# Brownian motion forex

- 04.12.2019
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It presented a stochastic analysis of the stock and option markets. Currency rates should behave very much as a pollen particle in water too. Brownian Spectrum An interesting property of the Brownian motion is its spectrum. This is called the Fourier series. The concept can be further extended to non periodic functions, allowing the period to go to infinite, and this would be the Fourier integral. Instead of a sequence of amplitudes for each multiple frequency you deal with a function of the frequency, this function is called spectrum.

Signal representation in the frequency space is the common language in information transmission, modulation and noise. Graphic equalizers, included even in the home audio equipment or PC audio program, have brought the concept from the science community to the household Present in any useful signal is noise.

These are unwanted signals, random in nature, from different physical origins. Thermal noise is approximately white , meaning that the power spectral density is equal throughout the frequency spectrum. It occurs in almost all electronic devices , and results from a variety of effects, such as impurities in a conductive channel, generation and recombination noise in a transistor due to base current, and so on.

The probability density of finding a random walker at position x after a time t follows the Gaussian law. Drift and Random Motions Motion of pollen particles can be said to have two components, one random in nature described above, but if the liquid has a flow in some direction, then a drift motion is superimposed to the Brownian.

The FOREX market presents both types of motion, a higher frequency random component and a slower drift motions caused by news affecting the rates. Random motion is bad for the speculation business; there is no way to average a profit on a perfectly random market.

Payoff of the European call option is max S T -K,0. This means if price of underlying stock or currency pair is above K only in that case we get a payoff otherwise we get zero. What this means we need to model the stock price or currency pair price if we want to calculate the options contract payoff at expiry.

In the coming decades this important breakthrough was forgotten but it was again discovered in s. First options pricing formula based on geometric Brownian motion was developed in by Fischer Black, Myron Scholes and Robert Merton. This is the famous Black Scholes options pricing formula. Then the discrete time Binomial options formula was developed by Cox, Ross and Rubenstein. Today Brownian motion is an important part of quantitative finance. Expected return is very difficult to measure statistically in the short term.

We can only measure it statistically in the long term. This is the basic stock price model that we use in developing quantitative trading strategies. Stock volatility has been assumed to be constant in Black Scholes options pricing formula. But in reality stock volatility is variable. Volatility can be measured in the short term. In the short term drift is not apparent and volatility dominates.

This explains why we find stock prices to become highly volatile at time and non volatile at other times. So essentially Brownian motion is a random process that makes stock price a random process. Brownian motion increment to stock price is continuous with the caveat that Brownian motion increments on two different non overlapping time intervals are independent random variables. There are three types of analyses that you can use when analyzing a stock, commodity or a currency. First type is known as fundamental analysis.

Fundamental analysis basically comprises reading balance sheet of companies and their quarterly earning reports when analyzing stocks. When it comes to commodities or currencies, fundamental analysis based on macroeconomic studies. Fundamental analysis is long term and difficult to quantity into actionable trades. Fundamental analysis talks about purchasing power parity and stuff like that when it comes to currency market.

Technical analysis is what most traders love to do. Technical analysis is just based on chart reading. All information is contained in the price and we believe that price patterns have predictive powers. In the last few decades a lot of studies have shown that technical analysis has no predictive power and chart patterns like Head and Shoulders will result in more lost trades as compared to winning trades.

Technical analysis is discretionary and subjective which makes it hard to quantify. What we need is something that we can quantity. Read this post on statistics the missing link between technical analysis and algorithmic trading.

This leads us to Quantitative Analysis. Quantitative Analysis is based on statistical principles and is now a days ruling Wall Street. As said in the start of this post, Wall Street is employing thousands of highly paid quantitative analysts known as Quants whose job is to develop quantitative trading strategies. Rather they have sophisticated quantitative models that use price volatility and returns in determining when is the best time to enter and exit a trade.

I have given you the basic stock price model. Stock returns is a random variable. So we use stochastic calculus to model the financial market randomness. You should keep this in mind that randomness plays a very large part in the financial market. We model randomness in the financial market returns and stock price or currency pair price with Brownian Motion. To keep it simple, we use probability a lot when it comes to modelling the financial markets.

You are standing at a major traffic hub where many roads are coming in and then exiting. You are watching thousands of cars coming. Some are turning right. Some are turning left. You have no information or knowledge that tells you why a particular car turned left. You can just watch the thousands of cars in a crowd and observe that majority are turning right.

Everything is random for you but on the macro level you have this idea that most drivers are turning right. For an individual driver things are no random at all. She knows why she is turning right or left. She has to go shopping or pick the kids or reach home. For the drivers that things are not random at all. But they are random for you as an observer. The same thing applies when you observe financial markets. For you price is random but for each individual players things are clear and not random at all.

Ponder over this example and things will become clear to you why financial markets are random to you. You can see volatility is associated with Brownian Motion which is totally random. In order to use volatility in our basic equation we need to know more about Brownian Motion. Brownian Motion is also know as Wiener Process.

You must have heard of random walk, Brownian Motion is the limiting case of a symmetric random walk. If stock price or currency price is a random walk, we have serious issues with technical analysis. If price is a random phenomenon in the short run than most of the chart patterns that we observe are just random patterns.

We will discuss this thing more. Random walk is a discrete time model that in the limiting case becomes the Wiener Process or Brownian motion. Brownian motion is more popular in quantitative finance as compared to Wiener Process. Both are same and nomenclature is used interchangeably. This is very important. Brownian motion price path is everywhere continuous but nowhere differentiable. Read this post on why I have decided to become a quant trader.

This is important for you to understand. This implies that the Brownian motion is a memory less process and the past information is irrelevant to the future stock price values. So our basic stock price model complies with the EMH. The problem with arithmetic Brownian motion is that it is normally distributed.

What this means is that stock prices can become negative over the long term. This is something impossible as stock prices cannot go below zero due to the limited liability concept. Stock holders are not responsible for the companies losses. Only thing that they can lose is the stock investment.

The same thing happens in the currency market. Currency pair prices cannot go negative. How to avoid prices going negative in the long run? By assuming stock prices to be log normally distributed we make sure that stock price never goes negative. The resulting Brownian motion is known as geometric Brownian motion. As I have said above, geometric Brownian motion is used extensively in modelling options pricing formula.

I will write a full post on how to derive the Black Scholes options pricing formula from first principles. Stochastic Volatility Models Volatility is not constant. Volatility is not predictable and directly observable. Returns on currencies, stocks and commodities are also not normally distributed. Financial time series returns have fat tails and high peaks which means we can frequently see very big moves in the market. Stochastic volatility models have been developed that use geometric Brownian motion to model returns and volatility as random variables.

We can then use the Ito calculus to develop a dynamic state space stochastic volatility model that can be used to predict the high and low of price in a certain time period using a particle filter. Did I mention that the famous Black Scholes options pricing formula also uses Brownian motion in its derivation with the assumption of constant volatility? As said above, volatility is not constant so we often see the options price to diverge from the price predicted by the Black Scholes options pricing formula with the famous smile effect.

I am not going to go into the details of the derivation of Black Scholes options pricing formula here and describe how the stochastic volatility models solve the smile problem of Black Scholes formula. Guess, what can be the reason? The reason is simple. We have assumed volatility and stock returns to be constant in the above R code that generated the stock price sample paths.

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In the derivatives markets, this question is of crucial importance. There are some exotic options, like lookback options, whose price actually depends on the path the asset takes. It seems obvious that path would be super important, though. You know what path it took to get there. It went down to in the financial crisis and has basically gone up for six years straight, until today. People are pretty bullish, right? The market has been going up every single year, for six years.

But what if the market had gone down to , then up to 4,, then crashed to 2,? This is a stupid example, but not really. Of course not. Path really matters. Because you remember. Did you see what happened next? Politicians started calling for an increase in gas taxes.

Because people were used to high gas prices and could easily absorb an extra 10 cents on the gallon. People would be furious. So gas prices are path dependent! Well… so is everything else. EMH Is Dead The Efficient Market Hypothesis is the idea that all information including the path of previous stock prices is reflected in current stock prices.

Even in the age of the Internet, this is not true. There was a 60 Minutes episode on curing cancer last Sunday. The biotech guys have known about this for years. It takes time for information to travel, sometimes a long time. Most options are priced with similar assumptions. The particle has no memory.

It is path independent. But unlike the particle, the market has memory. People have memory. And as we demonstrated, it is path dependent. So its behavior will not be a true random walk. People are slowly learning what the quant guys have known for years: the market is not random, and you can profit from that. There would be no commodity trading advisors CTAs if the market were random. But the guys who have disproved market efficiency are making too much money to bother filling out a Nobel Prize application.

This is the problem technical analysis claims to solve. The technical guys are half right. By repeating the experiment with particles of inorganic matter he was able to rule out that the motion was life-related, although its origin was yet to be explained. The first person to describe the mathematics behind Brownian motion was Thorvald N.

Thiele in a paper on the method of least squares published in This was followed independently by Louis Bachelier in in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. The Brownian motion model of the stock market is often cited, but Benoit Mandelbrot rejected its applicability to stock price movements in part because these are discontinuous.

Their equations describing Brownian motion were subsequently verified by the experimental work of Jean Baptiste Perrin in Statistical mechanics theories[ edit ] Einstein's theory[ edit ] There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the mean squared displacement of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities.

The number of atoms contained in this volume is referred to as the Avogadro number , and the determination of this number is tantamount to the knowledge of the mass of an atom, since the latter is obtained by dividing the molar mass of the gas by the Avogadro constant. The characteristic bell-shaped curves of the diffusion of Brownian particles.

As t increases, the distribution flattens though remains bell-shaped , and ultimately becomes uniform in the limit that time goes to infinity.

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