risk averse investors. From this set, we show how to construct option price bounds and recover the implied gamma: a parameter uniquely identifying the marginal investor pricing a given option. Empirically, we find that S&P 500 option prices are reconciled by heterogeneous marginal investors who differ in their assessment of tail risk. This heterogeneity is time-varying, decreases during financial crises, and provides novel insights into the skew patterns of index options. The recovered investors' preferences related to compensation for downside risk help predict future market returns.

The Deviation of the Behaviors of Rice Farmers from Their Stated Willingness to Apply Biopesticides-A Study Carried Out in Jilin Province of China.

The substitution of chemical pesticides by biopesticides is crucial to ensure the quality of agricultural products and to foster environmental sustainability. This study takes the willingness and the behaviors of rice farmers on the application of biopesticides as the research object. The survey questionnaire was designed based on the theory of rational small-scale farmers from three aspects: “individual and family characteristics of farmers”, “cognition of farmers” and “external factors”. The survey was then conducted on 163 rice farmers in seven prefecture-level cities in Jilin Province of China. The logistic model was used to analyze the influencing factors resulting in the deviation of the behaviors of the rice farmers from their initial willingness on the application of biopesticides. The explanatory structure model (ISM) was used to analyze the logical hierarchical relationship among various influencing factors. The results show that: (1) For 45% of the farmers surveyed, there’s a deviation between their willingness and behaviors regarding the application of biopesticides; (2) Among the significant factors leading to the deviation between farmers’ willingness and behaviors concerning the application of biopesticides, the surface-level direct factor is biopesticide awareness. The mid-level indirect factors are agricultural product quality and safety awareness and the deep-level root cause is farmers’ education level. (3) The primary reason for the deviation of the farmers’ behaviors from their willingness is their lack of knowledge about biopesticides and the biopesticides’ incomplete market structure. Based on the comprehensive analysis, it is recommended to improve the professionalization of the farmers, to strengthen the publicity of green production and to accelerate the formulation of the biopesticides market to further promote the usage of biopesticides.

Entangled risks in incomplete FX markets

We introduce the concept of risk entanglement in a preference-free setting to jointly explain the exchange rate volatility, cyclicality, and currency risk premia in the data. Risk entanglement specifies a subset of incomplete market models, in which nondiffusive or nonlog-normal shocks to exchange rates are not fully spanned by asset returns. When risks are entangled, there exist multiple pricing-consistent exchange rates, but none of them are equal to the ratio of the stochastic discount factors (SDFs) or their projections. Decoupling the exchange rate from the SDFs allows us to address key FX market patterns that are puzzling in international finance.

Real options for an entrepreneur with preferences for liquidity

This paper extends the classical real options model by taking into consideration the preferences for liquidity under incomplete market, and aims to examine the implications of liquidity preferences on the optimal investment timing and option value. The results show that preferences for liquidity have two opposite effects on optimal investment strategies, and significantly erode the option value.

A subdiffusive stochastic volatility jump model

Subdiffusions appear as good candidates for modeling illiquidity in financial markets. Existing subdiffusive models of asset prices are indeed able to capture the motionless periods in the quotes of thinly-traded assets. However, they fail at reproducing simultaneously the jumps and the time-varying random volatility observed in the price of these assets. The aim of this work is hence to propose a new model of subdiffusive asset prices reproducing the main characteristics exhibited in illiquid markets. This is done by considering a stochastic volatility jump model, time changed by an inverse subordinator. We derive the forward fractional partial differential equations (PDE) governing the probability density function of the introduced model and we prove that it leads to an arbitrage-free and incomplete market. By proposing a new procedure for estimating the model parameters and using a series expansion for solving numerically the obtained fractional PDE, we are able to price various European-type derivatives on illiquid assets and to depart from the common Markovian valuation setup. This way, we show that the introduced subdiffusive stochastic volatility jump model yields consistent and reliable results in illiquid markets.

Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering

<p style='text-indent:20px;'>This paper studies the hedging problem of unit-linked life insurance contracts in an incomplete market presence of self-exciting (clustering) effect, which is described by a Hawkes process. Applying the local risk-minimization method, we manage to obtain closed-form expressions of the locally risk-minimizing hedging strategies for both pure endowment and term insurance contracts. Besides, we demonstrate the existence of the minimal martingale measure and perform numerical analyses. Our numerical results indicate that jump clustering has a significant impact on the optimal hedging strategies.</p>

Optimal investment strategy in the family of 4/2 stochastic volatility models

This paper derives optimal investment strategies for the 4/2 stochastic volatility model proposed in [Grasselli, M., The 4/2 stochastic volatility model: a unified approach for the Heston and the 3/2 model. Math. Finance, 2017, 27(4), 1013–1034] and the embedded 3/2 model [Heston, S.L., A simple new formula for options with stochastic volatility. 1997]. We maximize the expected utility of terminal wealth for a constant relative risk aversion (CRRA) investor, solving the corresponding Hamilton–Jacobi–Bellman (HJB) equations in closed form for both complete and incomplete markets. Conditions for the verification theorems are provided. Interestingly, the optimal investment strategy displays a very intuitive dependence on current volatility levels, a trend which has not been previously reported in the literature of stochastic volatility models. A full empirical analysis comparing four popular embedded models—i.e. the Merton (geometric Brownian motion), Heston (1/2), 3/2 and 4/2 models—is conducted using S&P 500 and VIX data. We find that the 1/2 model carries the larger weight in explaining the 4/2 behaviour, and optimal investments in the 1/2 and 4/2 models are similar, while investments in the 3/2 model are the most conservative in high-variance settings (20% of Merton's solution). Despite the similarities between the 1/2 and 4/2 models, wealth-equivalent losses due to deviations from the 4/2 model are largest for the1/2 and GBM models (40% over 10 years). Meanwhile, the wealth losses due to market incompleteness are harsher for the 1/2 model than for the 4/2 and 3/2 models (60% versus 40% and 30% respectively), highlighting the benefits of choosing the 4/2 or the 3/2 over the 1/2 model.

Variable annuities: Market incompleteness and policyholder behavior

Variable annuities (VAs) are popular personal savings and investment vehicles with long-term guarantees. They include various exercise-dependent features, and the pricing, valuation and hedging of the guarantees depend critically on the investors’ decision making. I study whether the optimal exercise behavior of a VA investor is affected by market incompleteness, which arises since the VA generates payout profiles that intersect financial risk and idiosyncratic mortality risk and can thus not be (fully) replicated with traditional financial and insurance products.Implementing a generic VA policy with a withdrawal guarantee (GMWB) that reflects typical US product characteristics, I find that the optimal withdrawal behavior derived from a lifecycle utility model is closely approximated by the corresponding (after-tax) value maximization strategy. That is, from the perspective of the investor, the market around VA policies is sufficiently complete to justify the common approach of assessing optimal VA policyholder behavior with a risk-neutral valuation method.

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.

Variable Annuities: Market Incompleteness and Policyholder Behavior

Variable annuities (VAs) are popular personal savings and investment vehicles with long-term guarantees. They include various exercise-dependent features, and the pricing, valuation and hedging of the guarantees depend critically on the investors' decision making. I study whether the optimal exercise behavior of a VA investor is affected by market incompleteness, which arises since the VA generates payout profiles that intersect financial risk and idiosyncratic mortality risk and can thus not be (fully) replicated with traditional financial and insurance products. Implementing a generic VA policy with a withdrawal guarantee (GMWB) that reflects typical US product characteristics, I find that the optimal withdrawal behavior derived from a lifecycle utility model is closely approximated by the corresponding (after-tax) value maximization strategy. That is, from the perspective of the investor, the market around VA policies is sufficiently complete to justify the common approach of assessing optimal VA policyholder behavior with a risk-neutral valuation method.