By Ioannis Karatzas
This publication could be of curiosity to researchers wishing to determine complex arithmetic utilized to finance. the fabric on optimum intake and funding, resulting in equilibrium, is addressed to the theoretical finance group. The chapters on contingent declare valuation current innovations of sensible value, specifically for pricing unique concepts.
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197 empirical density and the fitted normal density for both daily and monthly data. We notice how the NIG distribution fits the empirical both in the tails and with respect to the skewness, contrary to the normal distribution. The four parameters of the NIG distribution can be transformed into the so-called shape triangle parameters (see ) : Since 0 :s; Ixi < ~ < 1, the parametrization (X,~) are coordinates in a triangle (inverted). In the limit (X,~) -+ (0,0), the NIG distribution converges to a Gaussian distribution.
The scale parameter is 8, which plays almost the same role as the standard deviation for the normal distribution. The tail heaviness of the distribution is modelled through a . 6]. 4) that the parameters 0: and 13 must satisfy 0 ~ 1131 ~ 0:. In addition, 8 > O. If the distribution of a random variable L is normal inverse Gaussian with parameters o:,j3,p, and 8, we write L rv NIG(o:,j3, p"8). 9 exhibit four examples of the NIG distribution for different choices of parameters. In Fig. 9 all the distributions are plotted using a logarithmic scale on the vertical axis.
Bm(t) be m independent Brownian motions. Define the price dynamics of stock i to be the solution of the stochastic differential equation m dSi(t) = aiSi(t) dt + Si(t) L aij dBj(t), i = 1, . 23) j=l which is called a multi-dimensional geometric Brownian motion. 24) for i = 1, ... , n. The number of independent Brownian motions m can be less than, equal to or greater than the number of stocks n . The volatility parameters aij describe the correlation among the logreturns of the stocks. 14. The conclusion from this section is that geometric Brownian motion has two representations.
Methods of Mathematical Finance by Ioannis Karatzas