Ekonomi Bukan Fisika

Jelas beda. Molekul, atom pergerakannya setidaknya bisa di track secara pasti ataupun secara probabilistik (quantum mechanics).  Interaksi manusia dengan lingkungannya (ekonomi, sosial, alam, dsb) adalah suatu sistim yang sangat kompleks sehingga saat ini adalah naif kalau kita mengharapkan interaksi kompleks ini bisa diterjemahkan dalam bentuk sistim ekuilibrium saja.

Economics and the Theory of Finance – It Ain’t Physics!
Rahul Bhattacharya
December 23, 2008

Recently, some economists, a few very well known amongst them, got very upset when they read our recent post on our site: Quantity Theory of Money – the Fisher-Friedman Illusion!.They wrote to us with lengthy explanation of Irving Fisher’s thesis on the equation of exchange and defending the name and the turf of the celebrated economist.

To all those people we have only one answer: It ain’t physics!

Of all the crises confronting us today, none is greater or more devastating than the crisis that is shaking the foundations of the Economic Theory and the Theory of Finance. As the titans of the banking and investment banking lie humbled and as thirty years of a secular bull market comes to an end, the winter of 2008 is finally drawing the curtains on one hundred and fifty years of economic thinking, which was predicated on the notions of equilibrium and linear dynamics.

The world, as economists are realizing, does not come wrapped in differential calculus.

Hundred fifty years of obsession with equilibrium and linear dynamics (CAPM, linear optimization, asset allocation models, Black-Scholes option pricing models) has resulted in a lot of Nobels, but pretty much nothing else

“Causality” and “correlation” are different concepts and whereas you can have “reverse causality” in financial markets and economies the notion of “reverse correlation” is not possible. How many economists understand this? Theory of probability is totally counter-intuitive and whereas one can apply probability theory and stochastic processes to quantum mechanical world in physics, it makes no sense to build models of a larger, real world of men and things based on these. If economists and financial theorists know that Economics is not physics, especially quantum mechanics, then why do they keep on applying probability theory and stochastic processes to economic systems?

In fact, economics should be studied more like electrical engineering or systems engineering. I believe Economics is an extremely complex subject, one that should borrow heavily from history, sociology, molecular biology and systems / electrical engineering.

As long as economists, and their closest cousins, the financial theorists, compete with the physicists in their usage of differential calculus, as long as economists delude themselves into thinking that they are scientists we will keep moving from crisis to crisis.

Ito Lemma Dan Ekspansi Taylor Series

Agak mirip. Yang satu untuk stochastic differential equation (Ito), yang lain untuk Ordinary Differential Equation.

Why so much fuss about Taylor Series Expansion?
Team latte
Oct 20, 2005

Taylor Series expansion is one of the most, if not the most, applied mathematical concept in financial engineering. It is a simple expansion series of any function around a small increment .

Where are higher order terms (greater than the power of three). If we ignore
the higher order terms as well as the third order term then we can write:

And in the limiting case when we have

If you carefully look at the above equation then we will see a striking resemblance
to the famous Ito’s stochastic equation, which is the basis of Black-Scholes option
pricing partial differential equation (PDE). The only difference is that in Ito’s
equation we have the term  instead of   , that is in an Ito process we have

The above equivalence has a great intuitive meaning with respect of geometric Brownian motion and asset price movement, but that is not the purpose of this article. What we wanted to show was that even without any knowledge of stochastic calculus and the corresponding Ito process one can approach the Black-Scholes option pricing PDE if one were to make the accommodation shown above. It also shows that any asset price movement, such as the movement of a bond, can be approximated by the Taylor series expansion and regardless of whether we know the exact closed form solution for the pricing of that asset, it is possible,

using first principles of differential calculus to approximate the movement of the asset over small increments of state space and time.

In keeping with our arguments above let us show you that the Taylor series expansion can be used to calculate the value of any variable with a great degree of precision and speed, which may not be possible numerically.

Say, you wanted to calculate the value of , that is eleven raised to the power of eight. Doing this manually could take you forever. Of course, you can do it in less than a second using a scientific calculator or an Excel spreadsheet. But suppose you don’t have an Excel spreadsheet (or a calculator near you) and you wanted a quick approximation of this number.

You can say that and that a small increment of   is one and therefore,
and . First we calculate the value of which is very
straightforward and simple: 100,000,000. We need to use the Taylor series to
calculate the change in value of this number around the point . (Remember
by making x = 10, we have made all our calculations very simple and that is where
the usefulness of Taylor series lies)

Now using Taylor series for calculation we get:

Thus the change in the value of the function f which is given by is:

And the value of the function is approximated as:

This is quite good an approximation given the fact that if you were to use your Excel spreadsheet then for 11^8 (eleven raised to the power of eight) you would get 214,358,881.

Barrier Option

Pernah dengar barrier option? Agak beda dengan option yang konvensional. Valuasinya juga tentunya beda dari sekedar menggunakan formula Black-Scholes. Berikut artikel mengenai produk ini.

A Probability Conundrum: First Passage time of a Brownian Motion
Rahul Bhattacharya
Jan 25, 2005

Recently we encountered a bit of a problem with the use of first passage time in our forecasting models. The problem appeared more fundamental in terms of the mathematical formulation. I asked a trader who trades currency options in a large bank in Hong Kong as to what is the probability that an asset, say a currency or a stock, which is currently trading at $100 with a historical volatility of 15% and a mean historical return of 10% will hit a barrier at $95. It took him ten seconds to plug it in his option pricing calculator the values and he gave me an answer of 57.3% (assuming the implied vol for 3 months is 15%).

I am sure this is correct given the fact that he had calculated the probability of hitting the barrier by using the pricing model of a one touch digital option. However, when we use a different formula for the probability of the first passage time, as the problem is known, in asset forecasting models we get a different result for the probability of the asset hitting the barrier.

First passage time is a central concept in the analysis of Brownian motions with absorbing barriers. Consider a Brownian particle undergoing a one-dimensional random walk within a domain between x=0 and x=L, the boundaries being such that the particle will be reflected at x=0 and absorbed at x=L. If the particle starts from some position within the domain when will it hit the absorbing boundary for the first time? This first-passage problem in stochastic processes, in various spatial dimensions and with a variety of geometries and boundary conditions, is of longstanding interest and has ramifications and applications in diverse areas.

In simple English, First Passage Time refers to the probability of a random particle hitting and being absorbed by a barrier or a boundary. This barrier or a boundary could be the barrier of an asset price as in a barrier option, or a predetermined band such as a government determined peg rate of a currency.

Mathematically, the probability that an asset following a geometric Brownian motion will hit a barrier is given by:

Probability that an asset following a geometric Brownian motion will hit a barrier

Let us assume that an asset, for example a stock or a stock index, follows a geometric Brownian motion. The stock is currently at $100, the annualized volatility of the stock is 15% and the mean historical return of the stock is 10%. Then the probability that in three months time the stock will hit a barrier at $95 will be given by the above formula and is simply equal to 38.46%.

This could explain the difference – fairly large – between our result and the result that my friend got from his digital option pricing formula. Later on he used Monte Carlo simulation and yet he came up with a probability number much greater than 40%.

We believe that the formula mentioned above gives the correct probability of a stock hitting a barrier provided it follows a geometric Brownian motion with a drift equal to the mean return. This could be viewed as the real probability of hitting the barrier as opposed to a risk neutral probability measure.