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**Lie-algebraic approach for pricing moving barrier options with time-dependent parameters.**
*(English)*
Zbl 1148.91020

In the previous communications: [C. F. Lo and C. H. Hui, Quant. Finance 1, 73–78 (2001); Int. J. Math. Math. Sci. 32, No. 7, 401–410 (2002; Zbl 1068.91035)] the authors have used the Lie-algebraic approach in financial problems. In this paper the authors extend the Lie-algebraic approach to the valuation of moving. Barrier options with time-dependent parameters. In the valuation of these moving barrier options, the value of the underlying asset is assumed to follow the constant elasticity of variance (CEV) diffusion process:
\[
DS= \mu(t)Sdt+\sigma(t)S^{\beta/2}dZ,\quad 0\leq\beta<2.
\]
Where \(\mu\) is the instantaneous mean, \(\sigma S^{\beta/2}\) is the instantaneous variance of the stock price, \(dZ\) is a Weiner process and \(\beta\) is the elasticity factor. The equation shows that the instantaneous variance of the percentage price change is equal to \(\sigma^2/S^{2-\beta}\) and is a direct inverse function of the stock price. In limiting case \(\beta=2\), the CEV model returns to the conventional Black-Scholes model in which the variance rate is independent of the stock price. Where as \(\beta=0\), it is the Ornstein-Uhlenbeck model. The authors generalize the Lie-algebraic technique to derive the analytical kernels of the pricing formulae of the moving barrier options with time-dependent parameters, and thus provide an efficient way for computing the prices of both up-and-out and down-and-out barrier options. It has been also shown that making use of the maximum principle for the parabolic partial differential equation this approach can be applied to yield very tight upper and lower bounds of the exact prices of CEV barrier options with feed barriers.

Reviewer: C. L. Parihar (Indore)

### MSC:

91B28 | Finance etc. (MSC2000) |

### Citations:

Zbl 1068.91035
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\textit{C. F. Lo} and \textit{C. H. Hui}, J. Math. Anal. Appl. 323, No. 2, 1455--1464 (2006; Zbl 1148.91020)

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### References:

[1] | Lo, C.F.; Hui, C.H., Valuation of financial derivatives with time-dependent parameters—lie-algebraic approach, Quantitative finance, 1, 73-78, (2001) · Zbl 1405.91637 |

[2] | Lo, C.F.; Hui, C.H., Pricing multi-asset financial derivatives with time-dependent parameters—lie-algebraic approach, Int. J. math. math. sci., 32, 401-410, (2002) · Zbl 1068.91035 |

[3] | Wei, J.; Norman, E., Lie-algebraic solution of linear differential equations, J. math. phys., 4, 575-581, (1963) · Zbl 0133.34202 |

[4] | Friedman, A., Partial differential equations of parabolic type, (1964), Prentice Hall New Jersey · Zbl 0144.34903 |

[5] | J. Cox, Notes on option pricing I: Constant elasticity of variance diffusions, Working Paper, Stanford University, 197 |

[6] | Cox, J.C.; Ross, S.A., The valuation of options for alternative stochastic processes, J. financial economics, 3, 145-166, (1976) |

[7] | Perelomov, A.M., Generalized coherent state and its applications, (1986), Springer New York |

[8] | Ban, M., Decomposition formulas for \(\operatorname{su}(1, 1)\) and su(2) Lie algebras and their applications in quantum optics, J. opt. soc. amer. B, 10, 1347-1359, (1993) |

[9] | Lebedev, N.N., Special functions and their applications, (1972), Dover New York · Zbl 0271.33001 |

[10] | Titchmarsch, E.C., Eigenfunction expansions associated with second-order differential equations, (1946), Clarendon Oxford |

[11] | Lo, C.F.; Yuen, P.H.; Hui, C.H., Constant elasticity of variance option pricing model with time-dependent parameters, Int. J. theor. appl. finance, 3, 661-674, (2000) · Zbl 1006.91050 |

[12] | Lo, C.F.; Yuen, P.H.; Hui, C.H., Pricing barrier options with square root process, Int. J. theor. appl. finance, 5, 805-818, (2001) · Zbl 1154.91461 |

[13] | Schmalensee, R.; Trippi, R.R., Common stock volatility expectations implied by option premia, J. finance, 33, 129-147, (1978) |

[14] | Beckers, S., The constant elasticity of variance model and its implications for option pricing, J. finance, 35, 661-673, (1980) |

[15] | Lauterbach, B.; Schultz, P., Pricing warrants: an empirical study of the black – scholes model and its alternatives, J. finance, 45, 1181-1209, (1990) |

[16] | Hauser, S.; Lauterbach, B., Tests of warrant pricing models: the trading profits perspective, J. derivatives, 71-79, (1996) |

[17] | Cathcart, L.; El-Jahel, L., Valuation of defaultable bonds, J. fixed income, 2, 65-78, (1998) |

[18] | Hui, C.H.; Lo, C.F., Valuation model of defaultable bond values in emerging markets, Asia-Pacific financial markets, 9, 45-60, (2002) · Zbl 1061.91027 |

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