Fundamental Solutions for Zero-Coupon Bond Pricing Models

Pooe, Charlemagne; Mahomed, F. M.; Soh, C. Wafo

doi:10.1023/b:nody.0000034647.76381.04

Cited by 35 publications

(32 citation statements)

References 11 publications

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“…Bakkaloglu et al [5] worked on optimal investment-consumption problem with CEV model by invariant approach. Pooe et al [6],İzgi and Bakkaloglu [7], [8], [9] investigated the fundamental solutions to the zero-coupon bond pricing equations with the Lie symmetry analysis. In recent times, the group approach has been widely applied to other partial differential equations of finance, for example, Naicker et al [10], Ivanova et al [11], Liu and Wang [12], Caister et al [13], Sinkala [14] and an interesting topical review work by Hernández et al [15].…”

Section: Introductionmentioning

confidence: 99%

Invariant criteria for the zero-coupon bond pricing Vasicek and Cox-Ingersoll-Ross Models

Aziz¹,

Mahomed²,

Bakkaloglu³

2017

ntmsci

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Abstract:The zero coupon bond pricing Vasicek and Cox-Ingersoll-Ross (CIR) interest rate models are solved using the invariant approach. The invariance criteria is employed on the linear (1 + 1) parabolic partial differential equations corresponding to the Vasicek and CIR models in order to perform reduction into one of the four Lie canonical forms. The invariant approach helps in transforming the partial differential equation representing the Vasicek model into the first Lie canonical form which is the classical heat equation. We also find that the invariant method aids in transforming the CIR model into the second Lie canonical form and with a proper parametric selection, the CIR equation can be converted to the first Lie canonical form. For both the Vasicek and CIR models, we obtain the transformations which map these equations into the heat equation and also to the second Lie canonical form. We construct the fundamental solutions for the Vasicek and CIR models via these transformations by utilizing the well-known 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 of the Vasicek and CIR models with suitable choice of terminal boundary conditions are also deduced.

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Section: Introductionmentioning

confidence: 99%

Invariant criteria for the zero-coupon bond pricing Vasicek and Cox-Ingersoll-Ross Models

Aziz¹,

Mahomed²,

Bakkaloglu³

2017

ntmsci

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“…In their work, they applied Lie symmetries to transform the BlackScholes equation into the heat equation and solved it. Pooe et al [24] deduced a fundamental solution to zero-coupon bonds. Since the introduction of Lie symmetries to finance, many models have been designed to price and hedge options accurately, and models are becoming more sophisticated with the development of new techniques and the evolvement of technology (see for example, [12,13]).…”

Section: Introductionmentioning

confidence: 99%

“…The derivation of the option pricing equation relies heavily on the random process followed by the underlying stock price. In the case of the celebrated Black-Scholes equation, one assumes that the stock follows the classical geometric Brownian motion [5,24]. This process assumes independent increments, which makes it different from the fractional Brownian motion (fBm) where there is a serial correlation in the increments.…”

Section: Introductionmentioning

confidence: 99%

Symmetry Analysis of Options Pricing with Transactions Costs Driven by Fractional Brownian Noises

Bf¹,

Pindza²,

Maré³

2017

J Appl Computat Math

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We provide a closed-form solution for the European and Asian option pricing models when the source of randomness is a fractional Brownian motion as opposed to the geometric Brownian motion. In addition to the source of randomness, transaction costs are considered to be non-negligible. For the case of the European option, proportional transaction costs hide in the volatility and do not change the form of the model. The construction of the solution is based on the symmetries of the model. The model for Asian options has an additional parameter that makes the volatility time-dependent, which complicates the solution process. However, we are still able to obtain solutions using Lie symmetry methods. Citation: Nteumagn BF, Pindza E, Mare E (2017) Symmetry Analysis of Options Pricing with Transactions Costs Driven by Fractional BrownianNoises. J Appl Computat Math 6: 356. doi: 10.4172/2168-9679.1000356 Page 2 of 8Volume 6 of the option pricing equation relies heavily on the random process followed by the underlying stock price. In the case of the celebrated Black-Scholes equation, one assumes that the stock follows the classical geometric Brownian motion [5,24]. This process assumes independent increments, which makes it different from the fractional Brownian motion (fBm) where there is a serial correlation in the increments. This correlation allows for predictability in the model. The similarity between the two processes is the fact that they are both geometric.It has been shown [8] that random changes can be accurately measured by the use of a parameter known as the Hurst parameter. The construction of a dam on the Nile river in the early 1950s saw the birth of the Hurst exponent. The measure of the water levels of increase or decrease was known to follow a random walk. In other words, the next level of increase or decrease is independent of the previous one. However, Edwin Hurst, hydrologist in the Nile dam project, showed that the long term memory of the system indicates an autocorrelation of the time series of water levels, and the rate at which these decrease as the lag between pairs of values increase [8]. The movement of stock price is similar to this model. We therefore exploit the results by Leland [11] for the derivation of our model. Earlier in 2014, we achieved an accurate numerical solution of our model with exponential convergence ([19]) by applying spectral methods to analyze the problem. In this paper, we use Lie symmetries to determine the exact solution, which sets us a step closer to the calibration of our model.We first give some preliminaries on Lie symmetries of differential equations in Section 2. Secondly, we determine the symmetries of the European and Asian option pricing models and obtain their analytical solutions in Section 3. In Section 4 we apply the Lie symmetry technique to reduce the PDEs and provide solutions. Finally, we provide concluding remarks in Section 5. Lie Symmetries and Equivalence TransformationsOur aim in this section is to provide a brief introduction to ...

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“…A number of studies have been devoted to the use of symmetry techniques for PDEs arising in the field of finance mathematics (see e.g. [4,8,12,14]). The theory and applications of symmetries may be found in excellent texts such as [3,7,10,15].…”

Section: Introductionmentioning

confidence: 99%

“…Pooe et al [12], assumed that the spot rate follows the stochastic process (see also [4,5]) given by dx(t) = a(x, t) dt + w(x, t) dZ(t), and ended up solving the model…”

Section: Introductionmentioning

confidence: 99%

Optimal Systems and Group Invariant Solutions for a Model Arising in Financial Mathematics

Nteumagne

Moitsheki

2009

Mathematical Modelling and Analysis

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Fundamental Solutions for Zero-Coupon Bond Pricing Models

Cited by 35 publications

References 11 publications

Invariant criteria for the zero-coupon bond pricing Vasicek and Cox-Ingersoll-Ross Models

Invariant criteria for the zero-coupon bond pricing Vasicek and Cox-Ingersoll-Ross Models

Symmetry Analysis of Options Pricing with Transactions Costs Driven by Fractional Brownian Noises

Optimal Systems and Group Invariant Solutions for a Model Arising in Financial Mathematics

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