In that case what would be the expression for pdf of log(S(t+1)/S(t)) for any t in terms of VGSSD characteristic function?

I gave that earlier: (**) on Dec 7 gives the c.f., and right below that is the expression for the pdf (requires a numerical integration)

Statistics: Posted by Alan — Yesterday, 6:00 pm

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I guess the right characteristic function for VGSSD would be:

CF_VG = @(u)(1./(1-1i.*u.*theta.*nu.*

Statistics: Posted by czar3k — Yesterday, 3:55 pm

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Your properties 1), 2), 3) are false for VGSSD because the daily return distribution changes each day; it is not a stationary increment process. Property 5) is true.

Re property 4), the distribution of the daily increments (daily returns) changes just enough each day to allow what I will call (4)*

(4)* distribution of aggregated returns (sum of daily returns from T=0 to N = any value) have the same skewness & kurtosis as the distribution

of the day 1 returns (the one from T=0 to N=1).

You might want to take a look at Ch, 14 in Cont & Tankov which discusses "additive processes". The VGSSD process is an additive process. For all additive processes, 1), 2), and 3) are generally false; 5) is true. However, as far as (4)*, while it holds for VGSSD, for other additive models, I think you probably have to do a model-by-model calculation from the characteristic function.

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p.s. Let me repeat my previous p.s. about the characteristic function for these VGSSD increments:

(*) [$]\phi_{s,t}(z) \equiv E[e^{i z (X(t) - X(s))}] = \frac{\phi_{X(t)}(z)}{\phi_{X(s)}(z)}, \quad\quad (t > s)[$],

where the numerator and denominator you have from the linked papers.

So, if you want to see how the various moments of the VGSSD increment distribution are supposed to change from day to day, it is just a tedious exercise in differentiation of the known characteristic function (*). (A good CAS like Mathematica might help, but it is certainly doable by hand). Again from the papers, you already have the moments for the first daily draw: s=0 to t=1. But, I suggest you work out what are the first 4 moments for the daily draw from an arbitrary s to t = s+1. Then, if you are able to develop a good approximate simulation, you can test if your moments change on a daily basis appropriately. Also, I will repeat my caution that the density function for the daily increments has a Dirac mass at zero (with a mass that changes daily), so that is another property I would want to see reproduced in any decent simulator.

Statistics: Posted by Alan — Yesterday, 1:42 am

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1) distributions of daily returns from T(0)->T(1), T(1)->T(2), T(end-1)->T(end) are all skewed with the same skewness

2) distributions of daily returns from T(0)->T(1), T(1)->T(2), T(end-1)->T(end) are all fat tailed with equal kurtosis

3) distributions of daily returns from T(0)->T(1), T(1)->T(2), T(end-1)->T(end) have the same mean and variance

4) distribution of aggregated returns (sum of daily returns from T=0 to N = any value), say early that is from T(0)->T365, have the same skewness & kurtosis as daily returns

5) daily returns are independent

Unless I'm wrong...

Statistics: Posted by czar3k — December 14th, 2017, 10:33 pm

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On the other hand, if you are trying to approximately simulate the VGSSD process, then the kurtosis shouldn't fall if the approximation is any good.

Statistics: Posted by Alan — December 13th, 2017, 1:47 am

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As I said I'm in a position to generate yearly returns / prices that are VG distributed I started with doing so. I ended with a vector of 100k final asset prices. Price at time 0 was 1 in each case.

The code is:

Next step was a little drift from the purely mathematical approach.

I simulated 100k paths of daily returns/prices from the same VG distribution (just scaled drift and sigma accordingly), each path starting

Next I looked what is the difference between the final price for each of those 100k x 365 days paths and 1 and "corrected" each of 365 prices in those paths to make sure that the final price in all 100k paths is = 1.

Finally I reversed those paths to have S1 = S365, S2 = S364 etc.

I will try with applying correction not to the prices but the returns themselves but that's a different story/

The code was:

After all I had 100k paths of 365 daily returns which when summed up along the math were VG distributed.

I'm aware that it's not the way I should proceed.

Anyway I checked the properties of the returns for 1st, 2nd,... 365day, they were almost identically distributed (with falling kurtosis as you go from day 1 to day 365, rising std deviation and stable mean). Overall the resuls seem to be quite satisfactory.

As a final check I drew an implied volatility plot which looks good.

As with real IV surfaces vol. falls as time to maturity gets longer and smile effect is stronger for short maturities. For reference I added a flat surface at 0.2 which is the sigma value of VG dist.

Below are some links to the plots so you can have a better picture of what was actually done.

daily paths for T = 365, no correction, yearly returns are normal

daily paths for T = 365 with correction, yearly returns are VG distributed

histogram of D1-D2 returns vs D364-D365 returns, diff. is slight

IV surface

I will appreciate your comments.

Does it make any sense (I'm sure from the mathematical point of view it's not the right attitude but if it is at least to some extend reasonable with small error margin I'll be satisfied)?

As the daily distribution gradually changes my next idea is to randomly permute daily returns for each path as the order of summing does not affect the sum.

Also I will try to adjust returns instead of prices and see what;s the difference.

I'll post when I'm done it will take a while.

Regards

Thanks for understaning & sorry for my math defficiencies & lack of rigour.

I'm really grateful for your thourough investigation and help, your post are inspiring even though I miss some points and I'm not able to check your suggestions.

Statistics: Posted by czar3k — December 12th, 2017, 11:03 pm

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https://people.maths.ox.ac.uk/trefethen/4all.pdf

Statistics: Posted by Cuchulainn — December 11th, 2017, 8:00 am

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I have a very general question (for the beginning, by an absolute beginner): what are the most common proofs of convergence of numerical algorithms? (I've encountered proofs based on energy minimisation so far.)

It depends on the problem (my very general answer ::). But in many cases you want to solve a problem [$]F(x) = 0[$] in some Banach space by replacing it by a computable discretized problem [$]G(x^h,h) = 0 [$] where [$]h[$] is a parameter. Then all convergence proofs centre around computing the norm of [$]x^h - x[$] as a function of [$]h.[$]

In all cases a problem in uncountable space must be posed in countable space and then projected to a finite-dimensional problem. You don't need to know the exact solution (and in most cases it's not feasible).

The methods for proving convergence is a part of numerical analysis:

1. Taylor expansions

2. Polynomial and rational function approximation.

3. Divided differences to approximate derivatieves.

4. etc. etc.

5. Iterative processes that depend on an integer parameter [$]n[$], e.g. Newton.

Classic intros are Dahlquist/Bjorck 1974 and esp. Conte/de Boor 1980. A good 101 case is numerical quadrature or [$]e^A[$].

Many numerical analysts in academia spend time establishing convegence results for numerical processes.

http://web.mit.edu/10.001/Web/Course_No ... ration.pdf

http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf

Do you have a specific problem in mind?

Statistics: Posted by Cuchulainn — December 11th, 2017, 7:36 am

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Do you/can you know the true value?

In maximum likelihood you can sometimes compute analytical derivatives of the likelihood function given your data, find the value where you derivatives is zero, proof that you have all the zeros, and then you can say the maximum must be at one of these positions where the derivatives is zero.

Statistics: Posted by outrun — December 11th, 2017, 6:55 am

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If you look at the characteristic function for the VGSSD increments from s to t, you see that it is the ratio of two VGSSD characteristic functions, one using s and one using t. That ratio tends to a constant as [$]z \rightarrow \infty[$]. That is a symptom that the distribution of the returns from s to t has a Dirac mass at 0 (assuming [$]r = \omega = 0[$] for simplicity). So, if x is the log-return, the returns density [$]p_{s,t}(x)[$] decomposes into [$]A \,\delta(x)[$] plus a regular part. At a minimum, you need to regularize the characteristic function that is being inverted by subtracting [$]\phi(\infty)[$]. This will yield a more suitable function for the integration with some decay as [$]z \rightarrow \infty[$].

You can play around with the special case [$]\theta = 0, \gamma = 1/2, \nu = 1[$], where the inversion is analytically simple and confirm, at least in that case, the Dirac mass. In that case, [$]p_{reg,s,t}(x) = a \, e^{- b|x|}[$], where [$](a,b)[$] are easily related to the model parameters and s and t.

Statistics: Posted by Alan — December 10th, 2017, 7:57 pm

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[$]\phi_{X(t+1) - X(t)}(u)[$] is the ratio of the expression on the r.h.s of (15) of O'Sullivan at t+1 and t.

Then, for the risk-neutral stock price process, which has a compensator drift, you need something like

(**) [$]f_{t,t+1}(u) \equiv \phi_{\log S(t+1) - \log S(t)}(u) = e^{i u r - i u[\omega(t+1) - \omega(t)]} \phi_{X(t+1) - X(t)}(u) [$] ,

where [$]\omega(t) = \log \phi_{X(t)}(u=-i)[$].

The pdf of the daily log-return [$]x = \log S(t+1) - \log S(t)[$] is given by the (numerical) Fourier inversion

[$]p_{t,t+1}(x) = \int e^{-i u x} f_{t,t+1}(u) \frac{du}{2 \pi}[$].

Finally, drawing from this pdf can be done by standard "cdf inversion sampling", This requires the cumulative distribution function (cdf), but that too can be obtained by a similar Fourier inversion. I don't have the formulas at hand for you, but you can google for something like "inverting a characteristic function to get a cdf". In this case the c.f. is (**).

Bottom line for the brute force method: set up a function in matlab that produces, via Fourier inversion, the cdf for day t to t+1 returns and sample from it using the cdf inversion sampling method. It won't be fast but it should work and you can then see if the skewness and kurtosis behave as you expect.

p.s. The word "inversion" is used above with two different meanings: "Fourier inversion" and "cdf inversion sampling". Also, (**) needs to be checked carefully. It looks OK to me now after a half-dozen edits, but no guarantees. It's a daily return version of (3.2) in Carr et al's "Self-Decomposability and Option Pricing"

Statistics: Posted by Alan — December 7th, 2017, 11:24 pm

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If the characterictic function of a return of an asset at time t follows VGSSD dist and is expressed as (1-sqrt(-1) *sqrt(t)*nu*theta+0.5*t*nu*sigma^2)^(1/nu) then the characteristic function of dist. of return between t and t+1 (daily return) will be expressed as the ratio of the two c.f., that is:

((1-sqrt(-1) *sqrt(t+1)*nu*theta+0.5*(t+1)*nu*sigma^2)/(1-sqrt(-1) *sqrt(t)*nu*theta+0.5*t*nu*sigma^2))^(1/nu)

I'm basing my intuition on the example with Gamma dist. from wikipedia on c.f.

If that's correct what transformation do I need to obtain pdf of return from t to t+1?

I'm feeling that we are reaching the boundary of my perception abilities

Statistics: Posted by czar3k — December 7th, 2017, 10:50 pm

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Thanks for your time on investigation!

I think (just a feeling) that it is impossible that increments (daily returns) are at the same time: independent of previous increments , identically distributed and have a let's call it "kurtosis & skewness" preservation property - central limit theorem forces it. Unless it is a brownian motion as you mentioned (as it has no excess kurtosis).

You're welcome.

Yes, i.i.d increments in continuous-time essentially define Levy process. The paper you posted cites another (Konikov & Madan 2002) proving that *all* Levy processes have c t^(-1/2) and d t^(-1) skewness and excess kurtosis decay. The constants (c,d) are model-dependent and can vanish. Likely true for i.i.d discrete-time as well because (while I haven't actually looked at the cite), the two decay properties likely follow from a short computation with the general form: [$]E[e^{i z X(t)}]= e^{-t \psi(z)}[$].

Of course, the point is that these SSD processes are *not* Levy processes and so need not follow these decay rules.

Statistics: Posted by Alan — December 7th, 2017, 10:19 pm

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