Sunday, September 25, 2011

Options - The Pricing Ways Continued.......

Today let’s take a break from economic issues and get back to option pricing. In continuation with my last piece on option pricing (Link) , wherein we explored a much easier way to price options and what was even better was that it allowed for the flexibility to have non-normal probability distribution in pricing. Today I would like to write about how we can develop the concept further to include in the money and out of money options. Well if one understands the basic premise of my last article on this topic it’s fairly simple.

As I wrote before, basically if I buy an “At the money” call and put, the price that I should pay is the expected movement of the stock in that period. So in essence
C+P = E[S] – Eq 1

Now for any option that is “In the money” for the buyer to make any money and for the seller to loose money would be when the Stock crosses the Strike price of the option, so in essence what we really care about are the probabilities wherein the Stock Price crosses the Strike Price of the option.
So the price of the option would be:

C = (S-K) +E[ x=kn px*S]

The above formula gives the flexibility to the reader to assume different probability distributions and use them in the option pricing.

Please Note that If I try to simplify the Black Scholes model i.e. take the assumption of Normal probability distribution I would be only be approximating the pricing, but nevertheless let me put it down anyways.

The call price (assuming interest rates as negligible) c = SN(d1) – KN(d2), the put formula is also accordingly KN(-d2) – SN(-d1),

c= SN(ln(S/K) + σ/2*√t) – KN(ln(S/K) - σ/2*√t), Since We are taking volatility for the entire duration of the option so I am going to replace σ*√t with σ .

Also I would express S in terms of K and σ as that’s what we are concerned about, i.e.
S = K+ K*σ*Ɵ

Putting these values into the Black Scholes formula and expanding the log portion using taylor equation the log portion would be reduced to Ɵ – Ɵ^2* σ/2

Finally adding both these terms and using the Normal expansion and exponential expansion by taylor series
N(x) =[∫-∞0exp^- (x)^2  + ∫0 σ/2exp^- (x)^2]/sqrt(2 π)

The first part is equal to 0.5, to integrate the second part of the expression let me expand it using the Taylor series. The Taylor series expansion of the exponential term is
ex = 1 + x + x^2/2! + x^3/3! ……

Replacing x by Ɵ – Ɵ^2* σ/2+σ/2 and then integrating we get
N(Ɵ – Ɵ^2* σ/2+σ/2) = 0.5+ [0 σ/2 ∑0n (-1) n (Ɵ – Ɵ^2* σ/2+σ/2 )2n+1/2nn!(2n+1)] /sqrt(2 π)

As SD is small number (in decimals) we can ignore the higher power of σ,

Please note that if σ is high the whole notion of Normality falls apart and as does the Black Scholes model.
So expanding this expression and putting it in Eq 1 we get

C = (2* Ɵ - Ɵ^2* σ/2 + σ) * √(1/2π)
Or more simply C = (2* Ɵ + σ) * √(1/2π)

Again this is simplying Black Scholes using mathematical tools what’s more important to understand is that the genesis of any option pricing i.e. using any probability distribution is this formula below:

C = (S-K) +E[ x=kn px*S]
Till Next Time………….

Monday, September 12, 2011

Failure is the option

Writing after a long time indeed, got stuck in some daily chores and also reduced travel in the last couple of months gave me less time to think. Anyways here it goes now; also I did put up this article at Business Insider yesterday.

The high unemployment rate that has engulfed the major economies of the world is not just serious but also bewildering and these moments of pain and awe among the population is when governments try to intefere and curb the freedom of the society, the case in point being the new excess stimulus plan of Obama. After all wasn't capitalism supposed to allocate the resources efficiently then why is that we have 16% unemployment in US (U6), over 30% youth unemployment in Spain, Greece, Italy. I mean the problem of this world is fall output and so isn't it ironic that with so much unemployment around we are taking about this problem. A garbage reason that I always listen on television by many analysts and CEO's is that of lot of policy uncertainty which is leading to this bemusing irony. Well this may be playing a very small part in e scheme of things as I find it hard to believe that policy makers of a few decades back were far idealistic, efficient and superior than what the are now. The real reason for this catharsis is that the world we are living in is no longer free markets or the economic order is no longer capitalist but "crony capitalist"

Recessions, fall in stock prices and asset prices in general and failure of big and mighty are features of capitalism wherein it reorganises resources and efficiently allocates it to new sectors, people and industries which then take it forward thus creating jobs and continuing the boom. Unfortunately now even a 10-15% stock market fall is followed up by printing of money by the Central Bank of US, companies like GE and GM are saved under the garb of avoid huge job losses and of course the banks and the financial institutions (the less said of those the better) these institutions that add very little value in this society but for continuing the ponzi currency scheme. What's worse is that with every time they being saved under the farce too big to fail these banks have raked in too much of the economic resources.

What's required is not another failed government induced spending plan but instead the government getting out of the way of this economic reset, let the markets crash if ey have to and the companies and banks go bankrupt... There will be brief period of chaos followed by a new reinvigorated order as new capable people take over those deflated resources and as the capital froth from those assets move to new places thus creating opportunities from our young entrepreneurs and providing jobs to millions. Socialism and today's crony capitalism suffer from the same problem "misallocation of resources" the only difference is that while socialism gives less to the competent, crony capitalism gives almost everything to the incompetent but not letting them fail.

In the next article (sometime this month only) probably we shall talk about how this increase in government debt ends up doing no good but preserving the existing world order of ponzi banks and its monetary system.