Delta hedging in discrete time under stochastic interest rate

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

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 of DC pension plan in a stochastic interest rate and stochastic volatility framework

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.

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

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.

Acknowledgments. Introduction. Chapter 1: Tools of the Trade. Basic Statistics. Regression The Fundamentals. Regression How Good a Fit? Principal Components Analysis. Scaling Through Time. Backtesting Strategies. Summary. Chapter 2: Introduction to Bonds. Basics of Bonds. Risks Embedded in Fixed Income Instruments. Discounting. Bond Pricing. Yield Curve. Duration. Convexity. Repo Markets. Bid Offer. Calculating Profit/Loss (P/L) of a Bond. Carry and Forward Rates. Forward Rates. Rolldown/Slide. Curves and Spreads. Butterflies and Condors. Summary. Chapter 3: Overview of Fixed Income Markets. Federal Reserve. Treasuries. STRIPS. TIPS. Mortgages. Agency Debt. Corporate Bonds. Municipal Bonds. Summary. Chapter 4: Introduction to Futures Markets. Basics of Futures Transactions. Eurodollar Futures. Convexity Bias (or Financing Bias). Creating Longer Dated Assets Using Eurodollar Futures. Treasury Futures. Fed Funds Futures. Futures Positioning Data. Summary. Chapter 5: Introduction to Swaps. Duration and Convexity. Uses of Swaps. Other Types of Swaps. Summary. Chapter 6: Understanding Drivers of Interest Rates. Supply and Demand for Borrowing. Flight to Quality. Components of Fixed Income Supply and Demand. Treasury Supply. Other Sources of Fixed Income Supply. Fixed Income Demand. Foreigner Holdings. Households. Federal Reserve. Mutual Funds. Banks. Pension Funds. Short Term Yield Drivers. Summary. Chapter 7: Carry and Relative Value Trades. Carry Trades. Carry Trade Set up and Evaluation. Pitfalls of the Carry Trade. Carry Efficient Directional Trades. Relative Value Trades. Setting Up Relative Value Trades. Treasury Bond Relative Value Par Curve. Other Treasury Relative Value Trades. Summary. Chapter 8: Hedging Risks in Interest Rate Products. Principles of Hedging. Choices of Instruments for Hedging. Calculating Hedge Ratios. Yield Betas. Convexity Hedging. Summary. Chapter 9: Trading Swap Spreads. How Swap Spreads Work. Why Trade Swap Spreads? Directionality of Swap Spreads to Yields. Futures Asset Swaps. Spread Curve Trades. Summary. Chapter 10: Interest Rate Options and Trading Volatility. Option Pricing and Fundamentals. Modifications for the Interest Rate Markets. Quoting Volatility. Measuring Risks in Option Positions. Put call parity. Implied and Realized Volatility. Skew. Delta Hedging. Interest Rate Options. Embedded Options and Hedging. More Exotic Structures. Yield Curve Spread Options (YCSOs). Forward Volatility. Volatility Trading. Interest Rate Skew. Volatility Spread Trades. Caps versus Swaptions. Summary. Chapter 11: Treasury futures basis and rolls. The Bond Basis. Calculating the Delivery Option Value. Option Adjusted and Empirical Duration. Treasury Futures Rolls. Summary. Chapter 12: Conditional Trades. Conditional Curve Trades. Conditional Spread Trades. Summary. References. About the Author. About the Website. Index.

We propose a dynamic equilibrium model with stochastic interest rates in which agents hold heterogeneous valuations for the same asset and take on positions against each other. The model shows that interest rate uncertainty and investor heterogeneity are key determinants of price dispersion. Higher search intensity reduces price dispersion, while raising volume, leading to a negative volatility-volume relation. The sensitivity of volatility to volume is high when liquidity is low, interest rate variations are high and investors’ valuations are more heterogeneous. Evidence supports our model’s predictions and shows that search frictions play an important role in driving the volatility-volume relation. (JEL G12, G13) Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

This paper presents a new discrete time approach to pricing contingent claims on a risky asset and stochastic interest rates. The term structure of interest rates is modeled so that arbitrage‐free bond prices depend on an observable initial forward rate curve rather than an exogenously specified market price of risk. A restricted binomial process is employed to model both interest rates and an asset price. As a result, a complete market valuation formula obtains. By choosing the parameters of the discrete joint distribution such that, in the limit, the discrete model converges to the continuous one, a model is obtained that requires the estimation of only three parameters. The approach is parsimonious with respect to alternative models in the literature and can be used to price contingent claims on any two state variables. The procedure is used to numerically analyze the effects of the volatility of interest rates on the determination of mortgage contract rates.

In a complete financial market we consider the discrete time hedging of the American option with a convex payoff. It is well known that for the perfect hedging the writer of the option must trade continuously in time, which is impossible in practice. In reality, the writer hedges only at some discrete time instants. The perfect hedging requires the knowledge of the partial derivative of the value function of the American option in the underlying asset, the explicit form of which is unknown in most cases of practical importance. Several approximation methods have been developed for the calculation of the value function of the American option. We claim in this paper that having at hand any uniform approximation of the American option value function at equidistant discrete rebalancing times it is possible to construct a discrete time hedging portfolio, the value process of which uniformly approximates the value process of the continuous time perfect delta-hedging portfolio. We are able to estimate the corresponding discrete time hedging error that leads to a complete justification of our hedging method for nonincreasing convex payoff functions including the important case of the American put. This method is essentially based on a new type square integral estimate for the derivative of an arbitrary convex function recently found by Shashiashvili.

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.

An evaluator of warrant models is essential for enhancing the current procedures used in the country's stock market and economy. This would provide additional insight for investors to make informed investment decisions and help to better understand the dynamics of warrant pricing in Malaysia. It has been established that the conventional process for valuing equity warrants using call option pricing models, like the Black-Scholes model, has numerous flaws, including the presumption of constant volatility and interest rates. As a matter of fact, existing alternative models were more concerned with providing techniques for pricing than with conducting empirical tests of warrant pricing models. Consequently, it is crucial to have a mathematical model that takes into account stochastic volatility and stochastic interest rates when pricing and analyzing equity warrants. Recent research by Roslan, Ibrahim, Jameel, and Ibrahim (2020) and Roslan, Ibrahim, Karim (2020) examined the pricing of hybrid equity warrants in the presence of stochastic factors. There was no comparative analysis conducted in these studies to ascertain the relative efficacy of the two methods used. This study's main goal is to empirically investigate how well equity warrant pricing models perform in the Malaysian market. This evaluation will take into account the impact of stochastic interest rates, using the Cox-Ingersoll-Ross (CIR) stochastic interest rate model, and stochastic volatility, using the Heston stochastic volatility model. To perform the analysis, the study will make use of a number of mathematical techniques, such as partial differential equations, Fourier transform approaches, and the change of measure method. The research methodology is divided into three main phases. It is anticipated that the research's findings will provide new perspectives that go beyond the domain of warrant pricing, creating chances for more widespread applications in fields like foreign exchange, insurance, and mortgage contracts.

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.

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

Accounting for Stochastic Interest Rates, Stochastic Volatility and a General Dependency Structure in the Valuation of Forward Starting Options