You can find the spreadsheets for this video and some additional materials here: drive.google.com/drive/folders/1sP40IW0p0w5IETCgo464uhDFfdyR6rh7 Please consider supporting NEDL on Patreon: www.patreon.com/NEDLeducation
Thanks much for this. This level of math is way beyond my skill range. I appreciate you showing the actual formula line you used to implement the Johnsson SB. I am trying to model and insect from an article that provides parameters (gamma, delta, Xi, lambda) using Excle. Hard to find concrete info on that, and some of what I found was wrong. Thanks for sharing your expertise.
Thanks for the amazing videos Savva. Do you see any issue with using this as the underlying distribution in the Kernel Density Estimation? If so, why? If not, what is the correct way to implement it given it has so many parameters and differs from the kernel subtracting the current return from the entire range of returns?
Hi in another video you used the kolmogorov smirnov p-value to illustrate that the probability of the observed returns are generated by a normal distribution model was 0% I was calculating the kolmogorov smirnov statistic using the the empirical and the Johnson SU cumulative distributions and the resulting p-value is 49% My understanding is p-value in this context refers to the probability of the observed data to be generated by the selected distribution. So am I right in observing that although the line on the graph seems to describe a very good fit there is only 49% probability that the distribution we have found has generated the observations?
Hi, and glad to see you have applied the procedure to your data with the Kolmogorov-Smirnov test! Yes, this is the correct interpretation of the result. Generally, a p-value of 49% corresponds to very good fit.
Hi, and thanks for the question! The quantile function for the Johnson SU can be easily derived by inverting the CDF function, generating the following: x = lambda*sinh((z-score - gamma)/delta) + ksi, where z-score comes from a standard normal distribution for a quantile of interest.
I think I like this distribution as it seems to be better than some of the others you've covered. Is there a distribution that expands on the theory in the Johnson SU distribution? If not, from your experience, what other types of distributions are better at modeling stock returns than even the Johnson SU?
Hi Stephen and thanks for the question! Johnson SU indeed seems to be the best distribution to represent the US stock market as the return distribution is relatively fat-tailed. For other markets, error, Student, or asymmetric Laplace distributions can fit better than Johnson SU. Johnson SU is one of the many distributions that uses the normal cumulative distribution function, but tweaks it slightly (it applies the hyperbolic arcsine, while the error function raises its argument to varying powers). For distribution modelling that is completely different conceptually, I would suggest reading more on power law distributions and Paretian tails. Hope it helps!
@@NEDLeducation thank you sir, I've started looking at power law distributions and paretian distributions. I didn't know that the other markets followed a laplace distribution
Hi Aniket, and glad you liked the video! As for your question, the distribution is first proposed in Johnson (1949): www.jstor.org/stable/pdf/2332669.pdf. I have got a working paper on the applications of this and many other distributions (basically following all the videos in my "Modelling stock returns" series), check it out as well if you are interested: papers.ssrn.com/sol3/papers.cfm?abstract_id=3847351. Hope this helps!
Thanks for the phenomenal videos Savva. How can we calculate VaR using this distribution? Also, I got a Tukey Lambda value of -0.33, what distribution should I fit, and as a result how should I measure its VaR?
Hi Ankit, and glad you liked the videos! Tukey lambda of -0.33 suggests we are in the "no man's land" between Laplace and Cauchy, so error or Johnson SU distributions should be a good fit (-0.33 is too low to suspect it would be a slash distribution). VaR for parametric distributions can be calculated using the quantile function (so the inverse of the cumulative distribution function). For Johnson SU we have: prob = F(gamma + delta*asinh((x - ksi)/lambda)) So the quantile function would be: x = ksi + lambda*sinh(z(prob) - gamma)/delta) Here you can calculate VaR based on the z-stat of the probability you are interested in (for example, 1% for the regular 99% VaR) and the estimated parameters of Johnson SU (gamma, delta, ksi, and lambda). Hope it helps!
@@NEDLeducation Hey, I'm interested for the same problem. But I have a question to you know how to pass from a daily VAR to a yearly VAR with this distribution ? (Because i fitted the distribution on daily return so the calculation of the VAR is only for 1 day and not for a larger horizon)
@@ankitsom3855 Hey, I'm interested for the same problem. But I have a question to you know how to pass from a daily VAR to a yearly VAR with this distribution ? (Because i fitted the distribution on daily return so the calculation of the VAR is only for 1 day and not for a larger horizon)
Hi Savva! I´d like to ask you for a video where we change the BSM model to fit the preferred distribution we find fitter to the stock return. I´m trying to use the Johnson Su on BSM, but without much success, would be glad if you can help!
Hi Arindam, and thanks for the suggestion! Gamma distribution is not the best to model stock returns, but we will certainly make a video on it sometime in the future.
@@NEDLeducation Thanks. Actually I was trying to model quoted bid ask spreads (A-B). I find it difficult to model this stationary non-normal leptokurtic time series. A quick sense is, it is a difference series / of bid ask prices. I have seconds data of commodity futures.
Sounds like exciting stuff! What is the sample excess kurtosis of your data? If it is around 6, I suspect an exponential distribution can do a decent job (gamma distribution will also be able to pick this up, as exponential distribution is a special case of gamma). Overall, I find surprisingly little research on how bid-ask spreads are distributed, so it can be an interesting topic to investigate. Most of the papers I found use exponential or gamma. I was planning to record some content on semi-infinite distributions regardless, so let me know if you would like to share some of your data with me, and I can make a series of videos on distributions based on it.
@@NEDLeducation excess kurtosis around 7 using 1 year tick by tick data (commodities). Alpha Beta of Gamma dist were 2.86 and 0.859. KS test indicated non exponential dist. Dont know if i missed somthing.
You can find the spreadsheets for this video and some additional materials here: drive.google.com/drive/folders/1sP40IW0p0w5IETCgo464uhDFfdyR6rh7
Please consider supporting NEDL on Patreon: www.patreon.com/NEDLeducation
Thanks much for this. This level of math is way beyond my skill range. I appreciate you showing the actual formula line you used to implement the Johnsson SB. I am trying to model and insect from an article that provides parameters (gamma, delta, Xi, lambda) using Excle. Hard to find concrete info on that, and some of what I found was wrong. Thanks for sharing your expertise.
Thanks for the amazing videos Savva. Do you see any issue with using this as the underlying distribution in the Kernel Density Estimation? If so, why? If not, what is the correct way to implement it given it has so many parameters and differs from the kernel subtracting the current return from the entire range of returns?
Spectacular video!!!
Hi in another video you used the kolmogorov smirnov p-value to illustrate that the probability of the observed returns are generated by a normal distribution model was 0%
I was calculating the kolmogorov smirnov statistic using the the empirical and the Johnson SU cumulative distributions
and the resulting p-value is 49%
My understanding is p-value in this context refers to the probability of the observed data to be generated by the selected distribution.
So am I right in observing that although the line on the graph seems to describe a very good fit there is only 49% probability that the
distribution we have found has generated the observations?
Hi, and glad to see you have applied the procedure to your data with the Kolmogorov-Smirnov test! Yes, this is the correct interpretation of the result. Generally, a p-value of 49% corresponds to very good fit.
Hi I'm having a hard time finding the formula to get the inverse cdf of the Johnson's SU distribution ... can you help? thanks
Hi, and thanks for the question! The quantile function for the Johnson SU can be easily derived by inverting the CDF function, generating the following: x = lambda*sinh((z-score - gamma)/delta) + ksi, where z-score comes from a standard normal distribution for a quantile of interest.
I think I like this distribution as it seems to be better than some of the others you've covered. Is there a distribution that expands on the theory in the Johnson SU distribution? If not, from your experience, what other types of distributions are better at modeling stock returns than even the Johnson SU?
Hi Stephen and thanks for the question! Johnson SU indeed seems to be the best distribution to represent the US stock market as the return distribution is relatively fat-tailed. For other markets, error, Student, or asymmetric Laplace distributions can fit better than Johnson SU. Johnson SU is one of the many distributions that uses the normal cumulative distribution function, but tweaks it slightly (it applies the hyperbolic arcsine, while the error function raises its argument to varying powers). For distribution modelling that is completely different conceptually, I would suggest reading more on power law distributions and Paretian tails. Hope it helps!
@@NEDLeducation thank you sir, I've started looking at power law distributions and paretian distributions. I didn't know that the other markets followed a laplace distribution
Thanks for the amazing video. Just one request, can you please share any research paper on Johnson's SU distribution?
Hi Aniket, and glad you liked the video! As for your question, the distribution is first proposed in Johnson (1949): www.jstor.org/stable/pdf/2332669.pdf. I have got a working paper on the applications of this and many other distributions (basically following all the videos in my "Modelling stock returns" series), check it out as well if you are interested: papers.ssrn.com/sol3/papers.cfm?abstract_id=3847351. Hope this helps!
Thanks for the phenomenal videos Savva. How can we calculate VaR using this distribution? Also, I got a Tukey Lambda value of -0.33, what distribution should I fit, and as a result how should I measure its VaR?
Hi Ankit, and glad you liked the videos! Tukey lambda of -0.33 suggests we are in the "no man's land" between Laplace and Cauchy, so error or Johnson SU distributions should be a good fit (-0.33 is too low to suspect it would be a slash distribution). VaR for parametric distributions can be calculated using the quantile function (so the inverse of the cumulative distribution function). For Johnson SU we have:
prob = F(gamma + delta*asinh((x - ksi)/lambda))
So the quantile function would be:
x = ksi + lambda*sinh(z(prob) - gamma)/delta)
Here you can calculate VaR based on the z-stat of the probability you are interested in (for example, 1% for the regular 99% VaR) and the estimated parameters of Johnson SU (gamma, delta, ksi, and lambda).
Hope it helps!
@@NEDLeducation Thank you so very much. I understood it. And keep up the great work.
@@NEDLeducation
Hey, I'm interested for the same problem. But I have a question to you know how to pass from a daily VAR to a yearly VAR with this distribution ? (Because i fitted the distribution on daily return so the calculation of the VAR is only for 1 day and not for a larger horizon)
@@ankitsom3855 Hey, I'm interested for the same problem. But I have a question to you know how to pass from a daily VAR to a yearly VAR with this distribution ? (Because i fitted the distribution on daily return so the calculation of the VAR is only for 1 day and not for a larger horizon)
Hi Savva! I´d like to ask you for a video where we change the BSM model to fit the preferred distribution we find fitter to the stock return. I´m trying to use the Johnson Su on BSM, but without much success, would be glad if you can help!
Can you kindly put up a video on Gamma distribution?
Hi Arindam, and thanks for the suggestion! Gamma distribution is not the best to model stock returns, but we will certainly make a video on it sometime in the future.
@@NEDLeducation Thanks. Actually I was trying to model quoted bid ask spreads (A-B). I find it difficult to model this stationary non-normal leptokurtic time series. A quick sense is, it is a difference series / of bid ask prices. I have seconds data of commodity futures.
Sounds like exciting stuff! What is the sample excess kurtosis of your data? If it is around 6, I suspect an exponential distribution can do a decent job (gamma distribution will also be able to pick this up, as exponential distribution is a special case of gamma). Overall, I find surprisingly little research on how bid-ask spreads are distributed, so it can be an interesting topic to investigate. Most of the papers I found use exponential or gamma. I was planning to record some content on semi-infinite distributions regardless, so let me know if you would like to share some of your data with me, and I can make a series of videos on distributions based on it.
@@NEDLeducation excess kurtosis around 7 using 1 year tick by tick data (commodities). Alpha Beta of Gamma dist were 2.86 and 0.859. KS test indicated non exponential dist. Dont know if i missed somthing.
This of course in bid ask spread dat