By Svetlozar T. Rachev, John S. J. Hsu, Biliana S. Bagasheva, Frank J. Fabozzi
Bayesian tools in Finance offers an in depth assessment of the idea of Bayesian tools and explains their real-world functions to monetary modeling. whereas the foundations and ideas defined through the e-book can be utilized in monetary modeling and determination making usually, the authors specialize in portfolio administration and marketplace danger management—since those are the parts in finance the place Bayesian equipment have had the best penetration to this point.
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Additional info for Bayesian Methods in Finance
The parameter σ is the scale parameter. For σ σ example, the normal density, N(µ*, σ 2 ), is a scale density when the mean is fixed at some µ*. 8 See Jeffreys (1961). In general, Jeffreys’ prior of a parameter (vector), θ , is given by π (θ ) = |I(θ )|1/2 , where I(θ ) is the so-called Fisher’s information matrix for θ , given by I(θ ) = −E ∂ 2 log f (x | θ ) , ∂θ ∂θ Prior and Posterior Information, Predictive Inference 27 prior distribution. Let us turn again to the example for the monthly returns for some financial asset we considered earlier and suppose that we do not have particular prior information about the range of typical values the mean monthly return could take.
8 See Jeffreys (1961). In general, Jeffreys’ prior of a parameter (vector), θ , is given by π (θ ) = |I(θ )|1/2 , where I(θ ) is the so-called Fisher’s information matrix for θ , given by I(θ ) = −E ∂ 2 log f (x | θ ) , ∂θ ∂θ Prior and Posterior Information, Predictive Inference 27 prior distribution. Let us turn again to the example for the monthly returns for some financial asset we considered earlier and suppose that we do not have particular prior information about the range of typical values the mean monthly return could take.
N−x)! During the sample x period, there are X = 176 trade-by-trade consecutive price increases. This information is embodied in the likelihood function for θ : L θ | X = 176 = θ 176 (1 − θ )55667−176 . 11) We would like to combine that information with our prior belief about what the probability of a consecutive price increase is. Before we do that, we recall the notational convention we stick to throughout the book. We denote the prior distribution of an unknown parameter θ by π (θ ), the posterior distribution of θ by π (θ |data), and the likelihood function by L(θ | data).