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I am starting to study mathematical finance. When I studied microeconomics and macroeconomics I studied utility functions, but I never saw how they are in the real world. I do not see how they can be used since a utility function is very personal, and there are many stereotypes of investors: HFT firms, swing traders, day traders, etc., and their behavior not necessarily match the risk-averse behavior represented with convex functions. As an example, it is the case of companies such as Apple Inc. and netflix.com Inc. with prices $225.74 \$ $ and $374.13 \$ $ respectively, but the solvency, profitability and operating efficiency of Apple Inc.'s are better than netflix.com Inc.'s.

I wonder if utility functions are really used by hedge funds, banks, etc., to compute the price of financial instruments?

Thanks in advance

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    $\begingroup$ Thanks to methods developed by Black Scholes Merton (and others) derivative instruments can be priced without using utility functions. Also, corporations maximize the value of the firm, not utility. So utilty fuctions are seldom used in Mathematical Finance (unless dealing with portfolios of individual investors). $\endgroup$ – noob2 Sep 29 '18 at 19:35
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See also utility indifference pricing (Henderson, V., & Hobson, D. (2004) Utility Indifference Pricing - An Overview http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.321.994&rep=rep1&type=pdf is a good reference) for examples where utility functions can be used to price derivatives on non tradable assets, such as stock options on non listed companies.

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Utility functions are used all the times in systematic hedge funds/systematic trading desks to perform portfolio optimization.

If you are still not convinced here's a nice little one:

Say you and I decide to play 100 instances of an even-money game of flip a coin where you have a probability advantage of winning (p > 1/2). Say you start with a capital of 1$ and the rule is that you need to choose in the beginning a fraction F of your portfolio which you will play on each run until the end of the 100 games or before if you are ruined (if your capital is reduced to 0 then the game stops). The question is to choose what is the "optimal" fraction F of your portfolio which you should choose to invest in this game on each turn.

Clearly if you choose the "expected pnl" as your utility then you will be willing to invest 100% of your capital on each turn because that's the highest expected return you could make. But at the same time it would be foolish because of the chance to ruin which is also almost 1.0. A more reasonable choice of utility is to choose the terminal log return of your investment strategy that leads to an investment fraction F=2p-1. This is called the Kelly criterion

https://en.wikipedia.org/wiki/Kelly_criterion

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