Asset Pricing: Modeling and Estimation by B.Philipp Kellerhals

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By B.Philipp Kellerhals

The smooth box of asset pricing asks for sound pricing versions grounded at the idea of economic economies a l. a. Ingersoll (1987) as weIl as for accu­ cost estimation recommendations a los angeles Hamilton (1994b) by way of empirical inferences of the desired version. the assumption at the back of this e-book to be had is to supply the reader with a canonical framework that indicates the right way to bridge the distance among the continuous-time pricing perform in monetary engineering and the capital marketplace information necessarily basically to be had at discrete time periods. 3 significant monetary markets are to be tested for which we decide upon the fairness marketplace, the bond industry, and the electrical energy marketplace. In each one mar­ ket we derive new valuation versions to cost chosen monetary tools in continuous-time. the choice criterium for selecting a continuous-time version­ ing framework is the richness of the stochastic concept to be had for non-stop­ time strategies with Merton's pioneering contributions to monetary economics, amassed in Merton (1992). The continuous-time framework, reviewed and as­ sessed via Sundaresan (2000), permits us to acquire analytical pricing formulae that might be unavailable in a discrete time atmosphere. besides the fact that, on the time of imposing the derived theoretical pricing versions on marketplace information, that's unavoidably sampled at discrete time durations, we paintings with so-called special discrete time equivalents a los angeles Bergstrom (1984). We express tips on how to very easily paintings inside astate house framework which we derive in a common surroundings as weIl as explicitly for every of the 3 applications.

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We model the market value of the closed-end fund share prices, denoted by P (X, 7r, t; tP M) with parameter vector tP M, as a derivative security dependent on the values of the N AVt and the premium 7rt. , p I/. = -1 [( I/. , - oP (X, 7r,t;tPM) ot +P(Jx(J", _ ~ 2) ßP(X,7r,t;tPM) + ßX·· 1 -(Jx 2 'fix - p(Jx 1 2 + 2(JX 02p(X,7r,t;tPM) oXß7r K (() - 7r ) ßP(X,7r,t;tPM) Eh 02p(X,7r,t;tPM) OX2 1 2 + 2(J", (j2p (X, 7r,t;tPM)] 07r2 ' oP (X, 7r, t; tPM) d _ ~ oP(X,7r,t;tPM) oX ' an 'fI", - p(J", 07r ' where the time indices are temporarily dropped for legibility.

17 For an exploration of quasi (or pseudo) maximum likelihood estimation methods see, for example, Gourieroux, Monfort, Renault, and Trognon (1984). Chapter 2. 20) is needed. In order to estimate the model parameters t/J M L we decided to choose a quasi-Newton method, which is sirnilar to the methods used by Lund (1997) and Nunes and Clewlow (1999) based on Dennis and Schnabel (1996). The iteration rule for Newton's method for unconstrained minimization is given by the parameter estimates at the ith iteration step of where the variable Si is found by solving For the construction of"\7 L (y; t/Ji) and "\72 L (y; t/Ji) we use numerical approximations.

Estimation Principles 36 which reflects how likely it would have been to have observed the data if 1jJ were the true values for the parameters. Using the definition of conditional probability, we can split the likelihood up into conditional densities using Baye8 '8 theorem recursively and write the joint density as the product of conditional densities I (Yi 1jJ) = P (YTIYT-l,'" ,Yli 1jJ) ..... P (YtIYt-l, ... ,Yli 1jJ) ..... 14) where we approximate the initial density function P (Yli 1jJ) by P (YlIYOi 1jJ).

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