Questions tagged [utility-theory]

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Expected Utility Theory

I have a question that Say that there is a random variable of return for a risky investment opportunity $ \bar{r_i}$, its expected return is $E(\bar{r_i})$ and variance $\sigma_i^2$. Now, let's assume ...
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2 votes
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131 views

Utility Theory and Mean Variance Analysis

I was wondering if it's pertinent to use this interpretation of the expected utility function given by the Taylor series expansion, $${E(U(W)}\approx{U[E(W)}]+\frac{U''[E(W)]\sigma^2_W}{2}\tag{1}$$ to ...
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Domains of Utility Functions

I am learning a mathematical finance course and the lecturer didn't provide us with a rigorous definition of utility functions. He just shows us (by simple Calculus) that the utility functions of ...
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A question about the Grossman-Miller Market Making Model

I don't have any solid background in finance, but I have a strong mathematics and physics background. I am reading Algorithmic and high-frequency trading from A.Cartea, S.Jaimungal and J.Penalva, CUP (...
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How to compute the mean for utility function? [closed]

Let $u(x)=x^{2/3}$, $x>0$ be the utility function, $X \sim U(0, 100)$ is loss, wealth $w=\\\$150$. Calculate $\mathbb{E}(u(w_r))$ if a coinsurance is $80\%$ and gross premium is $\\\$43$. My ...
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Epstein-Zin utility intuition

I working a lot with Epstein-Zin utility (standard in asset pricing models). But I am having some issues wrapping my head around some intuition for how this utility function works. Let's think about a ...
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Why does Cumulative Prospect Theory coincide with Choquet Integral for continuous outcomes?

In Tversky and Kahneman 1992, the authors state in a footnote that their cumulative version of Prospect Theory can be written as Choquet utility for continuous prospects. Is this statement due to the ...
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Expectation of the negative exponential utility function for a Grossman and Miller model

I have the standard $3$-period Grossman and Miller model with $2$ outside traders and $M$ market makers. I'm told: $W_t^{(1)}, W_t^{(2)}, W_t^{(m)}$ is the wealth of the first outside trader, second ...
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Portfolio Theory - Maximizing Expected Utility Function

I am trying to implement a portfolio selection tool based on utility functions. So, I should maximize the expected utility of a given utility function: $$ \begin{align} &\max_{w}\ E[u(W_0(1+w^TR))]...
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How to evaluate the following integral?

I stumbled across an expression and I wonder how to evaluate this: $-\int_ {0} ^ {+\infty} {v(x)} dw^{+} (1-p(x))$ where $v(x)$ is some utility function and $w(p(x)) $ is a decision weighting function,...
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Expected value of stepwise function of Prospect Theory as improper integral

I am looking for a sufficiently precise solution algorithm to evaluate the following integral: $$\int_ {-\infty} ^ {\infty} {v(x)} {f(x)} dx$$ where $v(x)$ is the Prospect Theory value function and $f(...
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How does this imply that a Pareto optimum maximizes a weighted average of utility functions?

I'm reading a passage from Asset Pricing and Portfolio Choice Theory by Kerry Back, and I don't understand some of it. I would appreciate any help anyone could provide me. In the passage, Back is ...
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3 votes
2 answers
299 views

Deriving the risk-aversion coefficient

By considering the parametrised formulation of the mean-variance criterion by Markowitz, the risk aversion coefficient $\lambda$ can be derived as follow. As suggested by Arrow and Pratt, given the ...
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finding optimal weight using Kelly criterion

Question: Suppose you have two strategies. Strategy 1 gains 8% with probability p, and loses 5% with probability 1-p, where p = 0.53. Strategy 2 gains 8% with probability q, and loses 5% with ...
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2 answers
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How to compare mean-variance-skewness-kurtosis portfolios obtained by expected utility maximization?

Suppose I have some portfolios which are the result of maximizing the expected utility of different approximations of a utility function, how do you test these portfolio's out-of-sample and how do you ...
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Anyone has detailed explanation on how to use epstein-zin preferences in asset pricing models

I'd be interested to know how Epstein-Zin preferences are used in, say, consumption-based asset pricing models. I'm looking for specific derivations (how you get the SDF) and possible numerical ...
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ETF pricing papers

May I request for research paper recommendations, if any, on existing models that study how the presence of ETFs affect equilibrium prices of the underlying assets? I am exploring a project on a ...
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1 answer
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Application of Ito's Lemma in expected utility theory

An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\...
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How do I maximize my expected utility of wealth?

Suppose I have a utility function say $U(p)=p^{1/2}$ and I bet on a basketball game. I have my initial investment, payouts and probabilities of winning, how can I determine the maximum I need to bet ...
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1 answer
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Closed Form Solution for Implied Risk Aversion with Two Assets under Quadratic Utility

So I believe there should be a closed form solution for implied risk aversion for two assets but I'm not sure how to get there. Say you have Quadratic Utility $U$ on a fully invested portfolio of two ...
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Fixes of quadratic utility when probability of decreasing utility is large

In finance and specifically portfolio theory, a popular utility function is quadratic utility $$ u(x)=x-\frac{\lambda}{2}(x-\mu_X)^2 $$ where $x$ is wealth and $\lambda$ is the parameter of risk ...
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Prove that the portfolio that maximizes utility lies on the efficient frontier

When maximizing mean-variance utility in a portfolio optimization framework $max \{R - \lambda \sigma ^2\}$ where R is portfolio return, $\lambda$ is a risk aversion parameter, and $\sigma^2$ is ...
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Compute Utility From Portfolio Holdings Over Time

I have a dataset comprising daily stock holdings for individual investors over a one year-period. I only know about the individuals' investment in stocks. I have no information on any other wealth of ...
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Risk-aversion parameters estimation in utility functions

Are there any "typical" risk-aversion parameters for power utility function and exponential utility function? Once I've seen an articel, in which author stated that for extremely risky person gamma in ...
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Utility-based portfolio optimization

I think I can't get the idea of optimization based on utility. For some reasons, I should choose one of several common utility functions (exponential, isoelastic function and some others). Obviously, ...
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Risk neutrality coherence with risk aversion

I haven't been able to find an understandable explanation why the risk neutrality is coherent with the risk aversion implication of the expected utility hypothesis. I can see that when using the risk ...
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5 votes
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Merton's portfolio problem with constraints

Suppose the investor can invest in a Black-Scholes market with one risky asset $S$ with drift $\alpha$ and volatility $\sigma$ and a riskless asset $B$ with a riskless rate of return $r$, and the ...
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3 votes
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Utility Maximization on a finite Probability Space. Possible mistakes in a paper?

I am currently reading this paper on utility maximization in a financial market model. On page 5 the author starts with the case of a finite probability space and on page 19 he considers the ...
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Is there a relationship between Risk Neutral Pricing framework and Nash Equilibria?

Based on the Fundamental Theorem of Asset Pricing, the risk neutral price of a contingent claim on an asset in a liquid, arbitrage free market can be determined by switching to an equivalent $Q-$ ...
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Is positive skewness preferences rational or irrational?

Is positive skewness preference rational or irrational? I have a great trouble understanding why investors should prefer positive skewness over negative one. Sometimes it is argued that preference ...
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Hedger's utility function associated with minimizing value at risk

What must be the general shape of a hedger's utility function if the hedger is minimizing value at risk? What is a simple example of such a utility function?
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6 votes
1 answer
709 views

What is the relation between Relative Risk Aversion and Market Price of Risk

If we assume that the preferences of investors in a market aggregate to display the following utility function $$u(W)=\dfrac{1}{1-\gamma}W^{1-\gamma},\quad \gamma>0,\quad \gamma\neq1$$ then from$...
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Does CRRA-utility imply higher risk-aversion for lower wealth?

Consider the utility function $u(W)=\dfrac{1}{1-\gamma}W^{1-\gamma}$, where $\gamma=0.5$ Since this function will exhibit decreasing marginal utility of wealth, is it correct to say that for any ...
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Debreu's Representation Theorem proof

In microeconomy this theorem states that : given a consumption set $X\subseteq\mathbb{R}^n$, if the preference relation $\succcurlyeq$ is complete, transitive and continuous there exist a utility ...
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1 answer
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Finding parameters of an utility function in a market making strategy to apply it in practice

I am reading this paper below about optimal bid-ask spread in a market making strategy. It finds an approximation for optimal solution, but I cannot understand how it's practice to set the parameters ...
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How to derive Expected utility approximation with power function

My question concern how to derive the expected utility of a power function in the following model: I have two normally distributed risky assets X and Y and a risk-free asset B, for which : rA = 0.5y ...
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1 vote
0 answers
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optimizing the expected utility

The market consist of one single stock and call options with different strike price based on the given stock.Suppose the market believes the stock follows the following GBM:$$dS_t=\mu S_tdt+\sigma ...
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3 votes
1 answer
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Why is this utility function not picking up its penalty?

I was reading this seminal paper by Infanger. On page 40, Figure 11. was quite interesting. In particular I was interested in the top one, 19 Years and I wanted to reproduce this plot. To give some ...
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2 votes
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Why is utility concave?

I have read that the utility function is usually concave. I assume this requirement arises in order to meet the diversification effect:$$f(\lambda_1c_1+\lambda_2c_2)\ge \lambda_2 f(c_1)+\lambda_2f(c_2)...
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3 votes
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Martingale method for utility maximization - is the optimal strategy also a martingale?

The Martingale Method for utility maximization (seen in e.g. Björk's book) is based on separating the optimization problem $E^\mathbb{P}[U(X_T)]$ over a class of admissible strategies into the static ...
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3 votes
2 answers
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Why are some utility functions widely used?

There are some von-Neumann utility functions that I come across quite often in different articles / books like: $ U(x)=\ln(x)$, $U(x)= \frac {1}{\gamma}x^\gamma$ with $\gamma <1$ and $U(x)=\frac {1-...
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1 answer
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Utility-optimal leverage with costs

Say I have a portfolio, $X_t$, using a leverage of $f$, such that the dynamics are given by \begin{equation} dX_t = \mu f X_t dt + \sigma f X_t dW_t \end{equation} I want to optimize the expected ...
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2 votes
1 answer
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Are Insurance and Risk premium totally different?

I've been studying various aspects of utility function and I came across the definition of risk premium and insurance, which are mathematically very different from each other. In the book "Theory of ...
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Examples of risk-seeking utility functions?

In the past, most literature assumed a risk-averse investor to model utility preferences. This includes the CRRA and CARA utility functions. In recent papers, researchers state that investors may be ...
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Utility function for avoiding investment

An investor has initial wealth $30000$ and utility function $\ln{x}$. He is planning to invest in a project where he has $60%$ chance of gaining $\alpha%$ and $40%$ chance of losing $\beta%$. Express ...
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Differentiating risk aversion based on utility theory

A utility function $U$ whose corresponding relative risk aversion function is a linear, increasing function satisfies the differential equation $$-x\frac{U''(x)}{U'(x)}=ax+b$$ for some constants $a&...
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3 votes
1 answer
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List of risk-averse utility functions

In the context of optimal portfolio allocation, I am looking for a (possibly exhaustive) list of risk-averse utility functions verifying part of the so-called Inada conditions. Essentially, I am ...
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1 answer
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Explain Four Basic Axioms of Maximising Expected Utility

I begin learn PRM , Someone help me understand Four Basic Axioms of Maximising Expected Utility most intuitive way .Thank you very much
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2 votes
1 answer
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example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$

I have a market with safe rate r and risky asset S $$ \frac{dS_t}{S_t}=(r+Y_t)dt+\sigma dW_t \quad \quad (1)$$ $$ dY_t = - \lambda Y_t +dB_t \quad \quad (2)$$ where W, B are Brownian Motions with ...
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1 answer
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optimal strategy problem (using Jensen's inequality)

I have a strategy in Samuelson model with zero safe rate defined as $$Z_t^{\Pi}=\frac{X_t^{\Pi}}{X_t^{\rho}} \quad \quad (1)$$ where $$\frac{dX_t^{\Pi}}{X_t^{\Pi}} = \mu \pi dt + \sigma \pi \ dW_t \...
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