Search Results (101)

This paper demonstrates that perfectly calibrating a multi-asset model to observed market prices of all basket call options is insufficient to uniquely determine the price of a best-of call option. Previous research on multi-asset option pricing has primarily focused on complete market settings or assumed specific parametric models, leaving fundamental questions about model risk and pricing uniqueness in incomplete markets inadequately addressed. This limitation has critical practical implications: derivatives practitioners who hedge best-of options using basket-equivalent instruments face fundamental distributional uncertainty that compounds the well-recognized non-linearity challenges. We establish this non-uniqueness using convex analysis (extreme ray characterization demonstrating geometric incompatibility between payoff structures), measure theory (explicit construction of distinct equivalent probability measures), and geometric analysis (payoff structure comparison). Specifically, we prove that the set of equivalent probability measures consistent with observed basket prices contains distinct measures yielding different best-of option prices, with explicit no-arbitrage bounds $[a_K,b_K]$ quantifying this uncertainty. Our theoretical contribution provides the first rigorous mathematical foundation for several empirically observed market phenomena: wide bid-ask spreads on extremal options, practitioners’ preference for over-hedging strategies, and substantial model reserves for exotic derivatives. We demonstrate through concrete examples that substantial model risk persists even with perfect basket calibration and equivalent measure constraints. For risk-neutral pricing applications, equivalent martingale measure constraints can be imposed using optimal transport theory, though this requires additional mathematical complexity via Schrödinger bridge techniques while preserving our fundamental non-uniqueness results. The findings establish that additional market instruments beyond basket options are mathematically necessary for robust exotic derivative pricing. Full article

We introduce a fairly general, recombining trinomial tree model in the natural world. Market completeness is ensured by considering a market consisting of two risky assets, a riskless asset and a European option. The two risky assets consist of a stock and a perpetual derivative of that stock. The option has the stock and its derivative as its underlying. Using a replicating portfolio, we develop prices for European options and generate the unique relationships between the risk-neutral and real-world parameters of the model. We discuss calibration of the model to empirical data in the cases in which the risky asset returns are treated as either arithmetic or logarithmic. From historical price and call option data for select large cap stocks, we develop implied parameter surfaces for the real-world parameters in the model. Full article

We introduce a new coherent risk measure, the minimal-entropy risk measure, which is built on the minimal-entropy $\sigma$-martingale measure—a concept inspired by the well-known minimal-entropy martingale measure used in option pricing. While the minimal-entropy martingale measure is commonly used for pricing and hedging, the minimal-entropy $\sigma$-martingale measure has not previously been studied, nor has it been analyzed as a traditional risk measure. We address this gap by clearly defining this new risk measure and examining its fundamental properties. In addition, we revisit the entropic risk measure, typically expressed through an exponential formula. We provide an alternative definition using a supremum over Kullback–Leibler divergences, making its connection to entropy clearer. We verify important properties of both risk measures, such as convexity and coherence, and extend these concepts to dynamic situations. We also illustrate their behavior in scenarios involving optimal risk transfer. Our results link entropic concepts with incomplete-market pricing and demonstrate how both risk measures share a unified entropy-based foundation. Full article

Environmental, social and governance (ESG) ratings (scores) provide quantitative measures for socially responsible investment. We consider ESG scores to be a third independent variable—on par with financial risk and return—and incorporate such numeric scores into dynamic asset pricing. Based on this incorporation, we develop the entire investment process for the ESG market: portfolio optimization and efficient frontier, capital market line (the market portfolio), risk-assessment measures and hedging instruments (options). There is currently no riskless asset available in such an ESG market; to address this, we develop the so-called shadow riskless rate, applicable to markets having only risky assets. We believe this to be the first paper that fully develops, under a single dynamic pricing framework, the entire investment process for an ESG market. As there are significant differences in methodologies developed by providers of ESG scores, we do not take the position that data from any single agency are to be favored. Consequently, we utilize ESG scores from Refinitiv in the manuscript’s empirical studies and redo all computations using S&P Global RobeoSAM ESG scores. Full article

This paper presents a method for hedging in markets of two-factor diffusion and jump diffusion models under the restriction of a specified probability of success. In addition, a method for hedging with a given shortfall amount is developed. A maximal perfect hedging set is constructed for options involving the exchange of one asset for another. The developed method is applied to the pricing of equity-linked life insurance contracts, such as “pure endowments with a guarantee”. Traditional pricing approaches for hedging options often yield minimal returns for investors. By accepting a predefined level of risk, investors can achieve higher returns. In light of this, this paper proposes risk management strategies applicable to these hybrid financial and insurance products. Full article

The pursuit of sustainable development in the implementation of EU energy policy concerns, among other things, the area of trading greenhouse gas emission allowances. The increasing price volatility in the European Union Allowances (EUA) market necessitates the implementation of hedging strategies to minimize the impact of price risk on the operational performance of European enterprises. An intriguing research goal (both in terms of cognitive and practical applications) was to compare the effectiveness of hedging strategies for purchasing EUA in three scenarios: (1) without hedging; (2) hedging based on an unconditional instrument; and (3) hedging based on a conditional instrument. The analysis was conducted on a theoretical-comparative variant and on the example of an entity operating in the real economy. The research objectives were supported by the following methods: 1. Data collection, which included a review of the literature on hedging EUA purchases in the context of connections with financial risk management theories and corporate responsibility, as well as connections with EU ETS policy regulations. 2. Data processing, which involved a quantitative analysis of data mainly from the ICE Endex exchange and its historical quotations (2016–September 2024), including the determination of option pricing using the Black–Scholes model. 3. Expert judgment was used to justify the time frames adopted for the research. The findings revealed that the use of hedging in EUA purchases was effective and led to a reduction in the overall cost of acquisition throughout the analyzed period. The effectiveness of hedging based on an unconditional instrument, such as a futures contract, was higher than that of hedging based on a conditional instrument, such as an option. The results obtained provide a good basis for continuing research on the effectiveness of EUA hedging in extreme scenarios and in conditions of increased volatility. This research approach is justified by the upcoming dismantling of climate initiatives starting in 2025, related to the USA’s withdrawal from the Paris Agreement. Full article

Risk is often the most concerning factor in financial transactions. Option Greeks provide valuable insights into the risks inherent in option trading and serve as tools for risk mitigation. Traditionally, scholars compute option Greeks through extensive calculations. This article introduces an alternative method that bypasses conventional derivative computation using the Mellin transform and its properties. Specifically, we derive Greeks for perpetual American options with general payoffs, represented as piecewise linear functions, and examine higher-order risk metrics for practical implications. Examples are provided to illustrate the effectiveness of our approach, offering a novel perspective on calculating and interpreting option Greeks. Full article

Classic options can no longer meet the diversified needs of investors; thus, it is of great significance to construct and price new options for enriching the financial market. This paper proposes a new option pricing model that integrates the logarithmic investment strategy with the classic Black–Scholes theory. Specifically, this paper focus on put options, introducing a threshold-based strategy whereby investors sell stocks when prices fall to a certain value. This approach mitigates losses from adverse price movements, enhancing risk management capabilities. After deriving an analytical solution, we utilized mathematical software to visualize the factors influencing new option prices in three-dimensional space. The findings suggest that the pricing of these new options is influenced not only by standard factors such as the underlying asset price, volatility, risk-free rate of interest, and time to expiration, but also by investment strategy parameters such as the investment strategy index, investment sensitivity, and holding ratios. Most importantly, the pricing of new put options is generally lower than that of classic options, with numerical simulations demonstrating that under optimal parameters the new options can achieve similar hedging effectiveness at approximately three-quarters the cost of standard options. These findings highlight the potential of logarithmic investment strategies as effective tools for risk management in volatile markets. To validate our theoretical model, numerical simulations using data from Shanghai 50 ETF options were used to confirm its accuracy, aligning well with theoretical predictions. The new option model proposed in this paper contributes to enhancing the efficiency of resource allocation in capital markets at a macro level, while at a micro level, it helps investors to apply investment strategies more flexibly and reduce decision-making errors. Full article

Background/Objectives: Primary dysmenorrhea (PD) is a common condition affecting up to 90% of menstruating women, which often results in significant pain without an underlying pathology. Zinc, recognized for its anti-inflammatory and antioxidant effects through inhibiting prostaglandin production and superoxide dismutase 1 (SOD1) upregulation, alleviates menstrual pain by preventing uterine spasms and enhancing microcirculation in the endometrium, suggesting its potential as an alternative treatment for primary dysmenorrhea. The goal of this systematic review and meta-analysis was to assess the efficacy and safety of zinc supplementation in reducing pain severity among women with PD and to explore the influence of dosage and treatment duration. Methods: Following the PRISMA 2020 guidelines, we conducted an extensive search across databases such as PubMed, Embase, Cochrane Library, Web of Science, and Google Scholar, up to May 2024. Randomized controlled trials assessing the effects of zinc supplementation on pain severity in women with PD were included. Pain severity was evaluated with established tools, such as the Visual Analog Scale (VAS). Risk of bias was assessed using the Cochrane Risk of Bias 2 (RoB2) tool. Two reviewers independently performed the data extraction, and a random-effects model was used for meta-analysis. Meta-regressions were conducted to examine the influence of zinc dosage and treatment duration on pain reduction. Adverse events were also analyzed. Results: Six RCTs involving 739 participants met the inclusion criteria. Zinc supplementation significantly reduced pain severity compared to placebo (Hedges’s g = −1.541; 95% CI: −2.268 to −0.814; p < 0.001), representing a clinically meaningful reduction in pain. Meta-regression indicated that longer treatment durations (≥8 weeks) were associated with greater pain reduction (p = 0.003). While higher zinc doses provided additional pain relief, the incremental benefit per additional milligram was modest (regression coefficient = −0.02 per mg; p = 0.005). Adverse event rates did not differ significantly between the zinc and placebo groups (odds ratio = 2.54; 95% CI: 0.78 to 8.26; p = 0.122), suggesting good tolerability. Conclusions: Zinc supplementation is an effective and well-tolerated option for reducing pain severity in women with primary dysmenorrhea. Doses as low as 7 mg/day of elemental zinc are sufficient to achieve significant pain relief, with longer durations (≥8 weeks) enhancing efficacy. The favorable safety profile and ease of use support the consideration of zinc supplementation as a practical approach to managing primary dysmenorrhea. Full article

This work explores a finite liquidity model to price spread options and assess the liquidity impact. We employ Kirk approximation for computing the spread option price and its delta. The latter is needed since the liquidity impact is caused by the delta hedging of a large investor. Our main contribution is a novel methodology to price spread options in this paradigm. Kirk approximation in conjunction with Monte Carlo simulations yields the spread option prices. Moreover, the antithetic and control variates variance reduction techniques improve the performance of our method. Numerical experiments reveal that the finite liquidity causes a liquidity value adjustment in option prices ranging from 0.53% to 2.81%. The effect of correlation on prices is also explored, and as expected the option price increases due to the diversification effect, but the liquidity impact decreases slightly. Full article

Based on the typical positions of S&P 500 option market makers, we derive a funding illiquidity measure from quoted prices of S&P 500 derivatives. Our measure significantly affects the returns of leveraged managed portfolios; hedge funds with negative exposure to changes in funding illiquidity earn high returns in normal times and low returns in crisis periods when funding liquidity deteriorates. The results are not driven by existing measures of funding illiquidity, market illiquidity, and proxies for tail risk. Our funding illiquidity measure also affects leveraged closed-end mutual funds and, to an extent, asset classes where leveraged investors are marginal investors. Full article

This paper investigates the efficacy of weather derivatives as a risk management tool in the agricultural sector of Naousa, Greece, focusing on tree crops sensitive to temperature variations. The specific purpose is to assess how effectively weather derivative options can mitigate financial risks for farmers by providing strategic solutions. The study assesses the strategic application of Heating Degree Days (HDD) index options and their potential to alleviate economic vulnerabilities faced by farmers due to temperatures fluctuations. Employing different strike prices in Long Call and Straddle options strategies on the HDD index, the research offers tailored risk management solutions that cater to varying risk aversions among farmers. Moreover, the study applies the Value at Risk (VaR) methodology to quantify the financial security that weather derivatives can furnish, revealing a significantly reduced probability of severe financial losses in hedged scenarios compared to no-hedge conditions. Results show that all implemented strategies effectively enhance financial outcomes compared to scenarios without hedging, highlighting the exceptional utility of weather derivatives as risk management tools in the agricultural sector. Strategy 4, which exhibits the lowest VaR, emerges as the most effective, providing substantial protection against adverse weather conditions. This research supports the notion that weather derivatives can substantially contribute to the economic sustainability of rural economies, influencing policy decisions toward enhancing financial instruments for risk management in agriculture. Full article

This study investigated the interplay between exchange rate volatility, inflation rates, and real interest rates on Zambia’s economic growth from 1992 to 2022, utilizing annualized time series data. The study was necessitated by the limited published literature and relatively varying findings on the variables’ relationships in resource-dependent countries, such as Zambia. Diagnostic tests, including stationarity and co-integration analyses, were employed to determine integration orders and potential long-run relationships. The linear and nonlinear autoregressive distributed lag models were employed to assess short- and long-run dynamics of the variables on economic growth. The results established a positive short-run relationship between inflation rates and Gross Domestic Product (GDP) growth in the linear autoregressive distributive lag model, while an inverse relationship was observed in the nonlinear autoregressive distributive lag model, suggesting that negative shocks in inflation rates had a highly significant positive impact on economic growth. Furthermore, interest rates exhibited a positive relationship with economic growth, further suggesting that positive shocks had a greater significant direct effect on economic growth in comparison to negative shocks in the short and long run, respectively. Finally, exchange rates in both models exhibited an inverse relationship with economic growth irrespective of positive or negative shocks in the long run, highlighting the adverse effect of exchange rate volatility on economic growth prospects in developing countries, such as Zambia. The speed of adjustment to convergence following any disruptions was determined to be 75.18% (ARDL) and 89.19% (NARDL), highlighting relatively fast speeds of adjustments from any short-run disruptions. Notably, some of the policy recommendations included regular assessments of exchange rate volatility influences on import prices, domestic inflation, and production costs in key sectors. Additionally, the implementation of currency hedging options and forwards as well as bulking of foreign exchange reserves will ensure the stability of exchange rates against other major currencies in various economic conditions. Full article

This study contributes to the discussion about a fair and balanced pension system with a collectively funded pension scheme or social security and a defined contribution pillar. With an invigorated risk approach using financial option positions, it considers the variance of socioeconomic interests of different society-earning cohorts. By that, it enables the assumption of un-uniformity in interests about the fair and sustainable pension design. Specifically, we claim that the alternative cost of hedging the ideal position to the counterparty position studies the implied risks and returns that participants are willing to absorb and hence may lead to a fair compromise when there are different interests. The novelty of the introduced method is mainly based on the variety of participants’ risks and not on the utility function. Accordingly, we spare the discussion about the right shape of the utility function and the proper calibrations. Full article

Volatility as a measure of financial risk is a crucial input for hedging, portfolio diversification, option pricing and the calculation of the value at risk. In this paper, we estimate the asymmetric and time-varying volatility for Bitcoin as the dominant cryptocurrency in the world market. A novel approach that explicitly separates the falling markets from the rising ones is utilized for this purpose. The empirical results have important implications for investors and financial institutions. Our approach provides a position-dependent measure of risk for Bitcoin. This is essential since the source of risk for an investor with a long position is the falling prices, while the source of risk for an investor with a short position is the rising prices. Thus, providing a separate risk measure in each case is expected to increase the efficiency of the underlying risk management in both cases compared to the existing methods in the literature. Full article