Dynamic Mean-Downside Risk Portfolio Selection Problem with Stochastic Interest Rate in Continuous-Time

An exhaustive survey of research on pricing foreign exchange options with stochastic volatility and stochastic interest rates is presented in this article. The article also includes contributions to pricing of currency options with fixed volatility and deterministic interest rates, domestic and foreign, and in a systematic way extend to results obtained for cases which include stochastic volatility and stochastic interest rates. Since the literature in pricing foreign exchange currency option is quite extensive, the article considers some of the more important contributions.

In this paper, we consider a model with stochastic interest rate and stochastic mortality, which is driven by a Lévy process. Under the assumption that the stochastic mortality and interest rate are dependent, we discuss the valuation of life insurance contracts. Employing the method of change of measure together with the Bayes’ rule, we present the pricing formulas in closed form for the survival and death benefit models. Finally, numerical experiments illustrate the effects of some parameters.

Optimal management under stochastic interest rates: the case of a protected defined contribution pension fund

Efficient procedures for the valuation and hedging of American currency options with stochastic interest rates

Term life insurance reserve calculations are strongly influenced by interest rate uncertainty, which affects the present value of long-term liabilities. This study estimates term life insurance reserves using a stochastic interest rate approach based on the Extended Vasicek model, chosen for its mean-reverting property and analytical tractability, which allow it to capture interest rate dynamics in a stable and interpretable manner. The model parameters are estimated using the Jackknife resampling method to improve stability. The data used consist of the annual average BI Rate from 2010 to 2025 and the Indonesian Mortality Table 2019 (TMI 2019). The results suggest that the Jackknife-based Extended Vasicek model produces stable mean-reverting projections of interest rates. These projections are then applied to reserve calculations using the Fackler and Full Preliminary Term (FPT) methods. The findings indicate that the Fackler method generates a gradual reserve pattern that peaks mid-term before declining to zero. In contrast, the FPT method yields zero reserves in the first year, followed by higher accumulation in subsequent years. Overall, this approach provides relatively stable and realistic reserve estimates under stochastic interest rate conditions. From a practical perspective, the Extended Vasicek model may be considered an alternative framework for actuaries to model interest rate uncertainty in reserve calculations. Future research could further examine its performance under different economic scenarios or compare it with other stochastic interest rate models to better understand its applicability and limitations.

In this paper we assess the partial hedging problems by formulating hedging strategies that minimize conditional value-at-risk (CVaR) of the portfolio loss under stochastic interest rate environment. The combination of stochastic interest and CVaR hedging method makes the valuing approach more complex than the existing model with constant interest rate. We take up two issues in searching the optimal CVaR hedging strategy: given the initial capital constraint we minimize the CVaR of the portfolio loss; by prescribing a bound on the risk, we also minimize the hedging cost. As an illustration of this hedging technique we derive hedging strategies for a European call option with the Black Scholes setting under HJM framework; explicit formulas are presented. We also investigate CVaR hedging problems by using the real financial data.

Pricing long-dated insurance contracts with stochastic interest rates and stochastic volatility

In this paper, we consider a finite-horizon Markov decision process (MDP) for which the objective at each stage is to minimize a quantile-based risk measure (QBRM) of the sequence of future costs; we call the overall objective a dynamic quantile-based risk measure (DQBRM). In particular, we consider optimizing dynamic risk measures where the one-step risk measures are QBRMs, a class of risk measures that includes the popular value at risk (VaR) and the conditional value at risk (CVaR). Although there is considerable theoretical development of risk-averse MDPs in the literature, the computational challenges have not been explored as thoroughly. We propose data-driven and simulation-based approximate dynamic programming (ADP) algorithms to solve the risk-averse sequential decision problem. We address the issue of inefficient sampling for risk applications in simulated settings and present a procedure, based on importance sampling, to direct samples toward the “risky region” as the ADP algorithm progresses. Finally, we show numerical results of our algorithms in the context of an application involving risk-averse bidding for energy storage.The online appendix is available at https://doi.org/10.1287/moor.2017.0872 .

PurposeThe aim is to study jump liquidity risk and its impact on risk measures: value at risk (VaR) and conditional VaR (CVaR).Design/methodology/approachThe liquidity discount factor is modelled with mean revision jump diffusion processes and the liquidity risk is integrated in the framework of VaR and CVaR.FindingsThe standard VaR, CVaR, and the liquidity adjusted VaR can seriously underestimate the potential loss over a short holding period for rare jump liquidity events. A better risk measure is the liquidity adjusted CVaR which gives a more realistic loss estimation in the presence of the liquidity risk. An efficient Monte Carlo method is also suggested to find approximate VaR and CVaR of all percentiles with one set of samples from the loss distribution, which applies to portfolios of securities as well as single securities.Originality/valueThe paper offers plausible stochastic processes to model liquidity risk.

We present option pricing under the double stochastic volatility model with stochastic interest rates and double exponential jumps with stochastic intensity in this article. We make two contributions based on the existing literature. First, we add double stochastic volatility to the option pricing model combining stochastic interest rates and jumps with stochastic intensity, and we are the first to fill this gap. Second, the stochastic interest rate process is presented in the Hull–White model. Some authors have concentrated on hybrid models based on various asset classes in recent years. Therefore, we build a multifactor model with the term structure of stochastic interest rates. We also approximated the pricing formula for European call options by applying the COS method and fast Fourier transform (FFT). Numerical results display that FFT and the COS method are much faster than the numerical integration approach used for obtaining the semi-closed form prices. The COS method shows higher accuracy, efficiency, and stability than FFT. Therefore, we use the COS method to investigate the impact of the parameters in the stochastic jump intensity process and the existence of the process on the call option prices. We also use it to examine the impact of the parameters in the interest rate process on the call option prices.

A fundamental challenge for insurance companies (insurers) is to strike the best balance between optimal investment and risk management of paying insurance liabilities, especially in a low interest rate environment. The stochastic interest rate becomes a critical factor in this asset-liability management (ALM) problem. This paper derives the closed-form solution to the optimal investment problem for an insurer subject to the insurance liability of compound Poisson process and the stochastic interest rate following the extended CIR model. Therefore, the insurer’s wealth follows a jump-diffusion model with stochastic interest rate when she invests in stocks and bonds. Our problem involves maximizing the expected constant relative risk averse (CRRA) utility function subject to stochastic interest rate and Poisson shocks. After solving the stochastic optimal control problem with the HJB framework, we offer a verification theorem by proving the uniform integrability of a tight upper bound for the objective function.

The purpose of this study is to compare two major risk measurement models in financial markets—the Value at Risk (VaR) model and the Conditional Value at Risk (CVaR) model—and to evaluate their effectiveness and limitations in stock market risk management. First, through an in-depth analysis of the theoretical foundations of stock market risk, the paper clarifies the importance of risk for investment decision-making, in particular its impact on asset allocation and risk management strategies. The findings show that VaR, as a traditional risk measurement tool, is simple and convenient and enables a rapid assessment of market risk; however, when confronted with extreme tail events it exhibits certain shortcomings and cannot fully capture the true risk of extreme losses. By contrast, CVaR, by focusing on tail risk, makes it possible to identify potential extreme losses more accurately; especially during periods of sharp market volatility, its early-warning function substantially outperforms VaR. In the empirical part of the article, historical data for the Shanghai A-share index are used. The GARCH-M model is applied to estimate VaR and CVaR, and their values are compared when measuring stock market risk. The results indicate that, at the same confidence level, CVaR is always greater than VaR, which further confirms the advantage of CVaR in reflecting risk losses more accurately, especially tail risk. Overall, this study not only provides a theoretical basis for risk management in financial markets, but also offers recommendations for improving risk measurement methods in the stock market, emphasizing the need to account for tail risk when making investment decisions. The conclusions have important reference value for investors seeking to develop more scientifically grounded risk control strategies in a dynamic market environment.

Modern Portfolio Theory (MPT) is based upon the classical Markowitz model which uses variance as a risk measure. A generalization of this approach leads to mean‐risk models, in which a return distribution is characterized by the expected value of return (desired to be large) and a “risk” value (desired to be kept small). Portfolio choice is made by solving an optimization problem, in which the portfolio risk is minimized and a desired level of expected return is specified as a constraint. The need to penalize different undesirable aspects of the return distribution led to the proposal of alternative risk measures, notably those penalizing only the downside part (adverse) and not the upside (potential). The downside risk considerations constitute the basis of the Post Modern Portfolio Theory (PMPT). Examples of such risk measures are lower partial moments, Value‐at‐Risk (VaR) and Conditional Value‐at‐Risk (CVaR). We revisit these risk measures and the resulting mean‐risk models. We discuss alternative models for portfolio selection, their choice criteria and the evolution of MPT to PMPT which incorporates: utility maximization and stochastic dominance. Copyright © 2009 Wilmott Magazine Ltd

We investigate the relation between various alternative risk measures and future daily returns using a sample of firms over the 1988-2009 time period. Previous research indicates that returns are not normally distributed and that investors seem to care more about downside risk than total risk. Motivated by these findings and mixed empirical evidence supporting theoretical positive risk-return relationship, we model the relation between future returns and risk measures and investigate the following questions: (1) Are investors compensated for total risk and/or asymmetric measures of risk? (2) How does the degree of risk aversion in the lower tail of the return distribution impact the predictability of future returns? (3) Is upside risk or downside risk a better predictor of future returns? We find that, although investors seem to be compensated for total risk, measures of downside risk, such as the lower partial moment, are better at explaining future returns. Further, when comparing downside risk to upside risk, we find that investors are more concerned about downside risk. That is, downside risk is a better predictor of future returns. Our results are robust to the addition of traditional control variables, including size, book-to-market ratio of equity (B/M), leverage, and market risk measures, including beta, downside beta and co-skewness. Our findings are an important contribution to the literature as we document a positive risk-return relationship, using both total and asymmetric measures of risk.

<em>The application of information technology platforms that appear in business transactions in Indonesia is a pattern of business models that cannot be avoided. One of the rapidly growing technology companies is an online loan service.The ease of borrowing funds through these services supports business growth. In contrast to the banking sector, which is regulated by a series of strict government regulations and the determination of interest rates that have been determined by looking at economic conditions. The absence of regulation in the legislation makes AFPI set the limit for determining the maximum interest rate, but the determination of interest rates by AFPI is suspected to lead to cartel actions. There is a close relationship between fair business competition and consumer protection. This research is a normative juridical research with a statutory approach and a comparative approach. In this case, the setting of interest rate limits will reduce consumers’ rights to get competitive prices. The existence of conscious parallelism by the AFPI on the LPMUBTI services can be categorized as a price fixing agreement. It is because the determination of the maximum interest rate limit is not regulated through direct government regulations.</em>