## 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.

### MSC:

 91B28 Finance etc. (MSC2000)

Zbl 1068.91035
Full Text:

### References:

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