Search Results (23)

This paper addresses a critical inconsistency in models of the term structure of interest rates (TSIR), where zero-coupon bonds are priced under risk-neutral measures distinct from those used in equity markets. We consider a unified TSIR framework that treats zero-coupon bonds as European options with deterministic payoffs, ensuring that they are priced under the same risk-neutral measure that governs equity derivatives. Using put–call parity, we extract zero-coupon bond implied yield curves from S&P 500 index options and compare them with the US daily treasury par yield curves. As the implied yield curves contain maturity time T and strike price K as independent variables, we investigate the $K$—dependence of the implied yield curve. Our findings, that at-the-money option-implied yield curves provide the closest match to treasury par yield curves, support the view that the equity options market contains information that is highly relevant for the TSIR. By insisting that the risk-neutral measure used for bond valuation is the same as that revealed by equity derivatives, we offer a new organizing principle for future TSIR research. Full article

In this research, we summarize the results of implementing the market risk premium into the option valuation formulas of the Black–Scholes–Merton model for out-of-the-money (OTM) options. We show that derivative prices can partly depend on systematic market risk, which the BSM model ignores by construction. Specifically, empirical studies are conducted using 50ETF options obtained from the Shanghai Stock Exchange, covering the periods from January 2018 to September 2022 and from December 2023 to October 2025. The pricing of the OTM options shows that the adjusted BSM formulas exhibit better pricing performance compared with the market prices of the OTM options tested. Furthermore, a framework for the empirical analysis of option prices based on the Capital Asset Pricing Model (CAPM) or factor models is discussed, which may lead to option formulas using non-homogeneous heat equations. The later proposal requires further statistical testing using real market data but offers an alternative to the existing risk-neutral valuation of options. Full article

Hirshleifer’s difference-form contest technology is a useful tool in the study of a class of conflict, especially military combats. We aim to highlight an important feature that the Hirshleifer contest model distinctively has, namely passivity (bidding zero effort) may stand as an effective choice in conflict even when the contest is highly deterministic (i.e., with small noise). For that purpose, we establish two propositions on the contest with $n\ge 2$ risk-neutral contestants under small noise. The first proposition states that every contestant bids arbitrarily close to zero (if not bidding zero with positive probability at all) under sufficiently small noise. The second proposition, more strikingly, states that every contestant either bids arbitrarily close to the second-highest valuation (among all the contestants’ valuations), or simply remains passive with certainty under any sufficiently small noise. We further show that the first proposition holds for the contest between risk-averse individuals endowed with constant absolute risk aversion as well, and illustrate by an example how quickly polarization in bidding among contestants, as is predicted by the propositions, may emerge as the noise of the contest abates. These results help pave the way toward a complete characterization of the difference-form contest. Full article

The untapped potential of wave energy offers another alternative to diversifying renewable energy sources and addressing climate change by reducing CO₂ emissions. However, development costs to mature the technology remain significant hurdles to adoption at scale and the technology often must compete against other marine energy renewables such as offshore wind. Here, we conduct a real option valuation that includes the uncertain market price of wholesale electricity and managerial flexibility expressed in determining future optimal decisions. We demonstrate the probability that the project’s embedded compound real option value can turn a negative net present value wave energy project to a positive expected value. This change in investment decision uses decision tree analysis, where real options are developed as decision nodes, and models the uncertainty as a risk-neutral stochastic process using chance nodes. We also show how our results are analogous to a financial out-of-the-money call option. Our results highlight the distribution of outcomes and the benefit of a staged long-term investment in wave energy systems to better understand and manage project risk, recognizing that these probabilistic results are subject to the ongoing evolution of wholesale electricity prices and the stochastic process models used here to capture their future dynamics. Lastly, we show that the near-term optimal decision is to continue to fund ongoing development of a reference architecture to a higher technology readiness level to maintain the long-term option to deploy such a renewable energy system through private investment or private–public partnerships. Full article

Simulating and predicting carbon storage under different development scenarios is crucial for formulating effective carbon management strategies and achieving carbon neutrality goals. However, studies that focus on specific regions and incorporate local policy context require further investigation. Taking Fujian Province as a case study, this research developed four policy-driven scenarios—natural development, farmland protection, urban development, and ecological protection—based on local policy frameworks. Using the PLUS (Patch-generating Land Use Simulation) and InVEST (Integrated Valuation of Ecosystem Services and Tradeoffs) models, the study simulated and predicted the carbon storage dynamics under each scenario. The results show that carbon storage declined from 1995 to 2020, mainly due to the conversion of forests and agricultural land into construction areas. The ecological protection scenario demonstrated the highest potential for carbon storage recovery, projecting an increase to 2.02 billion tons by 2030, driven by afforestation and conservation initiatives. Conversely, the urban development scenario posed the greatest risks, leading to substantial losses. Key conservation areas, including 12 priority districts, were identified in the western and northwestern regions, while coastal urban areas, comprising 31 vulnerable districts, face significant carbon storage losses. These findings emphasize the need for balanced land use policies that prioritize both urban development and ecological protection to achieve sustainable carbon management. Full article

We develop a binary tree pricing model with underlying asset price dynamics following Itô–McKean skew Brownian motion. Our work was motivated by the Corns–Satchell, continuous-time, option pricing model. However, the Corns–Satchell market model is incomplete, while our discrete-time market model is defined in the natural world, extended to the risk-neutral world under the no-arbitrage condition where derivatives are priced under uniquely determined risk-neutral probabilities, and is complete. The skewness introduced in the natural world is preserved in the risk-neutral world. Furthermore, we show that the model preserves skewness under the continuous-time limit. We provide empirical applications of our model to the valuation of European put and call options on exchange-traded funds tracking the S&P Global 1200 index. Full article

Defined benefit (DB) pension plans are a primary type of pension schemes with the sponsor assuming most of the risks. Longevity-indexed bonds have been used to hedge or transfer risks in pension plans. Our objective is to study an aggregated DB pension plan’s optimal risk management problem focusing on minimizing the solvency risk over a finite time horizon and to investigate the investment strategies in a market, comprising a longevity-indexed bond and a risk-free asset, under stochastic nominal interest rates. Using the dynamic programming technique in the stochastic control problem, we obtain the closed-form optimal investment strategy by solving the corresponding Hamilton–Jacobi–Bellman (HJB) equation. In addition, a comparative analysis implicates that longevity-indexed bonds significantly reduce solvency risk compared to zero-coupon bonds, offering a strategic advantage in pension fund management. Besides the closed-form solution and the comparative study, another novelty of this study is the extension of actuarial liability (AL) and normal cost (NC) definitions, and we introduce the risk neutral valuation of liabilities in DB pension scheme with the consideration of mortality rate. Full article

The celebrated Heston’s stochastic volatility (SV) model for the valuation of European options provides closed form solutions that are given in terms of characteristic functions. However, the numerical calibration of this five-parameter model, which is based on market option data, often remains a daunting task. In this paper, we provide a theoretical solution to the long-standing ‘open problem’ of characterizing the class of risk neutral distributions (RNDs), if any, that satisfy Heston’s SV for option valuation. We prove that the class of scale parameter distributions with mean being the forward spot price satisfies Heston’s solution. Thus, we show that any member of this class could be used for the direct risk neutral valuation of option prices under Heston’s stochastic volatility model. In fact, we also show that any RND with mean being the forward spot price that satisfies Heston’s option valuation solution must also be a member of the scale family of distributions in that mean. As particular examples, we show that under a certain re-parametrization, the one-parameter versions of the log-normal (i.e., Black–Scholes), gamma, and Weibull distributions, along with their respective inverses, are all members of this class and thus, provide explicit RNDs for direct option pricing under Heston’s SV model. We demonstrate the applicability and suitability of these explicit RNDs via exact calculations and Monte Carlo simulations, using already published index data and a calibrated Heston’s model (S&P500, ODAX), as well as an illustration based on recent option market data (AMD). Full article

Decentralized finance (DeFi) protocols use blockchain-based tools to mimic banking, investment and trading solutions and provide a viable framework that creates incentives and conditions for the development of an alternative financial services market. In this respect, they can be seen as alternative financial vehicles that mitigate portfolio risk, which is particularly important at a time of increasing uncertainty in financial markets. In particular, some DeFi protocols offer an automated, low-risk way to generate returns through a “delta-neutral” trading strategy that reduces volatility. The main financial operations of DeFi protocols are implemented using appropriate algorithms, but unlike traditional finance, where issues of value and valuation are commonplace, DeFis lack a similar value-based analysis. The aim of this study is to evaluate relevant DeFi performance metrics related to the valuations of these protocols through a thorough analysis based on various scientific methods and to show what influences the valuations of these protocols. More specifically, the study identifies how DeFi protocol valuations depend on the total value locked and other performance variables, such as protocol revenue, total revenue, gross merchandise volume and inflation factor, and assesses these relationships. The study analyzes the valuations of 30 selected protocols representing three different classes of DeFi (i.e., decentralized exchanges, lending protocols and asset management) in relation to their respective performance measures. The analysis presented in the article is quantitative in nature and relies on Granger causality tests as well as the results of a fixed effects panel regression model. The results show that the valuations of DeFi protocols depend to some extent on the performance measures of these protocols under study, although the magnitude of the relationships and their directions differ for the different variables. The Granger causality test could not confirm that future DeFi protocol valuations can be effectively predicted by the TVLs of these protocols, while other directions of causality (one-way and two-way) were confirmed, e.g., a two-way causal relationship between DeFi protocol valuations and gross merchandise volume, which turned out to be the only variable that Granger-causes future DeFi protocol valuations. Full article

We present the Generalized Gamma (GG) distribution as a possible risk neutral distribution (RND) for modeling European options prices under Heston’s stochastic volatility (SV) model. We demonstrate that under a particular reparametrization, this distribution, which is a member of the scale-parameter family of distributions with the mean being the forward spot price, satisfies Heston’s solution and hence could be used for the direct risk-neutral valuation of the option price under Heston’s SV model. Indeed, this distribution is especially useful in situations in which the spot’s price follows a negatively skewed distribution for which Black–Scholes-based (i.e., the log-normal distribution) modeling is largely inapt. We illustrate the applicability of the GG distribution as an RND by modeling market option data on three large market-index exchange-traded funds (ETF), namely the SPY, IWM and QQQ as well as on the TLT (an ETF that tracks an index of long-term US Treasury bonds). As of the writing of this paper (August 2021), the option chain of each of the three market-index ETFs shows a pronounced skew of their volatility ‘smile’, which indicates a likely distortion in the Black–Scholes modeling of such option data. Reflective of entirely different market expectations, this distortion in the volatility ‘smile’ appears not to exist in the TLT option data. We provide a thorough modeling of the option data we have on each ETF (with the 15 October 2021 expiration) based on the GG distribution and compare it to the option pricing and RND modeling obtained directly from a well-calibrated Heston’s SV model (both theoretically and also empirically, using Monte Carlo simulations of the spot’s price). All three market-index ETFs exhibited negatively skewed distributions, which are well-matched with those derived under the GG distribution as RND. The inadequacy of the Black–Scholes modeling in such instances, which involves negatively skewed distribution, is further illustrated by its impact on the hedging factor, delta, and the immediate implications to the retail trader. Similarly, the closely related Inverse Generalized Gamma distribution (IGG) is also proposed as a possible RND for Heston’s SV model in situations involving positively skewed distribution. In all, utilizing the Generalized Gamma distributions as possible RNDs for direct option valuations under the Heston’s SV is seen as particularly useful to the retail traders who do not have the numerical tools or the know-how to fine-calibrate this SV model. Full article

Achieving climate neutrality, as dictated by international agreements such as the Paris Agreement, the United Nations Agenda 2030 and the European Green Deal, requires the conscription of all parts of society. The business world and, in particular, large enterprises have a leading role in this effort. Businesses can contribute to this effort by establishing a reporting and operating framework according to specific Environmental, Social and Governance (ESG) criteria. The interest of companies in the ESG framework has become more intense in the recent years, as they recognize that apart from an improved reputation, ESG criteria can add value to them and help them to become more effective in their functioning. In particular, large European companies are legally obligated by the Non-Financial Reporting Directive (NFRD—Directive 2014/95/EU) to disclose non-financial information on how they deal with social and environmental issues. In the literature, there are discussions on the extent to which a good ESG performance affects a company’s profitability, valuation, capital efficiency and risk. The purpose of this paper is to examine empirically whether a relationship between good ESG performance and the good financial condition of companies can be documented. For a sample of the top 50 European companies in terms of ESG performance (STOXX Europe ESG Leaders 50 Index), covering a wide range of sectors, namely Automobiles, Consumer Products, Energy, Financial Services, Manufacturing, etc., we first reviewed their reportings to see which ESG framework they use to monitor their performance. Next, we examined whether there is a pattern of better financial performance compared to other large European corporations. Our results showed that such a connection seems to exist at least for some specific parameters, while for others, such a claim cannot be supported. Full article

We address the process of discounting in random environments, which allows valuation of the future in economic terms. We review several approaches to the problem regarding different well-established stochastic market dynamics in the continuous-time context and include the Feynman–Kac approach. We also review the relation between bond-pricing theory and discounting and introduce both the market price of risk and the risk neutral measure from an intuitive point of view devoid of excessive formalism. We provide the discount for each economic model and discuss their key results. We finally present a summary of our previous empirical studies for several countries on the long-run discount problem. Full article

When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each factor or cost in isolation, while seemingly ignoring the combined effects. Instead, risk factors and costs can be jointly accounted for ab initio in the pricing mechanism at the level of cash flows, and this “adjusted cash flow" approach leads to a nonlinear valuation formula. While for practitioners this made more sense because it showed which discount factor is used for which cash flow (recall the multi-curve environment post-crisis), for academics, the focus was on checking that the resulting nonlinear valuation formula is consistent with the theoretical arbitrage-free “replication approach” that we also analyse in the paper. We formulate specific reasonable assumptions, which ensure that the valuation formulae obtained by the two approaches coincide, thus reinforcing both academics’ and practitioners’ confidence in adopting such nonlinear valuation formulae in a multi-curve setup. Full article

Investors’ decisions on capital markets depend on their anticipation and preferences about risk, and volatility is one of the most common measures of risk. This paper proposes a method of estimating the market price of volatility risk by incorporating both conditional heteroscedasticity and nonlinear effects in market returns, while accounting for asymmetric shocks. We develop a model that allows dynamic risk premiums for the underlying asset and for the volatility of the asset under the physical measure. Specifically, a nonlinear in mean time series model combining the asymmetric autoregressive conditional heteroscedastic model with leverage (NGARCH) is adapted for modeling return dynamics. The local risk-neutral valuation relationship is used to model investors’ preferences of volatility risk. The transition probabilities governing the evolution of the price of the underlying asset are adjusted for investors’ attitude towards risk, presenting the asset returns as a function of the risk premium. Numerical studies on asset return data show the significance of market shocks and levels of asymmetry in pricing the volatility risk. Estimated premiums could be used in option pricing models, turning options markets into volatility trading markets, and in measuring reactions to market shocks. Full article

We consider the model of Antonacci, Costantini, D’Ippoliti, Papi (arXiv:2010.05462 [q-fin.MF], 2020), which describes the joint evolution of inflation, the central bank interest rate, and the short-term interest rate. In the case when the diffusion coefficient does not depend on the central bank interest rate, we derive a semi-closed valuation formula for contingent derivatives, in particular for Zero Coupon Bonds (ZCBs). By using ZCB yields as observations, we implement the Kalman filter and obtain a dynamical estimate of the short-term interest rate. In turn, by this estimate, at each time step, we calibrate the model parameters under the risk-neutral measure and the coefficient of the risk premium. We compare the market values of German interest rate yields for several maturities with the corresponding values predicted by our model, from 2007 to 2015. The numerical results validate both our model and our numerical procedure. Full article