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Utility Functions in Portfolio Optimization and Derivative Pricing

Article Quant Q&A · Author: Ivan
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Summary

The document describes two practical roles for utility functions in quantitative finance. Systematic funds and trading desks may use utility in portfolio optimization to represent how an investor values risk and return. The answer illustrates why maximizing expected profit alone can lead to ruinous decisions: in a repeated favorable coin game, committing all available capital each round maximizes expected profit but creates a very high chance of losing the capital. Maximizing expected terminal log return instead leads to the Kelly fraction, given in the example as a function of the win probability.

Utility can also support utility-indifference pricing for derivatives on assets that cannot be traded, such as options on private-company shares. These examples show that utility is a modeling choice used to express objectives and risk preferences, rather than a universal description of every trader’s behavior. The document does not supply real-world implementation details, calibration guidance, or evidence about how broadly any particular utility specification is used.

Key ideas

  • Utility functions can encode preferences over risk and return in portfolio optimization.
  • Maximizing expected profit can favor very large bets despite a substantial risk of ruin.
  • Maximizing expected log wealth yields the Kelly betting fraction in the example.
  • Utility-indifference methods can be used to price derivatives on nontradable assets.
  • The selected utility specification reflects modeling assumptions and is not universal.

Tags

Portfolio constructionRisk managementDerivatives pricing
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# Utility functions, are they used in the real world by hedge funds, banks, etc?


# Utility functions, are they used in the real world by hedge funds, banks, etc?












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

## Answer by Antoine Conze (score 3)

https://quant.stackexchange.com/a/42241

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.

## Answer by Ezy (score 1)

https://quant.stackexchange.com/a/42212

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.