Is it possible to “capture” the expectations of investors? How is it even possible if each investor has his own expectation of where the price should be?
Some Background
Determining expectations when investing is important in order to price financial instruments. Take for example pricing an option with a strike of 100. If I expect that the stock would be at 110 in 1 month with 50% chance and 90 with 50% chance, how would I price the stock? We have a 50% chance of earning 10 and 50% of not earning nothing. This means that on average we would earn 5 and that should be a fair price of the option.
No arbitrage Principle
Each investor in the market has his own expectations like the one above. So how can we capture all of these? it turns out, quite remarkably, that in a complete market, there is one expectation that is true for all investors. That is quite a statement and it is difficult to explain without getting into more technical details. You could read about pricing an option to get an idea about it. The point here to remember is that we assume the market is complete. In a complete market, options can be perfectly replicated by a combination of stocks and bonds.
Risk Neutral Density
These probabilities as in the simple example of the 50-50 chance, are known as the Risk Neutral Density function. In the Black and Scholes world, the density is bell-shaped with known average and variance. The problem is that the Black and Scholes world is very simplified and does not fit empirical evidence. This leads us to try to find the “real” density function, assuming it exists and it is unique.
Implied Density Function
One way to find out the implied density function is to derive it from current option prices. That is why it is called implied. There have several methods proposed to find the implied density function. The big question is if it adds any new information we can’t derive from other methods? This question is complex and there has been a lot of research on the topic. If you believe that market is incomplete than finding the implied density function in theory does not tell you anything.
I’m currently working on this topic myself so I’ll write more about this in the next post. Don’t worry if you did not understand everything above. It takes time to assimilate.
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