Contingent Claims Research Articles

Insurer green financing for a supply chain under cap-and-trade regulation: a capped call contingent claim analysis

Insurer green financing for a supply chain under cap-and-trade regulation: a capped call contingent claim analysis

Assessing insurer guarantee cover and risk retention toward SDG 3: a structure-break down-and-out call valuation

The paper develops a down-and-out call option to evaluate the equity of a life insurer. The COVID-19 pandemic challenges captured by the premature feature of the down-and-out call option are also identified as structural breaks incorporated into the contingent claim option. Hedging, shown by the guarantee cover ratio, reflects the insurer’s ability to meet policy commitments. Reinsurance, indicated by the risk retention ratio, distributes insurance risk between the insurer and reinsurer. The insurer engages in those practices aligned with Sustainable Development Goal 3 (SDG 3: good health and well-being). The main findings are as follows. The COVID-19 pandemic harms policyholder protection and deviates from SDG 3 due to financial instability. However, strategic hedging, reinsurance, and government capital injections enhance policy assurances, achieving SDG 3 by increasing payments to policyholders and improving financial stability. Moreover, the risk retention ratio increases during severe COVID-19 periods but decreases with more government capital injections, which can substitute for risk retention. Policy implications suggest that policymakers can mitigate crisis impacts on insurer capacity to fulfill commitments, thereby enhancing life insurance stability and well-being aligned with SDG 3.

Risk measures based on weak optimal transport

In this paper, we study convex risk measures with weak optimal transport penalties. In a first step, we show that these risk measures allow for an explicit representation via a nonlinear transform of the loss function. In a second step, we discuss computational aspects related to the nonlinear transform as well as approximations of the risk measures using, for example, neural networks. Our setup comprises a variety of examples, such as classical optimal transport penalties, parametric families of models, divergence risk measures, uncertainty on path spaces, moment constraints, and martingale constraints. In a last step, we show how to use the theoretical results for the numerical computation of worst-case losses in an insurance context and no-arbitrage prices of European contingent claims after quoted maturities in a model-free setting.

A new lattice approach for risk-minimization hedging under generalized autoregressive conditional heteroskedasticity models

This paper explores the calculation of risk-minimization hedging strategies, specifically local and global risk minimization strategies for contingent claims under affine and non-affine GARCH models with the known closed forms of its first four moments across times under the physical measure. A unified and efficient willow tree method is introduced for various GARCH models. Unlike methods that provide option values and hedging ratios solely at the inception time, the proposed willow tree method generates a comprehensive table of option values and hedging ratios at each discrete time step across possible asset prices. Additionally, the method showcases robust performance in hedging at lower frequencies than the underlying asset’s modeling frequency (e.g., weekly or monthly hedging using a daily GARCH model). Lastly, the willow tree method outperforms the Monte Carlo method, offering greater efficiency, accuracy, and flexibility in solving risk-minimization hedging problems.

Vertical acquisition and carbon capture and storage choices under cap-and-trade regulation with sustainable finance

In this paper, we establish a contingent claim framework for environmental management. The paper focuses on the vertical acquisition of a supply chain in the context of sustainable insurance (i.e., sustainable finance), specifically considering carbon capture and storage (CCS) technology choices and cap-and-trade regulations. The main results are as follows. Firstly, a more stringent cap-and-trade scheme increases equity efficiency in sustainable insurance, particularly when the green upstream manufacturer with advanced CCS technology acquires the brown downstream manufacturer, which also uses advanced CCS technology. Secondly, it harms risk efficiency, especially when the green upstream manufacturer with backward CCS technology acquires the brown downstream manufacturer, which uses advanced CCS technology. Thirdly, it increases the default risk in the insurer's equity, especially when the green upstream manufacturer with backward CCS technology acquires the brown downstream manufacturer, which also uses backward CCS technology. Overall, cap-and-trade's effects on equity and risk efficiency in vertical acquisition and default risk in insurer's equity within sustainable insurance depend on adopting CCS technology. Policymakers should incentivize advanced CCS technologies in vertical acquisitions to enhance equity and reduce default risks, supporting long-term financial stability and environmental sustainability toward Sustainable Development Goals.

Pricing and Hedging Contingent Claims by Entropy Segmentation and Fenchel Duality

We present a new approach to the problem of characterizing and choosing equivalent martingale pricing measures for a contingent claim, in a finite-state incomplete market. This is the entropy segmentation method achieved by means of convex programming, thanks to which we divide the claim no-arbitrage prices interval into two halves, the buyer’s and the seller’s prices at successive entropy levels. Classical buyer’s and seller’s prices arise when the entropy level approaches 0. Next, we apply Fenchel duality to these primal programs to characterize the hedging positions, unifying in the same expression the cases of super (resp. sub) replication (arising when the entropy approaches 0) and partial replication (when entropy tends to its maximal value). We finally apply linear programming to our hedging problem to find in a price slice of the dual feasible set an optimal partial replicating portfolio with minimal CVaR. We apply our methodology to a cliquet style guarantee, using Heston’s dynamic with parameters calibrated on EUROSTOXX50 index quoted prices of European calls. This way prices and hedging positions take into account the volatility risk.

Measuring ESG risk premia with contingent claims

We propose a contingent claims approach for estimating ESG risk premia from market information and market participants' decisions. To this end, we infer the asset value dynamics via the structural model of Merton [1974, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates.” Journal of Finance 29: 449–470.] for a large panel of S&P 500 firms using an estimation algorithm that utilizes the information embedded in stock market prices, CDS spreads, and default probabilities. We find a statistically significant relationship between the ESG score and the volatility and drift terms of the asset value process, suggesting that ESG factors are structurally connected to the value of the firm. We establish a mapping between ESG scores and the cost of equity and debt as implied by firm's contingent claims, and derive estimates of the ESG risk premium across different ESG and leverage profiles. In addition, we break down the ESG risk premia by industry, and demonstrate how practitioners can adjust the weighed average cost of capital of ESG laggard firms for valuation and decision making purposes.

Higher order approximation of option prices in Barndorff-Nielsen and Shephard models

We present an approximation method based on the mixing formula [Hull, J. and White, A., The pricing of options on assets with stochastic volatilities. J. Finance, 1987, 42, 281–300; Romano, M. and Touzi, N., Contingent claims and market completeness in a stochastic volatility model. Math. Finance, 1997, 7, 399–412] for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the case where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das and Langrené [Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility. Stochastics, 2022, 94, 745–788]. We obtain asymptotic results for the error of the N th order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein–Uhlenbeck process or a gamma Ornstein–Uhlenbeck process.

Neural network empowered liquidity pricing in a two-price economy under conic finance settings

In the article at hand neural networks are used to model liquidity in financial markets, under conic finance settings, in two different contexts. That is, on the one hand this paper illustrates how the use of neural networks within a two-price economy allows to obtain accurate pricing and Greeks of financial derivatives, enhancing computational performances compared to classical approaches such as (conic) Monte Carlo. The methodology proposed for this purpose is agnostic of the underlying valuation model, and it easily adapts to all models suitable for pricing in conic financial markets. On the other hand, this article also investigates the possibility of valuing contingent claims under conic assumptions, using local stochastic volatility models, where the local volatility is approximated by means of a (combination of) neural network(s). Moreover, we also show how it is possible to generate hybrid families of distortion functions to better fit the implied liquidity of the market, as well as we introduce a conic version of the SABR model, based on the Wang transform, that still allows for analytical bid and ask pricing formulae.

Information aggregation with heterogeneous traders

The efficient market hypothesis predicts that asset prices reflect all available information. Recent experimental work found the rational expectation model to outperform the prior information model in contingent claim markets when traders hold homogeneous values, despite the no trade equilibrium. However, recent experiments have also demonstrated the inability of contingent claim markets to successfully aggregate information when traders hold highly differentiated asset values. These prior findings beg the question of whether homogeneous values are a necessary condition for efficient market outcomes in contingent claim markets. The experiments reported in this paper show that homogeneous values are not a necessary condition for information aggregation.

Discounted nonzero-sum optimal stopping games under Poisson random intervention times

We present solutions to some discounted nonzero-sum optimal stopping games of two players related to the perpetual game-type contingent claims with payoffs representing linear functions of the running values of a geometric Brownian motion. It is assumed that the underlying process can be stopped by the both players only at certain random intervention times which coincide with the jump times of the two appropriate independent Poisson processes. The optimal stopping times forming a Nash equilibrium are shown to be the first times at which the underlying process is either below or above certain lower or upper constant boundaries at the jump times of the appropriate Poisson processes. The proof is based on the reductions of the original games to the associated coupled free-boundary problems and the solutions to the latter problems by means of the smooth-fit conditions at the optimal boundaries for every player. We show that the optimal stopping constant lower and upper boundaries are determined as (possibly multiple) solutions to the equivalent coupled systems of arithmetic equations. The obtained results can be interpreted as the rational valuation of some perpetual randomized Bermudian game-type contingent claims in the Black-Merton-Scholes model.

The Maximal and Minimal Distributions of Wealth Processes in Black–Scholes Markets

The Black–Scholes formula is an important formula for pricing a contingent claim in complete financial markets. This formula can be obtained under the assumption that the investor’s strategy is carried out according to a self-financing criterion; hence, there arise a set of self-financing portfolios corresponding to different contingent claims. The natural questions are: If an investor invests according to self-financing portfolios in the financial market, what are the maximal and minimal distributions of the investor’s wealth on some specific interval at the terminal time? Furthermore, if such distributions exist, how can the corresponding optimal portfolios be constructed? The present study applies the theory of backward stochastic differential equations in order to obtain an affirmative answer to the above questions. That is, the explicit formulations for the maximal and minimal distributions of wealth when adopting self-financing strategies would be derived, and the corresponding optimal (self-financing) portfolios would be constructed. Furthermore, this would verify the benefits of diversified portfolios in financial markets: that is, do not put all your eggs in the same basket.

International climate policy dilemmas: Examining effective carbon tariff and cap-and-trade regulation from a sustainable insurance perspective

This paper introduces a contingent claim model, offering a financial perspective on the global climate policy challenge. The model prioritizes the use of carbon tariffs to prevent carbon leakage and cap-and-trade initiatives to incentivize carbon emission reductions. In this model, manufacturers in an exporting country's supply chain collaborate with a life insurance company to acquire carbon allowances through the cap-and-trade mechanism. The importing country enforces carbon tariffs to protect its import-substitution industries and to prevent carbon leakage, assessed by the effective carbon tariff rate. Our findings show that increasing carbon tariffs on final products and intermediate goods reduces equity for both the life insurance company and the importer but strengthens carbon tariff protection. Conversely, a stricter regulatory cap within the exporting country's cap-and-trade system bolsters equity for the life insurance company and the importer but weakens carbon tariff effectiveness in the importing country. This dilemma underscores the urgent need for global collaboration in managing carbon emissions. Our research highlights the complex challenges of global climate policies. It is suggested to foster efficient and sustainable international cooperation in shaping climate policies to address these challenges.

Contingent claim analysis and Minsky’s financial instability hypothesis

Contingent claim analysis and Minsky’s financial instability hypothesis

Density analysis for coupled forward–backward SDEs with non-Lipschitz drifts and applications

We explore the existence of a continuous marginal law with respect to the Lebesgue measure for each component (X,Y,Z) of the solution to coupled quadratic forward–backward stochastic differential equations (QFBSDEs) for which the drift coefficient of the forward component is either bounded and measurable or Hölder continuous. Our approach relies on a combination of the existence of a weak decoupling field (see Delarue and Guatteri, 2006), the integration with respect to space time local time (see Eisenbaum, 2006), the analysis of the backward Kolmogorov equation associated to the forward component along with an Itô-Tanaka trick (see Flandoli et al., 2009). The framework of this paper is beyond all existing papers on density analysis for Markovian BSDEs and constitutes a major refinement of the existing results. We also derive a comonotonicity theorem for the control variable in this frame and thus extending the works (Chen et al., 2005; Dos Rei and Dos Rei 2013).As applications of our results, we first analyse the regularity of densities of solution to coupled FBSDEs. In the second example, we consider a regime switching term structure interest rate models (see for e.g., Ma et al. (2017)) for which the corresponding FBSDE has discontinuous drift. Our results enables us to: firstly study classical and Malliavin differentiability of the solutions for such models, secondly the existence of density of such solutions. Lastly we consider a pricing and hedging problem of contingent claims on non-tradable underlying, when the dynamic of the latter is given by a regime switching SDE (i.e., the drift coefficient is allowed to be discontinuous). We obtain a representation of the derivative hedge as the weak derivative of the indifference price function, thus extending the result in Ankirchner et al. (2010).

A pseudo-analytic generalization of the memoryless property for continuous random variables and its use in pricing contingent claims.

We explore an extension of the memoryless property for continuous random variables by using the concept of pseudo-sum. Subsequently, we demonstrate the practicality of this approach through two financial applications in which pseudo-sums characterize the values of arbitrage-free contingent claims. Moreover, we are able to establish new interesting connections between different probability distributions.

Contingent Claims and Hedging of Credit Risk with Equity Options

Abstract Using contingent-claims valuation, we introduce novel hedge ratios for credit exposures using put options. Option hedge ratios are generally in line with the empirical sensitivities of credit spread changes to put option returns and, relative to stock hedge ratios, produce further reductions in volatility for a portfolio of North American firms. We show that option hedge ratios capture option-specific credit exposure related to the VIX index and the default spread, which is unaccounted for by Merton’s (1974) equity hedge ratios alone. Combining stocks and put options for credit risk hedging can be done effectively using the volatility smirk. (JEL E43, E44, G10)

Hedging lookback-barrier option by Malliavin calculus in a mixed fractional Brownian motion environment

We present a Malliavin calculus approach to a mixed fractional Brownian motion option hedging model, that adequately describes, e.g., the hedging of a lookback-barrier option with the floating strike price. The Markovian setup and smooth stochastic differentials are necessary components in the payoff function for classical Δ-hedging of a contingent claim. This is in contrast to the Malliavin calculus approach, which may be used to any type of path-dependent options. Based on the fractional Clark–Ocone formula, an entirely probabilistic computation is developed to obtain the closed-form solution of the hedging portfolio for a lookback-barrier option with the floating strike price. Numerical experiments are performed to demonstrate the performance of our proposed hedging model, and we also conduct sensitivity analysis to investigate the correlation between model parameters and the hedging portfolio.

Quantile Hedging in the complete financial market under the mixed fractional Brownian motion model and the liquidity constraint

This article proposes a pricing framework for European option that utilizes a Quantile hedging strategy in a complete financial market. The methodology involves applying the long memory Geometric Brownian motion model, utilizing the generalized mixed fractional Girsanov theorem, and incorporating relevant findings related to quasi-conditional expectation. The first step in this framework is to derive the option price formula using the equivalent martingale measure. This approach enables investors to use the entire option price without any restrictions to hedge their contingent claim. Then, we add the initial wealth condition to hedge and determine the option price with the Quantile hedging strategy. To do this, we find the maximum probability of success in complete risk hedging for different amounts of model parameters and then use the probability values to determine the option price. Moreover, we apply the minimum variance algorithms and the error correction method to find the hedge ratio. Finally, we illustrate the hedging strategy and provide numerical results that assist investors in determining the required capital for covering the option risk under market data. This calculation takes into account both the level of risk tolerance and the anticipated return on investment.

Doubly reflected BSDEs driven by RCLL martingales under stochastic Lipschitz coefficient

In this paper, we study a backward stochastic differential equation driven by a Right Continuous with Left Limits (RCLL) martingale with two completely separated RCLL barriers. When the coefficient is stochastically Lipschitz, we demonstrate the existence and uniqueness of a square-integrable adapted solution using the penalization method. Additionally, we provide a fair price for a game contingent claim between two traders with additional exogenous knowledge of the same stock price in a public financial market driven by an Azéma’s martingale. We also determine a saddle point for the game when the obstacles are left upper semi-continuous along stopping times.