Friday, June 3, 2011

Statistical Follies

In the previous article I showed how the price of an At the money option is nothing but .8* σ or in other words more generally the expected price movement of the stock. Now I would like to show how we can compute price of any option with much more flexibility i.e “Out of money” or “In the money” and then finally introduce interest rates. The idea behind this whole song and dance is not to attack any existing model etc. because anyways in my opinion they are nothing more than epitome of Platonicity but to provide with sufficient flexibility to price these derivatives under the real practical market conditions which are highly volatile more often than not.
But before that let’s first have a look at what is fundamentally wrong with the financial modeling. As always if the design of the whole building is flawed but the problem lies in the basics or the groundwork and that is what we are going to explore in this article. Again the following content would be a little technical but not a whole lot, since this article series are going to continue for sometime, I have decided to alternatively put out more pleasant read articles after each one these.

With the progression of time it seems and more and more aspects of our society are becoming centrally planned rather than aligned to efficiencies of the free markets.  With everything denoted and measured in government controlled currencies it’s a big sham to call global economies as free markets. Closing the heels is education and knowledge. What awards like Noble Prize etc. ensures is that rather than the forces of free market discovering what type of knowledge/research paper/theory is good for the world, this whole notion ends up in the hands of a few people. Can you believe it a “few good men” I am sure deciding what the coming generations are going to study and based upon that would eventually the future of this planet would get shaped.

Let me pick up the most cardinal concept in statistics – “standard deviation (SD)”. In my last article I showed how the value of an At the Money option is nothing but equal to its Mean Absolute Deviation (MD), we eventually replaced it by 0.8*SD to get it inline with the Black Scholes. Infact as I would show in this article using Mean Absolute Deviation is a far superior method of use in social sciences than SD. So the question:
Why on earth is SD in widespread fashion?
Answer: Happens when few people decide the fate of learning.

In the field of statistics a “statistic” is used for measurement if it has the smallest probable error as an estimate of the population parameter. If the population is perfectly normal then the standard deviation of their individual mean deviations is 14% higher than the standard deviations of their individual standard deviations. Please note the bold italicized word, that’s it… so basically SD is a perfect lab rat however in the field of real social sciences it’s an abject failure. The icing on the cake is that this useless musings are imparted to millions of naïve souls in schools and colleges since almost a century.

Himanshu Jain: “SD as a useless measure in the field of economic science”
Wrong Knowledge is tantamount to a Fukushima whereby even the coming generations pay a heavy price. With these noble thoughts let’s have a look at it’s disastrous consequences. Since I started with the option valuations and the Black Scholes hoax let’s take this topic for example.
Thinking intuitively the price of an option should have been equal to the expected value of the stock or in other words mean deviation. In perfectly normal world of lab rats we replaced it with .8*SD, however stock market returns are anything but normal. Ofcourse with Central Planning of Stock Market these days there can be an illusion of normality. My point if I carry on with MD as the measure of option (instead of replacing it with .8*SD) we find that:
C+P = E[S]  - Eq 1
Expanding E[S] we get:
S*∑k=0n pk*abs(S-xk), Now if I assume that the distribution is leptokurtic (fat tailed – high kurtosis)
pkl lims->0(probability of leptokurtic distribution close to mean) ≤  pkm lims->0(probability of mesokurtic distribution (no excess kurtosis) close to mean).
-          In other words the option prices of At the Money options should be slightly less than what is being computed by using standard volatility measure i.e. SD (σ)
pkl lims->(probability of leptokurtic distribution far away from mean) >>  pkm lims->(probability of mesokurtic distribution (no excess kurtosis) far away from mean).
-          This implies that the prices of out of money options is far lower as estimated by using σ
For someone keeping a Platonic mindset let me put the proof below:
Since σ is calculated by squaring the movements of the deviations from mean, as the deviations increase σ basically explodes when compared to MD. To prove:
σ1  = √[∑k=0n ak^2+S^2]/n, S>> ak
MD1 (d1) = [∑k=0n ak + S]/n, S>> ak
σ2  = √[∑k=0n+1 ak^2]/n
MD2 (d2) = [∑k=0n+1 ak]/n
Since S>> ak d1 -d2 ≈ S/n
And σ1 - σ2 ≈ S/√n , Since n is large in other words it shows that σ explodes parabolically when the deviations  are more meaningful and if SD explodes parabollicaly then think what would happen to kurtosis!!!
The implications of the use of the most fundamental concept of standard deviation in option pricing are enormous. It leads to a clear under pricing of Out of money options while making At the money options slightly expensive and it is this very pricing that we are going to explore soon………..

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