Symmetry analysis of a model of stochastic volatility with time-dependent parameters

Sophocleous, Christodoulos; O’Hara, John G.

doi:10.1016/j.cam.2011.03.009

Cited by 12 publications

(12 citation statements)

References 22 publications

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“…Also, two equations of the class in eqn. (2.29) that can be mapped to each other via an equivalence transformation have isomorphic Lie symmetry algebras [17,2,16,3]. The latter results are often used to detect isomorphic equations within the class in eqn.…”

Section: Equation Depending On Arbitrary Parameters: Equivalence Tranmentioning

confidence: 99%

“…(2.29) is as difficult as finding all its symmetries. In some particular instance, one may find all the equivalence transformations through the use of the chain rule and reasoning about the differential structure of the family [2,3,16,17]. As for Lie symmetries, if we restrict ourselves to continuous groups of equivalence transformations, one arrives at an algorithm [16] which is similar to Lie's symmetry algorithm in its derivation and application.…”

Section: Equation Depending On Arbitrary Parameters: Equivalence Tranmentioning

confidence: 99%

“…(2.29), one uses the fact that two equation are equivalence if and only if their symmetry Lie algebras are isomorphic i.e. they have the same structure constants after an appropriate choice of their bases [2,3,16,17]. We shall take advantage of this result in the sequel.…”

Section: Equation Depending On Arbitrary Parameters: Equivalence Tranmentioning

confidence: 99%

“…For models encoded as differential equations, assumptions that translate into the invariance of underlying equations are particularly useful in the search of their solutions. This is the main thrust of Lie symmetry analysis of differential equations pioneered by Sophus Lie [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In such analysis, one algorithmically looks for infinitesimal transformations (i.e., transformations depending on a small parameter and enjoying additional properties [2,9,17] of both dependent and independent variables that does not change the underlying differential equation.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Equation Depending On Arbitrary Parameters: Equivalence Tranmentioning

confidence: 99%

Section: Equation Depending On Arbitrary Parameters: Equivalence Tranmentioning

confidence: 99%

Section: Equation Depending On Arbitrary Parameters: Equivalence Tranmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…In the case of a positive answer the BVP for the PDE was reduced to a BVP for an ODE. Using this technique a number of boundary-value problems were solved successfully (see, e.g., [18,19,23]). Consider the initial and boundary value problem…”

Section: Application Of Lie Symmetries To a Boundary Value Problemmentioning

confidence: 99%

Group Classification of Variable Coefficient K(m,n) Equations

Charalambous¹,

Vaneeva²,

Sophocleous³

2014

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Lie symmetries of K(m, n) equations with time-dependent coefficients are classified. Group classification is presented up to widest possible equivalence groups, the usual equivalence group of the whole class for the general case and conditional equivalence groups for special values of the exponents m and n. Examples on reduction of K(m, n) equations (with initial and boundary conditions) to nonlinear ordinary differential equations (with initial conditions) are presented.

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Solving a partial differential equation associated with the pricing of power options with time‐dependent parameters

Okelola

Govinder

O’Hara

2014

Math Methods in App Sciences

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Previous analysis and research on the power option – one of the exotic options – have focused on the interest rate of the stock and its volatility as constant parameters throughout the run of execution. In this paper, we attempt to extend these results to the more practical and realistic case of when these parameters are time dependent. By making no ansatz or relying on ad hoc methods, we are able to achieve this via an algorithmic method – the Lie group approach – leading to exact solutions for the power option problem. Copyright © 2014 John Wiley & Sons, Ltd.

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Symmetry analysis of a model of stochastic volatility with time-dependent parameters

Cited by 12 publications

References 22 publications

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

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

Group Classification of Variable Coefficient K(m,n) Equations

Solving a partial differential equation associated with the pricing of power options with time‐dependent parameters

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