I am trying to understand utility function and its application in quantitative finance. I have done some preliminary research on the same (have gone through Paul Wilmott on Quantitative Finance) but unable to capture the idea. Any theoretical (or mathematical) explanation will be appreciated!!.Thanks!!
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In the current, applied field of quantitative finance, the utility function has taken a back seat. Whether or not that is valid is a concerning issue.
If the Markowitz model and its children are correct, the utility function matters but is unnecessary to include in calculations, explicitly. In other words, people are driven by utility, but the movement of the market equilibrium does not depend on the utility function. The challenge to this viewpoint is that neither the CAPM group of models nor the Ito calculus group of models has ever passed a validation test. There is ample evidence that they are wrong.
If that is the case, then utility appears in two places. First, there is the utility of wealth. Second, there is the utility of the estimators.
Now, as in the rest of economics, the utility function should still fall out at the tangency of the indifference curve. In most cases in economics, the exact functional form of the utility function doesn't actually matter as long as it preserves preference ordering and is of the right general shape.
In that case, the marginal utility of a quantity of stock would be its price.
The second place it appears is in the choice of estimators. That has not received adequate discussion in economics due to the convention of using unbiased estimators. In particular, unbiased estimators with risk-neutral probabilities.
The problem is that the entire field of statistics can be derived by joining utility theory with probability theory. When you minimize quadratic loss, you are minimizing $$(\hat{\theta}-\theta)^2.$$ That is a negative utility function. Indeed, statistical loss functions are defined as the negative of a utility function.
The problem with unbiased estimators come along with distributions with heavy tails, such as the returns on equity securities.
There are two problems. First, the heavy-tailed distributions are outside the larger exponential-family of distributions. Second, these heavy-tailed distributions lack a first moment.
The exponential family of distributions, which include the normal distribution also contains tons of important other distributions such as the Weibull or the binomial when the number of trials is known. These distributions have a very nice property, $$\Pr(\theta|\hat{\theta})=\Pr(\theta|X),$$ where $X$ is the data, $\theta$ is the parameter and $\hat{\theta}$ is the estimator.
What that says is that if you know the parameter estimator, then you have all the same information as if you had the sample. There is no information in the data set that is not in the estimator. Likewise, there is no information in the estimator that is not in the data. They are perfect substitutes. That is what makes some estimators such as $\bar{x}$ when data is normally distributed so important. It is also why the median is so rarely used for such data.
That statement is not true for heavy-tailed distributions. There does not exist a Frequentist point statistic that is sufficient for the parameters.
It is true that you can condition a point statistic on an ancillary statistic for inferential purposes, but not for predictive or projective purposes.
That does not mean that a solution does not exist, but that solution isn't in use in quantitative finance. The Bayesian likelihood function is always a minimally sufficient statistic. It does not generate a point, except in degenerate cases. However, if you apply a utility function to the Bayesian predictive distribution, you can get a valid point statistic.
The second issue is that the bulk of the heavy-tailed distributions do not have a first moment, so quadratic loss cannot be used. The integrals diverge.
That brings up the question that has been dormant in economics, but alive in statistics, of what utility function should be used. That is not the utility of wealth, it is the utility of making an estimate, which in turn determines the allocation of wealth and pricing.
In practical quantitative finance, in other words "in order to get a job," utility does not matter.
If you are looking at where it will be in ten years, there is a good chance that utility will matter a great deal.
EDIT In response to the comment, the question is a bit more challenging to answer than it would appear.
In de Finetti's axiomatization of probability, there is no way to separate probability from utility. The $\mathcal{U}(\tilde{x})$, where $x$ is a random variable is really one single calculation. It is a function and not a function of a function. In technical terms, they are not separable. If you change your utility function, in de Finetti, you also change your probabilities and vice versa.
For the purposes of creating statistics, it doesn't matter. For purposes of building economic models it does.
If you look at Savage's axiomatization of probability, utility and probability are separable. Your utility function is independent of your assessment of probabilities. You still end up with a composite calculation, but it is two separate functions. With de Finetti, the brain is doing one inseparable calculation. With de Finetti, your brain cares as much about the probabilities as the outcomes but mixes them together as if they were one thing.
In de Finetti's world, probability does not exist. Merely, it and utility are ways economists think about the world as an explanation of events and behaviors. Probability isn't a real thing. Utility is not a real thing. They are abstractions that are not grounded in reality and so are not separable in his math.
A way to think about it is like this. Rocks do not obey the laws of gravity. That is an absolute fact because rocks cannot disobey the laws of gravity. Rocks themselves do not exist either. That is our name for them.
De Finetti used the words probability, price and prediction as interchangeable. Usually, the word prediction is translated into English as prevision. The disadvantage of de Finetti's conceptualization is obvious. Preferences yield both utility and probability without a way to separate them making life terrible for the economist. It has one advantage though. It only depends on observable prices in markets. By making price=probability=prediction, you have a great advantage because you can see prices. The person isn't going to let you inside their head to see their probabilities or their predictions.
In Savage's world, to the extent your utility is averse or loving to uncertainty, the maximization will take it into account without a special process. For example, $$\mathcal{U}(\tilde{x})=\sqrt{\tilde{x}}$$ is risk-averse and it is maximized with that risk aversion built into the calculations. In von Neumann's world maximizing utility is maximizing $$\int_{x\in\chi}\mathcal{U}(x)p_\theta(x)\mathrm{d}x$$ subject to some constraints. In Savage the $p_\theta(x)$ becomes $p(x_{future}|x_{past})$ and de Finetti would ask the simpler question, "do you want to buy this bet?" It would be a function for de Finetti.
De Finetti would ask you to maximize $$\mathcal{G}(\tilde{x}).$$ What is in $\mathcal{G}$ you ask? Everything.
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$\begingroup$ is unpredictability a concept separated from utility in that we would want to minimize unpredictability while we maximize utility? $\endgroup$ Commented Sep 8, 2020 at 15:10
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$\begingroup$ @develarist hopefully answered your question, at least part way. $\endgroup$ Commented Sep 8, 2020 at 15:59
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Regarding applications, utility functions have traditionally not been used much in quantitative finance, at least not explicitly. However, they are latent in areas such as portfolio optimization, where the objective function can be interpreted as a utility function.
No arbitrage is usually a necessary condition for the existence of an equilibrium in financial economics models with utility maximizing agents. Hence standard derivatives pricing models are consistent with such equilibrium based models with utility functions. However, the innovation of Black-Scholes-Merton was that solving for option price consistent with no-arbitrage does not require specifying specific utility functions. Hence you rarely see utility functions in the derivatives pricing side of quant finance.
