## Friday, March 30, 2012

### Without a reason?

Chart: M10, 5 days, 2012-03-30, stooq.com

M10 is a small boutique developing algorithmic trading strategies.

The company debuted on the NewConnect exchange in Warsaw on February 6th this year.

The offered price was 0.10 PLN, and it closed at 0.15 or +50% up on the first day of trading.

Since then, its price has risen to 0.38 or +380% in relation to the IPO price.

Just today, without apparent reason it went down by nearly -27% to 0.19.

As a result, the price is still +90% above its initial level.

But where it will go from here?

## Monday, March 26, 2012

### Diverging on the option prices

I'm quite surprised that two option pricers, ostensibly using identical closed end solving mechanisms, sometimes give different results.

You have at least two different packages offering option pricers based on the Black-Scholes formula in R:

I've generated some extremely wild option parameters for the the pricers:

> xSpot   = 601042.9378
> xStrike = 254417.3841
> xRf     = 0.14395813
> xDiv    = 0.11951086
> xTTM    = 6.22885299
> xSigma  = 1.04225573

(option type = put)

and got the following results:

> GBSOption(TypeFlag="p",S=xSpot,X=xStrike,Time=xTTM,r=xDiv,b=xRf,sigma=xSigma)@price
[1] 71863.77
>
> EuropeanOption(type="put",underlying=xSpot,strike=xStrike,dividendYield=xDiv,riskFreeRate=xRf,maturity=xTTM,volatility=xSigma)\$value
[1] 71998.46

One may say that the difference of 134.69 is not much for the option valued some 72.000 (134.69 / 71863.77 = 1.8742e-3). It's probably true.

But what's even more amazing is that both these results are different from what you can get from the "pure" Black-Scholes formula:

> BSPut.dividend(S=xSpot,K=xStrike,rf=xRf,q=xDiv,TTM=xTTM,sigma=xSigma)
[1] 71993.11

So, which value is correct?

### Simple multicriteria optimization with Pareto layers

Chart: Three layers of Pareto sets
The red elements dominate over the green, the green over the blue, the blue over the unlayered rest.

Pareto set contains elements dominating over others.

From the optimization perspective, they contain elements that should be considered equal.

Often it is possible to establish layers of Pareto sets subordinate to the higher order sets.

Pareto sets and layers are interesting way of selecting elements characterized by multiple criteria.

One of the other methods is convolution.

There is a number of R packages for creating and analyzing  the Pareto frontier:

## Tuesday, March 20, 2012

### Comparing probability distributions

Chart: Probability distribution functions for daily returns of S&P500, WIG20, NatGas, Gold and EURUSD
data from 2011-01-10 do 2012-03-19

We are going back to the subject of similarities of return characteristics of various assets.

I was looking for a way to compare various probability distributions and decide which ones are the most similar to each other.

I've found three methods that can be of help here:

Since results given by these methods return vary depending on the type of distributions we compare, deciding which of them and how to use requires further research.

## Sunday, March 18, 2012

### What is the probability that the market will raise 5% over 25 sessions?

Chart: Probability of at least a +5% change over 25 sessions for WIG20
sampled on the latest 250-1250 sessions

A couple of days ago I wrote a post about differences in return characteristics of various assets.

Now I'd like to take a look on the markets from a little different perspective.

I was asking myself a pretty straightforward question:

What is the probability that the market will raise at least x% over the next n sessions?

The so called "modern portfolio theory" takes expected return of assets constituting the portfolio as one of its three critical parameters (the others are asset correlations and their volatility).

Hence, the people constructing portfolios based on this theory must somehow guess what these parameters will look like in the future.

The problem is, all of these parameters are pretty hard to forecast...

Hence, I decided to explore the first of these parameters - the expected return - and asked the above mentioned question.

On the chart above you can see the probability of a 5% growth of the WIG20 index of the Polish stock exchange over 25 sessions (a month), calculated on the basis of the 25-days samples drawn from the previous 250 to 1250 sessions (1 year - 5 years).

As you can see, the probability fluctuates between 14.38% and 30.46%.

OK, and how it looks from the opposite side, i.e. what is the probability of a 5% decline?

The probability range is pretty similar: 14.24% - 30.68%.

But the shape of the graph is completely different, especially on the left side of the chart:

Chart: Probability of at least a -5% change over 25 sessions for WIG20
sampled on the latest 250-1250 sessions

Even if the overall probability of a +/- x% change seems similar, the specific realization can vary greatly depending on the particular moment in time.

## Friday, March 9, 2012

### The spring seems stretched to the maximum on S&P500 vs WIG20

Chart: WIG20 [in PLN] vs S&P500 since 2005, data: stooq.com

Chart: WIG20 [in USD] vs S&P500 since 2005, data: stooq.com

A couple of days ago I mentioned a visible decoupling of WIG20 and S&P500 indexes.

I noted that most probably it would lead to a contraction of the distance between the indexes by a larger correction in S&P500 than should be expected in the WIG20 case.

Here we have another reinforcement of this hypothesis.

The chart above shows the co-integrated trajectories of the indexes as well as their spread since 2005.

As you can see we are clearly in the lower range of the possible spread values, and at the two standards deviations away from the mean spread. This is quite an unusual situation.

Chart: WIG20-S&P500 spread distribution since 2005, data: stooq.com

The history suggests that the spread can widen by some additional 10-15 pct. points in really extreme situations before it starts to contract, although such development seems currently unlikely.

## Wednesday, March 7, 2012

### Characterizing financial assets by their Power Law alpha exponents

Chart: Distributions of the daily returns and their best fit Power Law alpha exponent for EURUSD, S&P500 and natural gas

The increasing number of financial markets participants start to understand that applying models based on normal distribution can be disastrous to their investment performance.

Pareto distribution based on the Power Law may be a better alternative.

Still, various financial assets may follow Pareto distributions with different parameters, especially different  alpha exponents.

Above you can see a chart of probability densities of daily returns for three assets: EURUSD, S&P500 index and natural gas, together with the alpha exponents fitted to them.

The alpha exponent describes the concentration and fatness of the tails of the distribution, which can be seen here: WolframAlpha - (1/x)^2.5062, (1/x)^2.078, (1/x)^1.6326, from 0 to 50

[ R code ]

## Saturday, March 3, 2012

### Is Apple share price reaching inflection point?

George Soros described the Boom-Bust Model in his book "The New Paradigm for Financial Markets" (pg. 65-66) published in 2008:
Chart: The Boom-Bust Model

The model consists of seven phases:
The Boom-Bust Model
In the initial stage (1) the trend is not yet recognized
Then comes the period of acceleration (2), when the trend is recognized and reinforced by the prevailing bias. That is when the process approaches the far-from-equilibrium territory.
A period of testing (3) may intervene when prices suffer a setback.
If the bias and trend survive the testing, both emerge stronger than ever, and far-from-equilibrium conditions, in which the normal rules no longer apply, become firmly established (4).
Eventually there comes a moment of truth (5), when reality can no longer sustain the exagerated expectations, followed by a twilight period (6), when people continue to play the game although they no longer believe in it.
Eventually a crossover point (7) is reached, when the trend turns down and the bias is reversed, which leads to catastrophic downward acceleration (8), commonly known as the crash.
It is quite possible that we are currently experiencing the late stage of phase four of Soros' model, i.e. severe detachment of Apple share price from the equilibrium conditions, which culminates in the peak of the cycle, followed by accelerating decline:

It seems that the growing trend in Apple shares was successfully tested at the turn of 2008 and 2009, and accelerated significantly according to the model, to enter the current far-from-equilibrium territory.

As the boom bust model suggests, the inflection point is reached when earnings per share are still growing. On January 24th, Apple presented stupendous Q1 2012 financial results. The company also issued extremely positive Q2 2012 earning guidance. Just recently, Apple market capitalization has exceeded \$500 billion. On March 7th, Apple is going to present its new tablet - iPad 3 / iPad HD, and possibly some other products.

There are no limits to Apple growth, right?
I'm currently betting the opposite.

I'm fully aware that the Soros' model may be wrong or inadequate for the Apple's situation. I also realize that prices in the far-from-equilibrium area do not follow any strict rules. Hence, there is a high risk in going against the prevailing bias.

The Apple share prices can still go significantly higher. As John Maynard Keynes said "markets can stay irrational longer than you can stay solvent". But they simply cannot go up forever.

## Thursday, March 1, 2012

### Eurogeddon week 1

The Eurozone collapse scenario fund Eurogeddon started just a week ago.

Today we get the first real weekly performance reading for it.

And it doesn't look very good.

Over the recent week the fund went down by -4.01%.

Don't get confused by the reported overall result of +29.30%. It is just a residue resulting from the creation of the Eurogeddon on the basis of the previous long biased fund called Universa-plus, which was +34.70% at the moment it was transformed into Eurogeddon.

A single data point is definitely too little to judge the project, but it gives a hint of variability you may expect from the fund.

BTW: I'm just reading a special report from Phoenix Capital Research about the potential implosion of the US debt market. Paradoxically its quite important in the context of the Eurogeddon strategy. The fund plans to safeguard its assets by investing them in the US treasuries. But while both Mr. Rybiński, who is the initiator of Eurogeddon, and Phoenix see the major market collapse, Phoenix's scenario assumes  a huge jump in the US interest rates that would automatically translate into corresponding fall of the US bonds prices...

UPDATE 2012-03-02: my comment on Mr. Rybiński's blog (in Polish)