We have seen how we expect stock prices to behave — returns being normally distributed and consequently prices distributed lognormally. If you recall, the process for the behavior of stock prices is explained as follows. The same process however cannot be applied to interest rates. This is because while stock prices may follow a random walk, interest rates are generally considered mean reverting. The models of short term interest rates help determine the shape of the yield curve, and option pricing on bond options. There are two broad categories of models of the short rate — Equilibrium models, and No-arbitrage models.
Discrete Dynamics in Nature and Society
We also point to other possible studies on these models using the symmetries admitted by the valuation partial differential equations. Citations References In Cases 2 and 3a 1. We proceed to find the solution in each of these cases as an invariant solution [7,8,17,20, 22]. One starts by finding a symmetry admitted by both 1. The free parameters in the general invariant solution are then chosen suitably so that the solution satisfies the auxiliary condition 1.
Case 3a we note that an obvious renaming of the parameters in 1. So, after adjusting the parameters in the CIR problem appropriately, we have the following solution to 1. Table 1 Scaling factors for " standardising " the commutation relations of the symmetries in Section 3. 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.
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. The bond- pricing equation via Lie group approach was investigated in [26,27]. The invariant analysis of some well-known fi- nancial mathematics models was presented   , are few important studies to mention.
We now obtain the closed-form group-invariant solu- tion for the PDE 3 by making use of the Lie symmetry al- gebra calculated in the previous section. Firstly we calcu- late symmetry Lie algebra admitted by 3 that satisfies the terminal condition 2 [26, 28]. Algebraic aspects of evolution partial differential equation arising in the study of constant elasticity of variance model from financial mathematics.
Full-text available. Mar The optimal investment-consumption problem under the constant elasticity of variance CEV model is investigated from the perspective of Lie group analysis. The Lie symmetry group of the evolution partial differential equation describing the CEV model is derived. The Lie point symmetries are then used to obtain an exact solution of the governing model satisfying a standard terminal condition.
Finally, we construct conservation laws of the underlying equation using the general theorem on conservation laws. The transformation we have used above can be explained in terms of Lie symmetry groups for differential equations [6,8]. These are well known tools from mathematical physics which have appeared on occasion in various financial contexts [13,3, The usual results in the literature are to use point symmetries to derive the transformational properties of given differential equations and to study the integrability properties of models.
Largely these have been concerned with local point symmetry properties of the underlying space [13,14,17, 18]. This is the calculation in  for Eq. A note on the integrability of the classical portfolio selection model. We revisit the classical Merton portfolio selection model from the perspective of integrability analysis. By an application of a nonlocal transformation the nonlinear partial differential equation for the two-asset model is mapped into a linear option valuation equation with a consumption dependent source term.
This result is identical to the one obtained by Cox—Huang [J. Cox, C. Huang, Optimal consumption and portfolio policies when asset prices follow a diffusion process, J. Theory 49 33—88], using measure theory and stochastic integrals. The nonlinear two-asset equation is then analyzed using the theory of Lie symmetry groups. We show that the linearization is directly related to the structure of the generalized symmetries. There is however a growing lists of papers in applying this method to Finance.
Equation 2. Pricing Asian Options: Aug We consider the valuation of path-dependent contingent claims where the underlying asset follows a geometric average process. Considering the no-arbitrage PDE of these claims, we first determine the underlying Lie point symmetries. We then transform the BS equation into the heat equation which is solved using Poisson s formula taking into account the payoff condition.
We thus obtain a closed-form solution for the pricing of asian options in the geometric average case. This procedure appears for the first time in the finance literature. This is called the full feedback and leads to the nonlinear PDE: We will apply Lie symmetry analysis to the first-order feedback model 11 , which is coupled with the standard Black- Scholes equation Investigation of solutions of differential equations via Lie symmetry analysis has been done to many problems in financial mathematics, for example, 23 The primary objective of the present study is to determine general solutions as invariant solutions of the first-order feedback model Jul A first-order feedback model of option pricing consisting of a coupled system of two PDEs, a nonlinear generalised Black-Scholes equation and the classical Black-Scholes equation, is studied using Lie symmetry analysis.
This model arises as an extension of the classical Black-Scholes model when liquidity is incorporated into themarket. We compute the admitted Lie point symmetries of the system and construct an optimal system of the associated one-dimensional subalgebras. We also construct some invariant solutions of the model. Jan The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification UQ method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations.
In the present paper, we use the PCE approach in order to analyze some equity and interest rate models. In particular, we take into consideration those models which are based on, for example, the Geometric Brownian Motion, the Vasicek model, and the CIR model. We present theoretical as well as related concrete numerical approximation results considering, without loss of generality, the one-dimensional case.
We also provide both an efficiency study and an accuracy study of our approach by comparing its outputs with the ones obtained adopting the Monte Carlo approach, both in its standard and its enhanced version. We carry out group classification of a general bond-option pricing equation. We show that the equation admits a three-dimensional equivalence Lie algebra.
We also show that some of the values of the constants which result from group classification give us well-known models in mathematics of finance such as Black-Scholes, Vasicek, and Cox-Ingersoll-Ross. For all such values of these arbitrary constants we obtain Lie point symmetries. Symmetry reductions are then obtained and group invariant solutions are constructed for some cases.
Financial derivatives and Lie symmetries. Derivatives in finance have become pervasive in recent decades. Tremendous impetus was given to this aspect of finance by the pioneering papers of Black and Scholes and Merton and has attracted considerable interest since. In general the evolution partial differential equations were solved by means of the traditional methods of partial differential equations and by ansatz previously useful in similar contexts.
Here we illustrate the benefits of an algorithmic approach using the method of symmetry analysis introduced by Sophus Lie in the nineteenth century. We demonstrate the utility of this analytic approach with several examples chosen from the field of financial mathematics. We use Lie symmetry analysis to solve a boundary value problem that arises in chemical engineering, namely, mass transfer during the contact of a solid slab with an overhead flowing fluid.
This problem was earlier tackled using Adomian decomposition method Fatoorehchi and Abolghasemi , leading to the Adomian series form of solution. It turns out that the application of Lie group analysis yields an elegant form of the solution. After introducing the governing mathematical model and some preliminaries of Lie symmetry analysis, we compute the Lie point symmetries admitted by the governing equation and use these to construct the desired solution as an invariant solution.
Sep We consider the path-dependent contingent claims where the underlying asset follows an arithmetic average process. We then transform the BS equation into the heat equation and considering appropriate boundary condition, we find an explicit solution of the option pricing equation. Classical Lie point symmetry analysis of the considered PDEs resulted in a number of point symmetries being admitted. Two ways to solve, using Lie group analysis, the fundamental valuation equation in the double-square-root model of the term structure.
Two approaches based on Lie group analysis are employed to obtain the closed-form solution of a partial differential equation derived by Francis A. Longstaff [J Financial Econom ; Embedding the Vasicek model into the Cox—Ingersoll—Ross model. Jan Math Meth Appl Sci.
Zero coupon bond price vasicek model
To receive news and publication updates for Discrete Dynamics in Nature and Society, enter your email address in the box below. Correspondence should be addressed to Lina Song ; moc. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This work deals with European option pricing problem in fractional Brownian markets. Two factors, stochastic interest rates and transaction costs, are taken into account. By the means of the hedging and replicating techniques, the new equations satisfied by zero-coupon bond and the nonlinear equation obeyed by European option are established in succession.
Bond price under Vasicek model
What is the difference between an equilibrium model and a no-arbitrage model? Equilibrium models usually start with assumptions about economic variables and derive the behavior of interest rates. The initial term structure is an output from the model. In a no-arbitrage model the initial term structure is an input. The behavior of interest rates in a no-arbitrage model is designed to be consistent with the initial term structure. Problem In the Rendleman and Bartter model the standard deviation is proportional to the level of the short rate.WATCH THE VIDEO ON THEME: Stochastic Calculus Revealed - [Vasicek Model] Bond Price under Forward Risk Neutral Measure
Bond price under Vasicek model
In finance , the Vasicek model is a mathematical model describing the evolution of interest rates. It is a type of one-factor short rate model as it describes interest rate movements as driven by only one source of market risk. The model can be used in the valuation of interest rate derivatives , and has also been adapted for credit markets. The model specifies that the instantaneous interest rate follows the stochastic differential equation:. This is clear when looking at the long term variance,. This model is an Ornstein—Uhlenbeck stochastic process. Vasicek s model was the first one to capture mean reversion , an essential characteristic of the interest rate that sets it apart from other financial prices. Thus, as opposed to stock prices for instance, interest rates cannot rise indefinitely.
We have derived Vasicek SDE on a previous pos t. To ease the calculations, we will use the following property from the Moment Generating Function of a Normal Random variable:. Using the above, we can simplify the Bond Price calculations using the Expectation and the Variance of the Stochastic Rate:. We need to calculate the Variance of a stochastic integral, remember that the Variance of a deterministic quantity is zero, therefore we only need the Variance of the Stochastic term, as shown below. We will first concentrate on the double integrals and then we will plug-in the result into the above formula. The inner integral is stochastic while the outer one is deterministic. There are two difficulties with this equation; first, the Brownian Motion is non-differentiable and also, we have a nested i.
Zero coupon bond prices for Vasicek model.
Figure 1 from Machine Learning Vasicek Model Calibration Thus specifying a model for the short rate specifies future bond prices. The yield of any zero-coupon bond is taken to be a maturity-. An alternative approach to the calibration of the Vasicek and CIR interest rate models via generating functions. Vasicek model is. Interest rates l Cutting edge Shadow interest. The interest rates implied by the zero coupon bonds form a yield curve or more precisely, a zero curve. Fun with the Vasicek Interest Rate Model. The model specifies that the instantaneous interest rate follows the stochastic differential equation. Interest Rate Models:
.VIDEO ON THEME: Calculating the Yield of a Zero Coupon Bond