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A Student’s
t-distribution is obtained from a weighted average over the standard deviation of a normal distribution, σ, when 1/σ is distributed as chi. Left truncation at q of the chi distribution in the mixing integral leads to an effectively truncated Student’s
t-distribution with tails that decay as exp (-q
^{2}t
^{2}). The effect of truncation of the chi distribution in a chi-normal mixture is investigated and expressions for the pdf, the variance, and the kurtosis of the t-like distribution that arises from the mixture of a left-truncated chi and a normal distribution are given for selected degrees of freedom
__<__5. This work has value in pricing financial assets, in understanding the Student’s
t--distribution, in statistical inference, and in analysis of data.

A Student’s t-distribution is used for statistical inference for small sample sizes [1,2]. Nadarajah [

In addition to statistical inference, the Student’s t-distribution has application in finance, wherein the t-distribution is found to fit the distribution of the logarithms of daily returns better than a normal distribution or indeed better than most any other distribution [4-9]. The number of degrees of freedom, , is found by least squares fitting to historical returns to be around 3 for daily returns of the DJIA and S&P 500 indices [

The Student’s t-distribution also has application wherever a Cauchy (Lorentzian) distribution is employed, since a Student’s t-distribution with one degree of freedom is a Cauchy distribution.

The Student’s t-distribution offers support from to and has tails that decrease as for. This causes problems in finance as integrals needed to price financial instruments diverge for the logarithm of returns distributed as a Student’s t-distribution [9-11] and the frequency of occurrence of the logarithms of daily returns is fit well by Student’s t-distribution.

Truncation, capping, and modification of the t-distribution have been put forth as means to deal with the divergence [8,9,12]. Moriconi [

Praetz [

Using chi as defined above and a normal distribution with zero mean and standard deviation of, the mixing integral, when evaluated from to yields a Student’s t-distribution

with a mean of zero, degrees of freedom, and a scale parameter of. The parameters for the chi distribution were chosen to yield a Student’s t-distribution, , with a mean of zero, degrees of freedom, and a scale parameter of.

If a chi distribution for the reciprocal of is left truncated, then the result of the mixing integral with the left-truncated chi distribution is a t-like distribution that has exponentially decaying tails. This result is demonstrated for. The integrals involved can be evaluated analytically for odd. Small values of are of interest.

The contribution to a Student’s t-distribution, , from values in the left-hand wing of the chi distribution for is given by

(3)

whereas the contribution from values in the right-hand wing of the chi distribution for is given by

It is interesting to note that the sum of the contributions from Equations (3) and (4) form a full Student’s t-distribution (in this case the mixing integral has been written as the sum of two exhaustive and exclusive regions: and), and that the right-hand contribution has an exponentially decaying tail. As a result, any pdf that does not include the left-hand contribution from the chi distribution for the reciprocal of the standard deviation in a chi-normal mixture will have tails that decay as exp.

tribution.

The slowly decaying power tails of the t-distribution with increasing t results in the divergence of moments and of integrals required to price financial instruments that are based on a log Student’s t-distribution. The results presented in

Left truncation of the chi distribution for the reciprocal of the standard deviation to allow for only physically possible values of the standard deviation will impart exponentially decaying tails to the resulting t-like distribution from the chi-normal mixture. It is unlikely that. A value of implies no variability and a value of implies infinite variability.

In the following sections, expressions for the probability density function (pdf), the variance, and the kurtosis are given for several small values of for t-like distributions that are obtained by left truncation of a chinormal mixture. In addition, the first several terms for power series expansions of the variance and kurtosis are given. These power series demonstrate the effect of the left truncation on the moments. Since the power series are valid only for small amounts of truncation,

for small values of the independent variable for the chi distribution in the mixing integral that yields a Student’s t-distribution with v = 1, 2, 3, 5 and 9 degrees of freedom. The long tic marks give the values (q = 0.0125, 0.10, 0.196, 0.333, 0.482) for which the CDF.

The probability that, , is needed to normalize properly a truncated chi distribution. A left-truncated chi distribution is zero for values:

is the pdf for a mixture of a left-truncated chi and normal distribution. If then would be a Student’s t-distribution with degrees of freedom and scale parameter. An explicit expression for is

is the variance of the pdf,

and is the kurtosis of the pdf,

This paper provides information on the effective truncation of a Student’s t-like distribution when the chi distribution for the reciprocal of the standard deviation is left-truncated. It is shown that the tails of the effectively truncated t-distribution go as exp. The contribution to an odd Student’s t-distribution from values of for a chi-normal mixture, i.e., Equation (4) evaluated for odd, is given by (see Equation (9) below).

The expression, which decreases with t as exp, was obtained from an examination of the expressions for for and by comparison with the general case for odd. Clearly, left truncation of the chi distribution for the reciprocal of the standard deviation in a chi-normal mixture removes the fat tails of the Student’s t-distribution.

Expressions for some of the low, effectively truncated Student’s t-distributions and moments are given in the following subsections. For a Student’s t-distribution the variance equals and exists only for. The kurtosis equals and exists only for. For effectively truncated Student’s t-distributions (i.e., truncation of the large values of the standard deviation in the chi mixing distribution), the moments exist for all.

The pdf for a mixture of a left-truncated chi distribution for and and a normal distribution is

The tails of the pdf decrease as exp for non-zero.

The first moment for exists for the effectively truncated t-distribution and is given by

where is Euler’s contant and Ei() is the exponential integral. The series expansion is valid for. The series expansion shows a logarithmic divergence as approaches zero, as expected for a Cauchy distribution.

The variance is given by

is proportional to, remains finite for, and diverges as as approaches zero.

An exact expression for the kurtosis can be found, but the expression is long and cumbersome. The series expansion

shows that the kurtosis is proportional to and hence stays finite for. Both the variance and the kurtosis are not defined for a Cauchy distribution, i.e., for a Student’s t-distribution with and a region of support from to.

The pdf, , for a mixture of a left-truncated chi distribution for and with a normal pdf is

The first term in the series expansion is equal to.

has tails that decrease as exp for non-zero, since for large x. For large qt,

Simple, analytic expression for the moments for even values of v could not be found.

The pdf, , equals

and has tails that decrease as exp for non-zero q.

The variance is given by

and approaches as q approaches zero. This is expected since the variance for a Student’s t-distribution exists for and equals or for v = 3.

A series expansion for the kurtosis (the exact expression is long and cumbersome) is

and diverges as as q approaches zero. The kurtosis for a Student’s t-distribution is defined only for.

The series expansion for the variance shows a weak dependence on the left truncation. To lowest order in q, the variance has a cubic dependence on q. The curves of

The series expansions for the kurtosis is given by

The kurtosis is finite for. The series expansion is a linear decrease in q for small q.

For, the first two terms of the alternating series for the expansion of the kurtosis are. For the kurtosis is not impacted by a truncation for small.

Application of the effectively truncated Student’s t-distribution to pricing a European call option is given in this section.

The value of a European call option at the time of expiration, , is the expectation of the maximum value of, , where is the price of a stock at time T, is the strike price at time T, T is the time when the option expires, and is the expectation operator [

Let be the value of the stock at time T where is a random variable, and where both and the volatility do not depend on the random variable.

In terms of the pdf for, , the value of the option at time T is

The value of is determined by the requirements that the process be fair (i.e., the process is a martingale) and that the development in time of the price include the time value of money [

where is the value of the stock at time 0.

If is normally distributed, then follows a log normal distribution, , and the price for the option is given by the Black-Scholes formula.

If follows a Student’s t-distribution, then is infinite. The exponential growth of with dominates the tails of the Student’s distribution, Equation (2). However, if an effectively truncated Student’s t-distribution is used, then the tails of the effectively truncated t-distribution diminish with as, and the value of the European call option remains finite. Both the effectively truncated Student’s t-distribution and the normal distribution have a multiplicative factor that dominates the exponential growth of for large. In general, the integral to price the European call option must be evaluated numerically. Equation (21) is perhaps the simplest manner in which to write the cost of the call option.

The Student’s t-distribution is found (over the subinterval of the infinite region of support where returns are observed) to fit the distribution of the logarithms of daily returns better than a normal distribution or indeed better than most any other distribution [4-9]. It would be prudent to price options using a distribution that matches the data. The effectively truncated t-distribution that is described in this paper is a distribution that matches the observed data, as demonstrated in

Adaptive bins widths were used to generate the frequency of occurrence data for

A fit of a Student’s t-distribution to the logarithm of the daily returns over the period of January 1950 to 27 July 2011 for the S&P 500 Index gives and a scale parameter. The data show a kurtosis of 25 and a maximum 22-day volatility of. The approximation for the kurtosis for a effectively truncated t-distribution equal to 25, , is solved for. The chi distribution in the mixing integral is, assuming that, truncated for volatilities to obtain the same kurtosis as the data. This level of truncation is roughly twice the maximum 22-day volatility of that was obtained from the daily returns. Note that the kurtosis exists only for for Student’s tdistributions. In contrast, the kurtosis is defined for for an effectively truncated t-distribution, in agreement with the data.

The area in the left wing for truncation of the chi distribution at to match the kurtosis of the observed daily returns of 25 is. This level of truncation of the chi distribution has a minor effect on the effectively truncated Student’s t-distribution for, as can be observed in

A Student’s t-distribution arises from an averaging over the standard deviation of a normal distribution when the reciprocal of the standard deviation is distributed as chi. The Student’s t-distribution offers support over to. The slowly decaying power tails and infinite support region mean some moments for the Student’s t-distribution do not exist. This divergence of integrals

can cause problems, particularly in the pricing of options where the logarithm of returns is distributed as a Student’s t-distribution. One approach to deal with the divergence is to truncate the Student’s t-distribution [

Expressions or power series expressions for the pdf, variance, and kurtosis for several low number of degrees of freedom, effectively truncated Student’s t-distributions are given. These expressions demonstrate the exponential tails of the effectively truncated t-distribution and show that the variance and kurtosis remain remain finite for the effectively truncated distributions. In addition, it is shown that it is the large values of the standard deviation in the mixing integral that give rise to the fat tails of the Student’s t-distribution. The small values of the standard deviation in the mixing integral do not contribute to the tails of the Student’s t-distribution and only weakly contribute to the core of the Student’s t-distribution.

The effective truncation of the Student’s t-distribution means that integrals required to price financial instruments such as European call options remain finite. This permits pricing with a distribution, the log Student’s t-distribution, that describes well returns for stocks.

Simple expressions were not found for the effectively truncated t-distributions with even numbers of degrees of freedom. Given the importance of the Student’s t-distribution in the description of returns and data in general, it would be helpful to find accurate approximations that smoothly approach the expressions for the odd numbers of degrees of freedom. These approximations could then be used to model data where it is unphysical to assume support over to. The envelope modification of Moriconi [

This work was funded by the Natural Science and Engineering Research Council (NSERC) Canada.