1. Introduction & Review of the Literature

2. Characteristics of the Generalized Hyperbolic Distribution (GHD) – Distributional Form and Corresponding Parameters

3. Parameter Estimation of the Symmetrical GHD

4. Discussion

4.2. Determining median-sorted outcome after N trials

4.2.1. Calculating the inverse CDF

To determine the median sorted outcome, we need to first construct the inverse CDF function. An inverse cumulative distribution function (inverse CDF) is a mathematical function that provides the value of the random variable for which a given probability has been reached. It is also referred to as the quantile function or the percent-point function. In simpler terms, the inverse CDF essentially allows you to determine the value at which a specified probability is attained in a given distribution.

To derive the inverse CDF of the Symmetrical GHD Distribution, we first start with the CDF of the Symmetrical GHD Distribution, which is given by:

$$CDF\left(x\right)=F\left(x\right)=\frac{1}{2}\left[1+\mathrm{sgn}\left(x-\mu \right){\left\{1-{\left(\frac{\delta }{\sqrt{{\delta }^2+{(x-\mu )}^2}}\right)}^{\alpha }\right\}}^{\frac{\lambda -1}{\alpha }}\right]$$

(1)

Where:

α = concentration

δ = scale

μ = location

λ = shape

$\mathrm{sgn}\left(x-\mu \right)$ = sign function, the sign of (x-u), or simply $\mathrm{sgn}\left(x-\mu \right)=\frac{x-\mu }{\left|x-\mu \right|}$

Rearranging the equation to isolate the term inside the curly braces:

$${\left\{1-{\left(\frac{\delta }{\sqrt{{\delta }^2+{(x-\mu )}^2}}\right)}^{\alpha }\right\}}^{\frac{\lambda -1}{\alpha }}=\frac{2\left(F\left(x\right)-1/2\right)}{\mathrm{sgn}\left(x-\mu \right)}$$

We raise both sides to the power of $\alpha /\left(\lambda -1\right)$:

$$1-{\left(\frac{\delta }{\sqrt{{\delta }^2+{(x-\mu )}^2}}\right)}^{\alpha }={\left[\frac{2\left(F\left(x\right)-1/2\right)}{\mathrm{sgn}\left(x-\mu \right)}\right]}^{\frac{\alpha }{\lambda -1}}$$

Solving for $\sqrt{{\delta }^2+{(x-\mu )}^2}$:

$$\sqrt{{\delta }^2+{(x-\mu )}^2}=\delta \left[1-{\left\{{\left[\frac{2\left(F\left(x\right)-1/2\right)}{\mathrm{sgn}\left(x-\mu \right)}\right]}^{\frac{\alpha }{\lambda -1}}\right\}}^{1/\alpha }\right]$$

Solving for $x$:

$$x=\left\{\begin{array}{ll}\mu -\delta \left[1-{\left\{{\left[\frac{2\left(F\left(x\right)-1/2\right)}{-1}\right]}^{\frac{\alpha }{\lambda -1}}\right\}}^{1/\alpha }\right]\hfill & \mathrm{if}F\left(x\right)<\frac{1}{2}\hfill \\ \mu +\delta \left[1-{\left\{{\left[\frac{2\left(F\left(x\right)-1/2\right)}{1}\right]}^{\frac{\alpha }{\lambda -1}}\right\}}^{1/\alpha }\right]\hfill & \mathrm{if}F\left(x\right)\ge \frac{1}{2}\hfill \end{array}\right.$$

The inverse CDF function takes a probability value $p$ as input and returns the corresponding value of $x$ such that:

$$F^{-1}\left(p\right)=\left\{\begin{array}{ll}\mu -\delta {\left(1-{\left(\frac{2\left(1-p\right)}{-1}\right)}^{1/\alpha }\right)}^{1/\lambda }\hfill & \mathrm{if}p<\frac{1}{2}\hfill \\ \mu +\delta {\left(1-{\left(\frac{2\left(p-1/2\right)}{1}\right)}^{1/\alpha }\right)}^{1/\lambda }\hfill & \mathrm{if}p\ge \frac{1}{2}\hfill \end{array}\right.$$

Simplifying the expressions inside the parentheses:

$$F^{-1}\left(p\right)=\left\{\begin{array}{ll}\mu -\delta {\left(\frac{2p-1}{2p-1}\right)}^{1/\alpha \lambda }\hfill & \mathrm{if}p<\frac{1}{2}\hfill \\ \mu +\delta {\left(\frac{2p-1}{2-2p}\right)}^{1/\alpha \lambda }\hfill & \mathrm{if}p\ge \frac{1}{2}\hfill \end{array}\right.$$

Simplifying the expressions using the fact that $\mathrm{SIGN}\left(p-0.5\right)=\pm 1$:

$$x=F^{-1}\left(p\right)=\mu +\mathrm{SIGN}\left(p-0.5\right)\times \delta \times {\left(\frac{2\left|p-0.5\right|}{1-2\left|p-0.5\right|}\right)}^{1/\alpha \lambda }$$

This is our inverse CDF for the symmetrical GHD.

4.2.2. Worst-case expected outcome

Since we seek to construct a "representative" stream of outcomes, a stream of outcomes we would "expect" to be typical (half better, half worse) that what we would expect happen over N trials, and given the critical nature of "worst case outcome" in [18],[25],[30],[31],[12],[32] for which we wish to apply this expected stream of outcomes to, we thus wish to find that point on the distribution that we would expect be seen, typically, over N trials.

Using the inverse CDF, we can find this point, this "worst-case expected outcome" as that value of the inverse CDF for the input:

worst-case expected outcome = ( 1 / N ) ^N

We can use this to examine against the results of our representative stream of outcomes at the median sorted outcome.

4.2.3. Calculating the representative stream of outcomes from the median-sorted outcome

We proceed to discern the median sorted outcome. We will generate a large sequence of random variables (0..1) to process via our inverse CDF to give us the random outcomes from our distribution which we will take N at a time.

• For each of N outcomes, we determine the corresponding probability by taking the value returned by the inverse CDF as input to the CDF. When using the inverse CDF function, the random number generated is not the probability of the value returned by the inverse CDF. Instead, the random number generated is used as the input to the inverse CDF function to obtain a value from the distribution. This value is a random variable that follows the distribution specified by the CDF. To determine the probability of this value, you would need to use the CDF function and evaluate it at the value returned by the inverse CDF function. The CDF function gives the probability that a random variable is less than or equal to a specific value. Therefore, if you evaluate the CDF function at the value returned by the inverse CDF function, you will get the probability of that value.
• For our purposes however, we must now take any of these probabilities that are > .5 and take the absolute value of their difference to .5. Call this p`.

• p`= p if p <= .5
• p` = |1 - p| if p > .5

c.

We take M sets of N outputs, corresponding to outputs drawn from our fitted distribution as well as the corresponding p` probabilities.

d.

For each of these M sets of N outcome-p` combinations, we will sum the outcomes, and take the product of all of the p`s.

e.

We sort these M sets based on their summed outcomes.

f.

We now take the sum of these M product of p`s. We will call this the "SumOfProbabilities."

g.

For each M set, we proceed sequentially through them where each M has a cumulative probability specified as the cumulative probability of the previous M in the (outcome-sorted) sequence, plus the probability off that M divided by the SumOfProbabilities.

h.

It is this final calculation where we seek that M whose final calculation is closest to 0.5. This is our median sorted outcome, and the values for N which comprise it are our typical, median-sorted outcome, our expected, finite-sequence set of outcomes.

4.2.4. Example - calculating the representative stream of outcomes from the median-sorted outcome

We demonstrate the steps from 4.2.2 now. We will assume a finite stream of 7 outcomes in length (since we are using a distribution of daily S&P500 returns for the first half of 2023, we will assume we are modelling an existential contest of seven upcoming days in the S&P500). Thus, N=7.

We will calculate 1,000 such sets of these 7 outcomes, and therefore, M=1,000, and we therefore proceed to create, for each M, seven random values (0..1) representing the inputs to the inverse CDF, and the output for that N, then deriving the probability by taking the output for that N as input to the CDF. This final probability number we will process into a p`.

For each set of 7 probability-outcome pairs, we take the product of the p`s and the sum of the outcomes.

Table 3. Sorted outcomes. Outcomes labelled as columns o1..o7, each row one of 1,000 of the M sets of N=7 outcomes.

Table 3. Sorted outcomes. Outcomes labelled as columns o1..o7, each row one of 1,000 of the M sets of N=7 outcomes.

We next calculate the Cumulative Sum of the Probabilities for each M. This is simply the previous M in the sorted sequence, plus the p` product of the N p`s in the outcome-p` pairs, divided by the sum of such p`s over all M ("SumOfProbs").

Lastly, as shown in Table 4, we find that M in the sorted set of M sets that corresponds to a Cumulative Sum of Probabilities value closest to 0.5. This is the median sorted outcome. It's output values (o1..o7) represent the "typical" expected outcomes over N trials.

Table 4. Sorted outcomes with median selected.

Table 4. Sorted outcomes with median selected.

In this instance of 1,000 samples of sequences of N outcomes per the parameters for our symmetrical GHD representation of S&P500 closing percentage changes, we find the sample set, the "typical" set of seven closing prices – what we would expect, where half the time we would see a better sequence, half the time a worse one, as:

-0.006670152, -0.00638097, -0.006615831, -0.005151115, 0.013180558, 0.01693477, 0.011466985

Of note, our expected worst-case outcome we have calculated as -0.00642. As we have an outcome that is at least as negative as this, we are not deluding ourselves on what might be the worst-case outcome for the "typical," i.e., the median outcome stream of 7 outputs.

This setoff outcomes represents the expected, the typical set of outcomes from data distributed as the first half of 2023 for daily percentage changes in the S&P500.

Further, we can expect for 7 trials in this distribution of outcomes to see the ∑ o outcome of 0.016764 as our non-asymptotic expectation. Of note, we are using a symmetrical distribution in this paper and example, and as mentioned earlier, the divergence between the actual, finite expectation and the classical, asymptotic expectation is a function of the skewness of the distribution, which is 0 in our example. Nevertheless, the concept remains, the procedure illuminates this for more perverse and skewed distributions.

5. Conclusions

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