Some thoughts on option

Just checked: the underlying has almost picked up but the call option price remained less than halved. I hope I learned a hard lesson over "time value" of options. Some of the basic things I want to keep in mind: never hold an option if not for hedging purposes, never buy an option when volatility is perceived high, and never just bet with gut feeling.

Bankroll management

Read Kelly Criterion.

( thanks to Junyi for valuable discussion )

Betting

So I took a hard beat yesterday because I bet on the wrong side regarding Fed decision (with a long position in index call). In the mean time, I started to think of a very basic yet interesting problem: let us say participants in the market are betting on a single event of two outcomes, and they know where one asset price would be for either outcome of this event. What would the asset price be right before the event?

For simplicity, let us say there will be two outcomes: H & L. For H, the price of the asset would go up to H, for L it goes to L. Assume betters can both long & short their positions. Assume they are risk-neutral. Confidence over H outcome is denoted by x, and the pdf of betters (weighed with their capitals, of course) with x confidence is p(x). We have \Integrate p(x) dx = 1 as the constraint. The market price of the asset P right before the event, is determined by the following equation:

\Integrate [0,x₀] p(x) = 1/2,

in which, x₀=(P-L)/(H-L), because whoever has a confidence over x₀ would long the asset while others short it, and long positions should be equal to short ones.

PS: Thinking again, it might not be this simple. The final price could in some degree depend on the path - I shall do a simulation some day. It is an easy job. Keep bid-ask spread in mind.

Risk-Neutral Measure

Okay, I admit that I never carefully thought this through - basically, why should a derivative's price process be a martingale under the so called risk neutral measure?

Here is the answer: at the first place: there isn't any probability issue involved in the pricing of a derivative! The risk neutral measure is an artificial measure assigned to different outcomes of all the underlying assets, such that the discounted of them are martingales. Because these individual martingales, in their differential form, can be expressed as X_i(t)*dW_i(t), it follows that any martingale process can be expressed as a linear combination of these individual underlying assets. In other words - this derivative can be perfectly hedged.

So what happened when there are more than one risk neutral measure? It means that there are more "fundamentals" than actual assets available. The conclusion is that a position in the derivative can not be hedged by the existing underlying. However the market still does not admit any arbitrage (proof is easy).

What happened when there is no risk neutral measure? The market admits arbitrage because we can find a linear combination of the underlying to form a portfolio that drives non-negative returns in all scenario while getting positive returns for some. The proof is not included here. Intuitively a lack of risk-neutral measure would mean some assets are having an unfair price for their risks - fully expressible by that of other assets and can be properly hedged, therefore profits can be received from a proper portfolio.

PRACTICE: Solve Hull-White interest rate model by classical PDE ways (Green's function maybe).

Localization and/versus Chaos?

Talked to TW today at RZ's defense reception and got a bit inspired. I had this idea but it was rather vague, TW liked it a lot and helped me walk through some details. Our caged particle has a reasonable high zero point energy (of course check if it is sufficient to liberate the particle) yet still the ground state gets effectively localized. I want to relate that to the deterministic chaos in the classical limit - inject K.E. and perturb the potential, what would the phase space density look like? BTW I do not think Anderson has anything to do with it despite SKG's obsession with the idea.

Edit: Essentially it is a 3d problem (2d peripheral + 1d radial). Be wary of the limited exchange between them.

Benford's Law

Let's say you are given a batch of real-world data. Tell me what the distribution of the first digit is like. Evenly distributed between 1 and 9? Wrong! Benford's Law states that 30% of the cases the leading digit is 1.

Before Sunrise / Before Sunset

So I was watching the film Before Sunrise on this Friday night and at the same time text messaging with D. I told her I was watching this film. D SMS-ed: "... the two have wonderfully interesting conversation". I replied: "It's funny to watch a movie of just talking. Wouldn't it make more sense to write a novella?" And then I found it strikingly coincident with the plot in the sequel, which I watched right after and particularly didn't like. In fact I felt very bad after watching it, not because I felt sorry for the characters, well actually I thought they deserved it, but because the way that everything was meant. Good story(-ies), but so sad because every detail in it was so real and inevitable.

Randy Pausch: Achieving dreams

Huyue introduced to all her guests this speech which she considered her best b-day gift received.

I think you should watch it when you think you can spare one hour of your life and you can spare it to let someone touch you.

http://video.google.com/videoplay?docid=362421849901825950&hl=en

Sometimes I came to think why we would be better off with traditions, or even religions.

Because they teach you the ways that you might be struggling for decades to learn, and I assure you that (right) principles unlike knowledge, serve you better if they are learned before you can think critically.

I know this' geeky

The Situation:
The fan of my T60 did not work well. CPU temperature was as high as 88 Celsius at full load. System froze at times when it passed 90.

The Solution(s):

1. Called IBM/Lenovo Tech support and requested fan replacing. Figured I wouldn't be able to use the laptop for one week. No-go.


2. Vacuum from the exhausts (there are two on Thinkpad, be sure to tape one while vacuuming the other). Fan now runs freely. Temperature is anywhere from 76 to 80 degrees Celsius. I might need to open the case, cleaning or replacing the fan module to fix the problem for real. Will need to call IBM/Lenovo later for parts.
   


3. Undervolt the CPU by Windows-NHC/Linux-PHC tools. Further 10-12 Celsius down! Now at full load CPU temperature is only 57 degrees Celsius (idle at 48 Celsius). For starters (T2400 Core Duo):
   
   Linux: # echo '26 22 19' > /sys/devices/system/cpu/cpu0/cpufreq/phc_vids
   Windows: 11x: 1.0375 Volt; 6x: 0.9500 Volt.

If you have a laptop with Pentium-M / Core Duo / Core 2 Duo, you can use the same methods although exact specifications might (almost surely) vary.

Edit: I laughed while I wrote almost surely (a.s.) ... It was almost surely a misuse of the term!

IN A MUCH QUOTED PASSAGE in his inaugural address, President Kennedy said, “Ask not what your country can do for you—ask what you can do for your country.” It is a striking sign of the temper of our times that the controversy about this passage centered on its origin and not on its content. Neither half of the statement expresses a relation between the citizen and his government that is worthy of the ideals of free men in a free society. The paternalistic “what your country can do for you” implies that the government is the patron, the citizen the ward, a view that is at odds with the free man’s belief in his own responsibility for his own destiny. The organismic, “what you can do for your country” implies that the government is the master or the deity, the citizen, the servant or the votary. To the free man, the country is the collection of individuals who compose it, not something over and above them. He is proud of a common heritage and loyal to common traditions. But he regards government as a means, an instrumentality, neither a grantor of favors and gifts, nor a master or god to be blindly worshipped and served. He recognizes no national purpose except as it is the consensus of the purposes for which the citizens severally strive.

Capitalism and Freedom, Milton Friedman

How beautifully spoken. I have always disliked JFK's quote, strongly reminding me of the well-you-know-what-I-am-referring-to, but I can never elaborate like this as regards why it is so fundamentally wrong. I also admired Friedman for his pure audacity to say this at 1960s.

Idea

Had this idea when I was thinking how many people I can invite to my home party.

N*N grid, x of them are occupied by coins, asking the chance P that all of them are able to reach any unoccupied grid without moving others (this is equal to say any two of them is “connected”). P=P(x). Obviously P(x>0.5)=0.

Or alternatively, Avg(free space of each coin)/((1-x)N^2) as a function of x.

Applications include granular physics, glass transition etc. Any way, now, get back to paper drafting.

Edit Saturday: Party went nice. I also figured the maximal number of people allowed in my place is twenty... Obviously I did overlook the heat dissipation problem...

I made it

I finished the half-marathon, even though I had knee injury. By my watch it was 2:05. It was such a great feeling.

Reputation, Cooperation, Evolution, Repeated Games, Optimal Play etc.

I shall find time to write something on this topic. I might also take time to investigate into the “the meek/submissive mindset” of people (I am right now thinking of it as a form of risk-aversion).

For starters, read this article:

In Games, an Insight Into the Rules of Evolution

Self-control depletion

After receiving a number of (friendly) criticisms on my unnecessary self-awareness, I sought for help. I borrowed a book from the library on this subject. I skimmed it in one hour and found most of the text dry, boring and rather academic (read useless), except for one chapter on self-control, another favorite subject of mine precisely because I have problems with it. There is a notion called “ego depletion”, whose essense is that you just have that much of psychological power to force yourself to do things that are right but unpleasant, and although this power is replenishable usually on a daily basis, its depletion simply means you are out of power on yourself, like it or not.

Honestly this would not be the first time I saw this phrase, however a systematic examination of the subject was still informative and enlightening. I promptly linked this to my lack of motivation to study after I completed a long run that was beyond my comfort zone (for example, I slacked off for almost two days after I ran from U of C to the river on Wednesday under the scorching sun, something I never did before). The will power was used up as I forcing myself to keep running. It is then not surprising that excessive self-training on will power would always backfire, in the same fashion that you only get injuries if you overtrain your muscles. To rest your nerves and be easy with yourself when you have to, is as important as to push yourself when needed. Do NOT waste on your will power - spend them economically. Additionally, just as you do not get your iron pecs and abs overnight, you will not get your formidable will power in short term. Everything should follow a schedule. A firm success should override any self-pride. When your reserve is low, try ways that require less consciousness involvement, such as staying away from the pleasure stimuli, which I in the past would always resist doing, because I regarded it as an acknowledgement of lack of self-control, which hurts my ego. For instance, I would deliberately put a shortcut to my favorite game on the desktop when I was trying to stop playing it. Sounds silly isn't it. That was the way I perceived it. As the book points out, pride often leads us the wrong way.

Lastly: focusing on such matters once again reveals my excessive interest in “self”. Hopefully it is not restricted to “myself” this time.

The Yen Carry Trade

So the other day it crossed me the idea of making easy money by exploiting the difference in interest rates of yen and US dollar, as the latter is always higher. One simply borrows yen, exchanges and deposits the money in a USD account to generate risk-free income - or the yen has to go up. However the arbitrageur should hedge against the risk in movements of exchange rate - say by a forward contract of exchange rate. Then I did google a bit, and read the interest rate parity (which correlates the two interest rates and spot & forward exchange rates), whose “intention” is to prevent such opportunities of arbitrage. However, I also read that, in practice it is not working. In fact, there has been a term coined for such practices, called “The Yen Carry Trade”, where investors borrow yen and invest them in securities of other currencies. It could even be linked with the recent crisis of global financial markets. As subprimal markets turn sour, investors with investment backed by borrowed yen, a lot of which are with high leverage but without proper hedging, are forced to sell them off quickly and pay back their debts in yen, therefore putting a stress on the demand of yen, effectively drives yen up - and an expensive yen is bad news for anyone whose investment is backed by borrowed yen, and they are going in panic to do the same thing - sell the investment, get USD, exchange for yen, and pay off their debts, forming a vicious circle. The central bank of Japan can provide more supply of yen to break the chain, which they just tried (BoJ injected 400b yen into the banking system as it observes the interbank rate increasing). Oh - do not forget that strong yen hurts Japanese economy (Nikkei received heavier loss at least yesterday - did not keep track of it though).

I admit that I speculated most of the above - however who has the data of historical exchange rates can prove or disprove it easily: we all read yen went up against USD on the news. What about yen against other currencies? My suspicion is that yen would be generally driven up in such a global crisis. BoJ served for too long a period as the “generous lender of capital” and only Holy Moly would know how much yen is still out there.

Edit: Actually I got the data: Exchange rates for JPN Yen. See for yourself. USD rose (not as much as JPY did) against most currencies too - because of the difficulty now to get dollars. Well there is always the exception and mystery about RMB and China stock market. Leave them alone.

Edit²: An article titled Currency ‘Carry Trade’ Becomes Harder Play Amid Aversion to Risk appeared on WSJ front page just now. Also read Minsky's Theory.

Heads up strategy (tournament)

Assuming the following winning opportunity P(x) as a function of your stack proportion x:


Figure 1. P(x) vs. x

One would expect the deviation from P₀(x)=x grows larger as the BB size, R, grows larger.

Situation: after you placed BB of size R, your opponent pushes all-in. Should you call it, or fold?


Figure 2. When you should call the all-in (R=0.05)

One interesting finding is that one should be most cautious when he is 70-30 dominating the table, folding even when the odds are 3:2.

Voluntary all-in is different because your gain of winning is lowered by your opponent folding his inferior hands, in other words, you win small and lose big (somehow reminds me of winner's curse).

Linear P(x)?
If players are only exchanging blinds, i.e. the winning/losing is a complete random walk, one can show rigorously that P₀(x)=x. In other words, the chance for W(t) to reach +B before -A is exactly A/(A+B), if W(t) is a Brownian motion. Obviously the leading player would have a higher winning chance than this, that is why we would have the S-shape for P(x).

Quantum scar references

Three papers regarding "quantum scar and phase space" (Ref 72-74 in Rice' review)

E.J. Heller Phys. Rev. Lett. 53, 1515 (1984)
M.V. Berry Proc. R. Soc. Lond. A 423, 219 (1989)
E.B. Bogomolny Physica D 31, 169 (1988)

How language has helped us think

So on Friday I had dinner with RY and we talked about what he learned from his ILW classes. Mckinsey & gave a list of methods of persuasion, like 'by way of authority', 'alliance building' etc. I found it helpful when analyzing the experience I had before (and of course giving better illustrations to the interviewers).

Then we talked about why, although we had already been using those skills involuntarily, we felt a better grasp of them when they were explicitly spoken out, structured and made into a list. It is the power of language. Without a proper language, the instrument of abstract thinking, everything is vague and intangible, and you stay this way, call it natural or 'in a mess'. Intuitions without concepts are blind.

Drivels: Then I thought of density matrix in QM: basically every mixed state is a reduced state of a pure state. Or I say, everything you found hard to express is something clear with a broader language. Yep, all you lack is the language, "the basis set". Creating and extending the concepts thus the proper language, a process analytic a priori, is sometimes more important than any work that would follow.

I've been watching too much lately

I finally resisted the temptation of upgrading my Netflix membership. I do not want to become Tom in The Glass Menagerie. :P And life sucks when I saw stupid Mencia every time I turned on TV to see Eric Cartman (on this matter of taste I am with feir).

Let me say this to myself: once I turned the endo project to publish, I am going to award myself a new GAMING DESKTOP. Make it happen very soon.

Monodromy

So to verify that monodromy actually exists in the control process - we ought to examine the EM geometry. Ideally we will have only two parameters for the potential. See Efstathiou et al, PRA 69, 032504 (2004).

For Monodromy, see also
Dullin et al, Physica D 190, 15-37 (2004),
Cushman et al, Phys. Rev. Lett. 93(2), 024302 (2004),
Giacobbe et al, J. Math. Phys. 45(12), 5076 (2004).

Marketing hoax

I noticed a funny marketing trick played by Tropicana (orange juice) the other day. They released the new 'Light N Healthy' juice, which really pulled me to read the fine prints, which said '43% of orignal juice'. So intrisically you are paying $1 to buy $.43 worth of juice plus the water. LOLz!

Bruce Lee

Just watched the Biography DVD of Bruce Lee by A&E (hah, b-day gift from Y). One line of Lee impressed me the most, which can be paraphrased as 'martial art is about expressing yourself freely and truly, not the fancy movements'. I can not agree more. As a matter of fact, life is about being true and expressive of oneself, not about impressive or influential to others - which, has become a commonly accepted goal of today's power-thirsty humanity. One step further: I believe if one is truly expressive of oneself, he will be more impressive and influential than otherwise; but how many people have truly accepted and followed it? Very few.

Free rotation in 3d space

The problem arises as I have to rotate (randomly) the linear molecule for initial conditions. There have been established ways to do this: Euler angles, unit vector + rotation (essentially the same thing), SO(3) matrices, quaternions etc. I chose Euler angles for my previous simulations, however there is a serious issue: how to make the orientations evenly distributed in the rotation space? Obviously generating evenly distributed φ, θ, ψ is not the answer. Down to the ground, Euler angles → SO(3) is not even a bijective map; for example, (-φ,π,φ) all gives the same rotation.

The proper probability density function: ρ(φ,θ,ψ)=cos(θ).