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Murmuring about a martingale

1909_SS

Murmuring about a martingale

In the field of financial mathematics, for 50 or so years we have essentially been restating and coming up with different ways to answer the same question; if I know a stock price today, can I know the stock price tomorrow?

Naturally, you would expect to hear a resounding “No!” from the audience you asked. Yet in financial mathematics, we do this all the time. In fact, most models including that of the famous Black Scholes Merton model imply it (although you’d need to run through the derivations to see where).

Mathematically, if there exists a random variable that is real (below infinity) and fluctuates over time, we call this variable a martingale. By definition, for a martingale we expect that tomorrow’s value will be equal to today’s value. Applied to the stock market, we can theoretically say that by knowing today’s stock price, we now know tomorrow’s price.

The key word here is ‘expect’, as in; we believe it’s most likely that tomorrow’s value will be equal to that of today’s. This is an issue because there is an infinite array of factors that can influence the value of a stock beyond those that are random in nature that the definition of a martingale cannot incorporate.

Without the martingale assumption however, many financial models either cannot be derived or produce strange results. Yet their application by the financial industry on the markets is a double edged sword; in times of normality, the models work reasonably well to price risk and allow commerce to function in an orderly manner, yet as left-tailed events enter the fray, chaos can ensue – just ask Long-Term Capital Management about this one…

So let’s look at two different perspectives; i) the more traditional discounted cash flow model and ii) a more mathematical expression that states that stock prices rise over time at a known rate, but are also subject to random fluctuations at each time step. (I’ll leave out the equations so as to not overcomplicate the matter, but for those interested, Google ‘Geometric Brownian motion’).

Both models are essentially trying to explain how a financial instrument can be expected to act in the future, yet both approach the issue from different angles. And both are based on assumptions (however good or poor).

This post was contributed by a representative of Montgomery Investment Management Pty Limited (AFSL No. 354564). The principal purpose of this post is to provide factual information and not provide financial product advice. Additionally, the information provided is not intended to provide any recommendation or opinion about any financial product. Any commentary and statements of opinion however may contain general advice only that is prepared without taking into account your personal objectives, financial circumstances or needs. Because of this, before acting on any of the information provided, you should always consider its appropriateness in light of your personal objectives, financial circumstances and needs and should consider seeking independent advice from a financial advisor if necessary before making any decisions. This post specifically excludes personal advice.

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Comments

  1. In normal markets, i think the arguement that the fluctuations are random can hold as although every decision would be based on some type of rationale, they are likely to even themselves out. It is equally likely that someone might come up with a buy decision or a sell decision. However, for whatever random reason, one will out number the other slightly.

    In less normal times, the rationale behind buy/sell decisions will be shared by a disproportionate number and there for it can be argued that movements aren’t random and any model which assume randomness might find itself a bit confused.

    May not be really what you are discussing above but just a thought to consider.

    On a different note, although linked, i still believe a lot of investors need to understand the extent to which assumptions are used in finacnial markets and be comfortable with them. Going through another valuation assignment at university and seeing the same confused looks and questions when people find out there is no clear exactly correct answer. I believe understanding the flaws of financial models will allow you to make better decisions as you will find better ways to overcome them and reduce the impact of biases. Also reinforces one more lesson, margin of safety is important.

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