Using Continuous Markov Chain to Find Average Amount
Markov chains are something unheard of amongst finance enthusiasts. But I will change the whole mindset after you read this article to the end. Markov chains are extremely vital in finance domains, be it risk management, stock market predictions or analyzing prospective debtors.
These models are operated by mathematicians on the backend, so it remains behind the doors for financial analysts because on the frontend they happen to study the patterns, so they don't give much heck about it. I believe these concepts should be taught to a greater audience and especially to non mathematicians, so they deliver more efficient results with improved performances.
The concept of Markov chains was developed by a Russian mathematician Andrey Markov in 1906 out of disagreement with Pavel Nekrasov who claimed independence was necessary for the weak law of large numbers to hold. A Markov chain is based on a Stochastic model where future actions are governed by present events.
A typical example would include weather predictions where the future conditions are predicted by a lot of factors relying on a Stochastic model. These chains are governed on two models –
a) countably infinite sequence, where the chain moves at discrete time steps giving us a discrete time Markov chain model and
b) continuous time sequence, where the chain moves at a continuous pace giving us a continuous time Markov chain model.
Markov chains have a lot of practical applications in today's world. Almost all statistical models have Markov concepts stuck to them in their operating systems. Markov models are used to govern the queues at the airports, cruise control systems in motor vehicles, currency exchange rates and stock market predictions and what not.
These models have also found application in textbooks of physics, mathematics, statistics to impart knowledge of their know-how. Markov chains are also briefly described in Gregory Zuckerman's book ' The man who solved the markets; how Jim Simons launched the quant revolution'. The main character of the book , Simons, was also in complete awe of the theory behind these models and used them to launch the quant revolution.
This article will shed light on the practicalities of Markov models in finance and before jumping on to the applications of these models we will have a short look at the process and the concept on which it is based.
Concept of Markov chains
A Markov chain is a stochastic process that satisfies the Markov property, which means that the past and future are independent when the present is known. This means that if one knows the current state of the process, then no additional information of its past states is required to make the best possible prediction of its future. This simplicity allows for great reduction of the number of parameters when studying such a process.
In mathematical terms, the definition can be expressed as follows: A stochastic process X = {Xn , n ε N} in a countable space S is a discrete-time Markov chain if: For all n ≥ 0, Xn ε S For all n ≥ 1 and for all i , … i , i ε S, we have 0 n−1 n : P{ X i | X i , … , X } { X | X }.
This mathematical pinch would not have tasted well for not so mathematics friendly ones. Let's not complicate things further and understand by an example on how things work in a Markov model.
If we arbitrarily pick probabilities, a prediction regarding the weather can be the following: If it is a sunny day, there is a 30% probability that the next day will be a rainy day, and a 20% probability that if it is a rainy day, the day after will be a sunny day. If it is a sunny day, there is therefore a 70% chance that the next day will be another sunny day, and if today is a rainy day, there is a 80% chance that the next day will be a rainy day as well. This is a simplistic way of working of Markov models, although the equations are a lot more perplexed than this.
In my other example, I will tell you how Markov models work in the stock market.
Markov analysis can be used by stock speculators. Suppose that a momentum investor estimates that a favorite stock has a 60% chance of beating the market tomorrow if it does so today. This estimate involves only the current state, so it meets the key limit of Markov analysis. Markov analysis also allows the speculator to estimate that the probability the stock will outperform the market for both of the next two days is 0.6 * 0.6 = 0.36 or 36%, given the stock beat the market today. By using leverage and pyramiding, speculators attempt to amplify the potential profits from this type of Markov analysis.
I am sure that so far you have understood the basis of a Markov model and the importance it has in the practical world. In the next paragraphs, I will discuss the usage of these models particularly for the financial sectors.
- Credit risk management – in credit risk management, the transition matrix plays a crucial role. This matrix is also called stochastic matrix which is a square matrix used to determine transitions of a Markov chain. This matrix will formulate probabilities that which company will remain in its current state or transition to new states. The transition data is derived from rating agencies like Moodys, S&P, Dow jones. The data is studied and analyzed to conclude prospective results. A non homogenous model is used in this case because it seems more realistic. A sample of transition matrix is attached below –
- Prediction of market trends – Markov chains and its models are more often used in determining stock market activities. The present data of a market is used to deliver future results. The actions of buyers and sellers are studied thoroughly to identify a pattern and draw models and derive results out of them. In fair markets, it is assumed that the market information is distributed equally among its actors and that prices fluctuate randomly. This means that every actor has equal access to information such that no actor has an upper hand due to inside-information. Through technical analysis of historical data, certain patterns can be found as well as their estimated probabilities. For example, consider a hypothetical market with Markov properties where historical data has given us the following patterns: After a week characterized by a bull market trend there is a 90% chance that another bullish week will follow. Additionally, there is a 7.5% chance that the bull week instead will be followed by a bearish one, or a 2.5% chance that it will be a stagnant one. After a bearish week there's an 80% chance that the upcoming week also will be bearish, and so on. This data is compiled to form a matrix and then the results are drawn thereof.
These were the most vital uses of a Markov model. These models are excessively used in stock market predictions in financial domains. They have delivered excellent results over time and we have Mr. Markov to thank for. I hope this clears your mind on these models and their importance.
~ Niharika Gupta
Source: https://thefinancepedia.in/financial-applications-of-markov-chains-post/
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