Bond Pricing Equation Research Articles

Lie Symmetry Analysis for the Fractal Bond-Pricing Model of Mathematical Finance

The classical bond-pricing models, as important financial tools, show strong vitality in bond pricing. However, these models also expose their theoretical defects, which leads to inconsistencies with the actual observation results and usually causes the theoretical prices of bonds to be lower than the actual market prices in the financial market. In order to change this situation, considering that the price change of the underlying is regarded as a fractal transmission system, the fractal derivative is introduced into the bond-pricing equation. In order to solve the fractal bond-pricing equation, we first convert it into an equivalent equation by using a fractal two-scale transform. Only in this case can we start to study it by means of the Lie symmetry analysis method. Then the geometric vector fields, the symmetry reductions, and the exact solution to the equations are obtained. Furthermore, the dynamic behaviors of the fractal bond-pricing equation are discussed. The results show that the fractal dimension bond-pricing formula can better explain price changes in the capital market than the classical one. That is to say, the classical bond-pricing equation is only a special case of the fractal-bond pricing equation, which makes up for the defect that the theoretical bond price given by the classical bond-pricing equation is often lower than the actual market price. The results of this paper provide a basis for bond pricing in the financial market in order to seek a more appropriate and real 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.

Lie Symmetries and Partial Differential Equations(PDEs)

There are many well-known techniques for obtaining exact solutions for differential equations, but many of them are simply special cases of a few powerful symmetry methods. Lie symmetry methods are techniques for finding exact solutions of a wide variety of differential equations. In this paper, we discuss the use of Lie symmetries on PDEs of order two in two independent variables and show how they can be used to transform the governing equation into another equation with one less independent variable. In addition, for our case study we consider the PDE(the bond-pricing equation) and solve the equation using Lie symmetry methods.

Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance

AbstractThe invariant approach is employed to solve the Cauchy problem for the bond-pricing partial differential equation (PDE) of mathematical finance. We first briefly review the invariant criteria for a scalar second-order parabolic PDE in two independent variables and then utilize it to reduce the bond-pricing equation to different Lie canonical forms. We show that the invariant approach aids in transforming the bond-pricing equation to the second Lie canonical form and that with a proper parametric selection, the bond-pricing PDE can be converted to the first Lie canonical form which is the classical heat equation. Different cases are deduced for which the original equation reduces to the first and second Lie canonical forms. For each of the cases, we work out the transformations which map the bond-pricing equation into the heat equation and also to the second Lie canonical form. We construct the fundamental solutions for the bond-pricing model via these transformations by utilizing the fundamental solutions of the classical heat equation as well as solution to the second Lie canonical form. Finally, the closed-form analytical solutions of the Cauchy initial value problems for the bond-pricing model with proper choice of terminal conditions are obtained.

Enhanced group analysis of a semi linear generalization of a general bond-pricing equation

The enhanced group classification of a semi linear generalization of a general bond-pricing equation is carried out by harnessing the underlying equivalence and additional equivalence transformations. We employ that classification to unearth the particular cases with a larger Lie algebra than the general case and use them to find non trivial invariant solutions under the terminal and the barrier option condition.

ON THE FOUR-PARAMETER BOND PRICING MODEL

A four-parameter random walk model for the short rate of interest is described in Wilmott et al. (15). For pricing zero-coupon bonds from the resulting partial differential equation based on this short rate model, a certain form of solution requires the solution of two first-order nonlinear ordinary dif- ferential equations. In the present paper we show the interesting result that, for obtaining solutions of the bond pricing equation, neither of these two equa- tions requires any differential equation solving techniques; in fact, both these first-order nonlinear differential equations can be solved simply by elementary integration. We include the corresponding yield curve and its asymptotic be- havior. We identify our results obtained here for the general four-parameter model in the two special cases of Vasicek (14) and Cox, Ingersoll and Ross (4) with those given by these authors.

The role of uncertainty in the term structure of interest rates: A GARCH-ATSM approach

This article examines the roles of uncertainties regarding various macro-variables in determining risk premiums of bond yields. We develop a multivariate GARCH-VAR to quantify uncertainties regarding inflation, real activities and monetary policy as time-varying conditional variances. We jointly estimate the multivariate GARCH and no-arbitrage bond pricing equations using a maximum likelihood method. The results indicate that the inflation uncertainty is the largest contributor to the dynamics of long-term yields since the 1980s, while the monetary policy uncertainty also plays noticeable roles.

Deriving the bond pricing equation

Deriving the bond pricing equation

Invariant Approaches to Equations of Finance

We firstly show how effective it is to utilize the invariant criteria for scalar linear (1+1) parabolic equations in order to perform reductions to the three Lie canonical forms of a bond-pricing model from finance. As a consequence we arrive at new results on bond-pricing equations that admit four nontrivial symmetries. In the second part we draw attention to a new method developed for equations of economics.

Lie-Algebraic Approach for Pricing Zero-Coupon Bonds in Single-Factor Interest Rate Models

The Lie-algebraic approach has been applied to solve the bond pricing problem in single-factor interest rate models. Four of the popular single-factor models, namely, the Vasicek model, Cox-Ingersoll-Ross model, double square-root model, and Ahn-Gao model, are investigated. By exploiting the dynamical symmetry of their bond pricing equations, analytical closed-form pricing formulae can be derived in a straightfoward manner. Time-varying model parameters could also be incorporated into the derivation of the bond price formulae, and this has the added advantage of allowing yield curves to be fitted. Furthermore, the Lie-algebraic approach can be easily extended to formulate new analytically tractable single-factor interest rate models.

Lie-Algebraic Approach for Pricing Zero-Coupon Bonds in Single-Factor Interest Rate Models

The Lie-algebraic approach has been applied to solve the bond pricing problem in single-factor interest rate models. Four of the popular single-factor models, namely the Vasicek model, Cox-Ingersoll-Ross model, double square-root model, and Ahn-Gao model, are investigated. By exploiting the dynamical symmetry of their bond pricing equations, analytical closed-form pricing formulae can be derived in a straightfoward manner. Time-varying model parameters could also be incorporated into the derivation of the bond price formulae, and this has the added advantage of allowing yield curves to be fitted. Furthermore, the Lie-algebraic approach can be easily extended to formulate new analytically tractable single-factor interest rate models.

Pricing Defaultable Bonds Based on BSDEs in Incomplete Markets

In this paper, we treat defaultable bond as a contingent claim in incomplete markets. By setting up an investment portfolio with risky asset and construct two backward stochastic differential equation (BSDE), we develop the bond pricing equation by finding the optimal investment strategy with minimum risk through linear-quadratic hedging method.

“Extended Black” term structure models

This paper examines “Extended Black” term structure models (EBTSM), which are multi-factor extensions of the one-factor Black model (Black, F., 1995. Interest rates as options. Journal of Finance 50, 1371–1376). EBTSM are not affected by the admissibility restrictions that plague canonical affine models. EBTSM encompass quadratic models, but unlike in quadratic models bond yields are sufficient statistics to infer the latent factors driving the short interest rate. EBTSM are amenable to econometric estimation despite the need to solve bond pricing equations through finite difference numerical methods. Estimation through the Iterated Extended Kalman filter reveals that a two-factor EBTSM fit well the observed cross section and time series of Japanese Government bond yields. A three-factor EBTSM is also proposed.

Lie group classifications and exact solutions for two variable-coefficient equations

In this paper, the Lie symmetry analysis and group classifications are performed for two variable-coefficient equations, the hanging chain equation and the bond pricing equation. The symmetries for the two equations are obtained, the exact explicit solutions generated from the similarity reductions are presented. Moreover, the exact analytic solutions are considered by the power series method.

Invariance properties of a general bond-pricing equation

We perform the group classification of a bond-pricing partial differential equation of mathematical finance to discover the combinations of arbitrary parameters that allow the partial differential equation to admit a nontrivial symmetry Lie algebra. As a result of the group classification we propose “natural” values for the arbitrary parameters in the partial differential equation, some of which validate the choices of parameters in such classical models as that of Vasicek and Cox–Ingersoll–Ross. For each set of these natural parameter values we compute the admitted Lie point symmetries, identify the corresponding symmetry Lie algebra and solve the partial differential equation.

Tractable forms of the bond pricing equation

So far, a small number of analytically tractable single-factor models have been devised for the well-known bond pricing equation (BPE). In this paper, new tractable models are formulated in a systematic manner. First, the BPE is transformed to a standard canonical form in which only one coefficient function appears. In some interesting cases, this single coefficient function is identically zero, leaving nothing more to solve than the classical heat equation. In many cases, the canonical form allows a general solution by standard mathematical techniques such as separation of variables and Laplace transforms. In other cases, the general solution of the BPE is reduced to a single inverse Laplace transform.

Fundamental Solutions for Zero-Coupon Bond Pricing Models

The transformation approach is employed to reduce one-factor bond-pricing equations into the heat equation for which the fundamental solution is wellknown. These transformations are subsequently used to construct the fundamental solutions of two zero-coupon bond-pricing equations. The closed-form analytical solutions of the Cauchy initial value problems of the two bond-pricing model equations are then obtained.

A Model for Pricing Stocks and Bonds with Default Risk

This paper develops a tractable, dynamic, no-arbitrage model for the pricing of bonds and stocks that are subject to default risk. The model produces the bond pricing equations of the Duffie and Singleton (1999) framework. It is then shown that a particular choice of dividend process, characterized by affine dividend yields, along with the Duffie and Singleton (1999) default specification, produces stock prices that are exponential affine in the model's state variables. Importantly, the model allows for quite general interdependence between the prices of risky debt and equity. This, along with the model's tractability, makes it a natural platform for empirical investigations into the pricing of a firm's capital structure.

Pricing Interest Rate Derivatives: A General Approach

The relationship between affine stochastic processes and bond pricing equations in exponential term structure models has been well established. We connect this result to the pricing of interest rate derivatives. If the term structure model is exponential affine, then there is a linkage between the bond pricing solution and the prices of many widely traded interest rate derivative securities. Our results apply to m-factor processes with n diffusions and l jump processes. The pricing solutions require at most a single numerical integral, making the model easy to implement. We discuss many options that yield solutions using the methods of the article.

New solutions to the bond-pricing equation via Lie's classical method

Using Lie's classical method of group invariants, we provide new and simple solutions to the bond-pricing partial differential equation. All of our solutions satisfy the final condition for the bond price. In finding these solutions, we used one-factor interest-rate modelling, in which our short-term interest rate follows a random walk which allows the volatility of interest rate changes to be highly sensitive to the level of riskless rate, and the market price of risk is arbitrary. As well, in one of our simple solutions, the nonlinear drift of our risk-neutral interest rate contains an arbitrary function of time, which may be freely chosen.