Zero-coupon Bond Prices Research Articles

Evaluation of bitcoin options with interest rate risk and systemic risk

This study introduces closed-form formulas for valuing European call options, assuming that Bitcoin follows a compound Poisson process. Additionally, instantaneous forward interest rates are considered in the Heath-Jarrow-Morton model, which includes a jump component. To address the impacts of systematic risk on Bitcoin price and interest rate, we model two stochastic processes using a correlated bivariate jump-diffusion model to capture individual jumps and systematic co-jumps. This study provides analytic formulas for pricing Bitcoin call options and zero-coupon bonds under the correlated jump-diffusion Heath-Jarrow-Morton model. Numerical analysis shows how co-jump intensity affects the prices of both zero-coupon bonds and Bitcoin call options. We specifically look at how these prices change in response to co-jump intensity across three different instantaneous forward rate term structures. The findings show that the prices of Bitcoin call options are contingent on the term structure types of zero-coupon bonds. In addition, the interaction of co-jump intensity and types of term structure also affects Bitcoin option prices. The practical significance of this study is to provide a comprehensive model to evaluate Bitcoin call options and enhance risk management strategies in the Bitcoin market when the Bitcoin market encounters changes in monetary policy or changes in macroeconomic conditions.

A Quantum Mechanics for interest rate derivatives markets

We present a model-free axiomatic formulation of Option Pricing Theory for interest rate derivatives. In this setting, completely analogous to axiomatic Quantum Mechanics, the role of the wave function is played by the discounted zero-coupon bond price. The theory is linked to term structure models through the Hamiltonian operator, and we show that its associated Schrödinger equation is consistent with the [25] model. We also find the quantum-mechanical equivalent of the standard risk-neutral option pricing formula.In order to take advantage of this quantum framework, we develop a time-dependent perturbation theory and we apply it to obtain an approximate closed-form expression for the price of a call option under the [26] model with a small local volatility perturbation.

The Gauss2++ model: a comparison of different measure change specifications for a consistent risk neutral and real world calibration

Especially in the insurance industry interest rate models play a crucial role, e.g. to calculate the insurance company’s liabilities, performance scenarios or risk measures. A prominant candidate is the 2-Additive-Factor Gaussian Model (Gauss2++ model)—in a different representation also known as the 2-Factor Hull-White model. In this paper, we propose a framework to estimate the model such that it can be applied under the risk neutral and the real world measure in a consistent manner. We first show that any time-dependent function can be used to specify the change of measure without loosing the analytic tractability of, e.g. zero-coupon bond prices in both worlds. We further propose two candidates, which are easy to calibrate: a step and a linear function. They represent two variants of our framework and distinguish between a short and a long term risk premium, which allows to regularize the interest rates in the long horizon. We apply both variants to historical data and show that they indeed produce realistic and much more stable long term interest rate forecast than the usage of a constant function, which is a popular choice in the industry. This stability over time would translate to performance scenarios of, e.g. interest rate sensitive fonds and risk measures.

A mixed fractional Vasicek model and pricing Bermuda option on zero-coupon bonds

This paper considers the problem of pricing of Bermuda options on zero-coupon bond in which the dynamics of the interest rate model follows the mixed fractional Vasicek model. The strong convergence of the Euler discretization scheme for the mixed fractional Vasicek model is analysed. Specifically, we find an approximate formula for zero-coupon bond price. Numerical experiments are provided and compared for Bermuda-style call and put options with the Monte Carlo simulation approach.

Time integral about solution of an uncertain fractional order differential equation and application to zero-coupon bond model

Uncertain fractional order differential equation is a significant tool for modeling the uncertain dynamic system. First, we consider solutions of an uncertain fractional order differential equation with the Caputo type and investigate inverse uncertain distributions of their time integral. On the basis of α-path, two different time integral theorems for inverse uncertain distributions are given. Second, in uncertain financial markets, the interest rate is considered as an uncertain process. As the application of the time integral, we present a novel zero-coupon bond model and derive a pricing formula of zero-coupon bond under this model. Last, by the predictor-corrector method, the numerical algorithm is designed. Analytic expressions and numerical calculations of the zero-coupon bond price are illustrated for fractional order mean-reverting model and standard deviation model, respectively.

A Quantum Mechanics for Interest Rate Derivatives Markets

We present an axiomatic formulation of the Option Pricing Theory for interest rates, completely analogous to axiomatic Quantum Mechanics. The role of the wave function is played in the financial theory by discounted zero-coupon bond prices. The theory is model-free and is linked to term structure models through the Hamiltonian operator. Under the quantum framework, we prove the consistency of the Schrodinger equation with the dynamics of the stochastic string model. We also find an expression for pricing options under this quantum theory, show its consistency with standard option pricing and apply the quantum formalism to Price calls and swaptions.

A BSDE approach for bond pricing under interest rate models with self-exciting jumps

In this article, we consider zero-coupon bond pricing problems for the stochastic interest rate model with clustering effects of self-exciting jumps. We first develop the evolution of the interest rate model under the equivalent martingale measure. Then we characterize the bond price in terms of a backward stochastic differential equation (BSDE). Closed-form solution of the BSDE is expressed as an exponential affine function of the interest rate and the intensity of jumps when the coefficients of interest rate model have affine structures.

A Kesten-type bound for sums of randomly weighted subexponential random variables

Sums of randomly weighted subexponential random variables have become an important research topic, but most works on the topic consider randomly weighted sums of finitely many terms. To extend the study to the case of infinitely many terms, we establish a Kesten-type upper bound for the tail probabilities of sums of randomly weighted subexponential random variables. As an application, we derive a precise asymptotic formula for the tail probability of the aggregate present value of subexponential claims, where the present value factor is determined according to the zero-coupon bond price.

Perturbation solutions for bond-pricing equations under a multivariate CIR model with weak dependences

In this article, we derive the explicit perturbation solutions for bond-pricing equations under a multivariate Cox–Ingersoll–Ross model whose factors have weak dependences between each other. The error bounds of the perturbation approximations are demonstrated for the Riccati equation and zero-coupon bond price. These results provide the accuracy order for the perturbation approximation. Since the perturbation approximations are constructed based on the exact solution of the Riccati equation, this computational approach can be applied to deal with the stiffness problem. Numerical results are presented to show that this computation approach is tractable and accurate for evaluating the bond price.

Pricing Longevity Bonds Under the Uncertainty Theory Framework

This paper aims to develop a new pricing approach for longevity bonds under the uncertainty theory framework. First, we describe the life expectancy by a canonical uncertain process and illustrate the dynamic of short interest rate via an uncertain Vasicek interest rate model. Then, based on these descriptions, we construct an uncertain survival index model and present its procedure for parameter estimation. By applying the chain rule, we derive a pricing formula of the uncertain zero-coupon bond. Considering that the financial market is incomplete, we put forward an uncertain distortion operator. Furthermore, based on the uncertain survival index, the uncertain zero-coupon bond pricing formula and the uncertain distortion operator, we develop a pricing formula of the uncertain longevity bond and its calculation algorithm. Finally, a numerical example is shown to illustrate the influence of parameters on the price of the uncertain longevity bond.

Bond and option pricing for interest rate model with clustering effects

This paper analyzes an interest rate model with self-exciting jumps, in which a jump in the interest rate model increases the intensity of jumps in the same model. This self-exciting property leads to clustering effects in the interest rate model. We obtain a closed-form expression for the conditional moment-generating function when the model coefficients have affine structures. Based on the Girsanov-type measure transformation for general jump-diffusion processes, we derive the evolution of the interest rate under the equivalent martingale measure and an explicit expression of the zero-coupon bond pricing formula. Furthermore, we give a pricing formula for the European call option written on zero-coupon bonds. Finally, we provide an interpretation for the clustering effects in the interest rate model within a simple framework of general equilibrium. Indeed, we construct an interest rate model, the equilibrium state of which coincides with the interest rate model with clustering effects proposed in this paper.

Approximate Closed Formulas for Zero-Coupon Bonds in the Zero Lower Bound Framework

Since the 2007 financial crisis, many central banks adopted policies to lower their interest rates, whose dynamics can not be captured using classical models. Recently, Meucci and Loregian (2016) proposed an approach to estimate nonnegative interest rates using the inverse-call transformation. Despite the fact that their work distinguishes from others in the literature by their consideration of practical aspects, some technical difficulties still remain, such as the lack of the analytic expression for the zero-coupon bond (ZCB) price. In this work, we provide approximate closed formulae for the ZCB price in the zero lower bound (ZLB) framework, when the underlying shadow rate is assumed to follow the classical one-factor Vasicek model. Then, a filtering procedure is performed using the Unscented Kalman Filter (UKF) to estimate the unobservable state variable (the shadow rate), and the model calibration is proceeded by estimating the model parameters using the Particle Swarm Optimization (PSO) algorithm. Lastly, empirical illustrations are given and discussed using (as input data) the interest rates of AAA-rated bonds compiled by the European Central Bank ranging from September 6, 2004 to June 21, 2012.

Bond pricing under the generalised Black-Karasinski models

Due to the lognormality of the short rate under the Black-Karasinski interest model, closed-form expressions for zero-coupon bond prices are not available. Existing methods for computing approximate prices include perturbation methods for solving the reaction-diffusion equation satisfied by the bond-price and the exponent expansion for computing the bond price via Arrow-Debreu prices. Perturbation methods are accurate for small volatility problems whereas the exponent expansion is accurate for small maturities. This work proposes a high-order computational method that works for all parameter settings. Several numerical examples are described to illustrate the high accuracy and rapid computation of bond prices.

SHORT RATE FORECASTING BASED ON THE INFERENCE FROM THE CIR MODEL FOR MULTIPLE YIELD CURVE DYNAMICS

In this paper, we propose a strategy to extract the information on the market participants’ expectation of the future short rate from the cross-sectional zero coupon bond prices. In line with the current market practice of building different yield curves for different tenors, we construct multiple one-factor short rate processes to pin down the salient features of the yield curve at different tenors. We represent this information in the form of the Cox–Ingersoll–Ross model implied parameters, and show that this information can be used to forecast the future short rate. This approach of representing the information on the market participants’ consensus in the form of implied model parameters and using these implied parameters for forecasting purposes resembles the approach of representing the market expectation of the underlying asset volatility reflected by stock option prices in the form of implied volatility, and using it to forecast the realized volatility. We illustrate the implementation of this method using historical US STRIPS prices and effective Federal Funds rate.

Calibration of Implied Volatility in Generalized Hull-White Model

This paper concerns a problem of calibrating implied volatility in generalized Hull-White model from the market prices of zero-coupon bonds. By using the regularization method, we establish the existence and stability of the optimal solution, and give the necessary condition that the solution satisfies. Finally numerical results show that the method is stable and effective.

Forecasting of the Convexity Adjustment and the Short Rate Based on the Inference from the CIR Model

We propose a strategy to extract the information on the market participants' expectation of the future short rate from the cross-sectional zero coupon bond prices. In line with the current market practice of building different yield curves for different tenors, we construct multiple one-factor short rate processes to pin down the salient features of the yield curve at different tenors. We represent this information in the form of the Cox-Ingersoll-Ross model implied parameters. We show that this information can be used to forecast the future short rate. This resembles representing the market expectation of the underlying asset volatility reflected by stock option prices in the form of implied volatility, and using it to forecast the realized volatility. We illustrate the implementation of this method using historical US STRIPS prices and effective Federal Funds Rate.

Explicit Solutions of Quadratic FBSDEs Arising From Quadratic Term Structure Models

We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to similar results on affine term structure models of Hyndman (Math. Financ. Econ. 2(2):107–128, 2009) due to the relationship between quadratic functionals of Gaussian processes and linear functionals of affine processes. Similar to the affine case a sufficient condition for the explicit solutions to hold is the solvability in a fixed interval of Riccati-type ordinary differential equations. However, in contrast to the affine case, these Riccati equations are easily associated with those occurring in linear-quadratic control problems. We also consider quadratic models for a risky asset price and characterize the futures price and forward price of the asset in terms of similar FBSDEs. An example is considered, using an approach based on stochastic flows that is related to the FBSDE approach, to further emphasize the parallels between the affine and quadratic models. An appendix discusses solvability and explicit solutions of the Riccati equations.

Path-Dependent Multicurrency Interest Rate Derivatives

This paper derives the arbitrage-free pricing formulae and hedging ratios of path-dependent multicurrency interest rate derivatives in continuous time. The exchange rates and the zero-coupon bond prices are modeled by multivariate log-normal processes with an arbitrary number of random factors. The exact pricing formulae are obtained by solving the time-evolution equation of the contingent claims described by a multi-dimensional Kolmogorov field equation. Applications to the Asian options and Bermudan options are given as examples. Several European options on currency swaps, forwards and futures contracts are also illustrated as special cases. These exact solutions are factor-independent and explicitly depend on the instantaneous covariances of the return rates of the exchange rates and the zero-coupon bonds. By scaling the historical covariances obtained from the market data, one can easily calibrate the volatility surface to match the vanilla options.

Liquidity risk and the term structure of interest rates

This paper develops an arbitrage-free pricing theory for a term structure of fixed income securities that incorporates liquidity risk. In our model, there is a quantity impact on the term structure of zero-coupon bond prices from the trading of any single zero-coupon bond. We derive a set of conditions under which the term structure evolution is arbitrage-free. These no arbitrage conditions constrain both the risk premia and the term structure’s volatility. In addition, we also provide conditions under which the market is complete, and we show that the replication cost of an interest rate derivative is the solution to a backward stochastic differential equation.

Explicit Solutions of Quadratic FBSDEs Arising from Quadratic Term Structure Models

We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to similar results on affine term structure models of Hyndman (Math. Financ. Econ. 2(2):107-128, 2009) due to the relationship between quadratic functionals of Gaussian processes and linear functionals of affine processes. Similar to the affine case a sufficient condition for the explicit solutions to hold is the solvability in a fixed interval of Riccati-type ordinary differential equations. However, in contrast to the affine case, these Riccati equations are easily associated with those occurring in linear-quadratic control problems. We also consider quadratic models for a risky asset price and characterize the futures price and forward price of the asset in terms of similar FBSDEs. An example is considered, using an approach based on stochastic flows that is related to the FBSDE approach, to further emphasize the parallels between the affine and quadratic models. An appendix discusses solvability and explicit solutions of the Riccati equations.