Questions tagged [utility-theory]
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67
questions
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20 views
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 ...
1
vote
1answer
70 views
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 ...
0
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0answers
73 views
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:
$$
\begin{align}
&\max_{w}\ E[u(W_0(1+w^TR))]...
0
votes
1answer
106 views
How to evaluate the following integral?
I stumbled across an expression and I wonder how to evaluate this: $-\int_ {0} ^ {+\infty} {v(x)} dw^{+} (1-p(x))$ where $v(x)$ is some utility function and $w(p(x)) $ is a decision weighting function,...
0
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0answers
41 views
Expected value of stepwise function of Prospect Theory as improper integral
I am looking for a sufficiently precise solution algorithm to evaluate the following integral:
$$\int_ {-\infty} ^ {\infty} {v(x)} {f(x)} dx$$
where $v(x)$ is the Prospect Theory value function and $f(...
0
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0answers
14 views
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 ...
3
votes
2answers
213 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 ...
0
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0answers
42 views
Iso-expected-value locus
I was considering the following indifference curve/Bernoulli utility function u:
For $ p_1 x_1 + p_2 x_2 = w $ it says, that this is an iso-expected-value locus. Can somebody explain what an iso-...
1
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0answers
68 views
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 ...
3
votes
2answers
293 views
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 ...
3
votes
1answer
661 views
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 ...
3
votes
1answer
214 views
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 ...
3
votes
1answer
211 views
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 $\...
1
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0answers
69 views
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 ...
0
votes
1answer
45 views
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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1answer
287 views
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 ...
0
votes
1answer
137 views
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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0answers
31 views
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 ...
2
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0answers
130 views
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 ...
2
votes
0answers
128 views
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, ...
2
votes
1answer
90 views
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 ...
5
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0answers
281 views
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 ...
3
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0answers
39 views
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 ...
24
votes
0answers
465 views
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-$ ...
5
votes
1answer
308 views
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 ...
1
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0answers
36 views
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?
6
votes
1answer
687 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$$
then from$...
2
votes
1answer
279 views
Does CRRA-utility imply higher risk-aversion for lower wealth?
Consider the utility function $u(W)=\dfrac{1}{1-\gamma}W^{1-\gamma}$, where $\gamma=0.5$
Since this function will exhibit decreasing marginal utility of wealth, is it correct to say that for any ...
1
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0answers
652 views
Debreu's Representation Theorem proof
In microeconomy this theorem states that : given a consumption set $X\subseteq\mathbb{R}^n$, if the preference relation $\succcurlyeq$ is complete, transitive and continuous there exist a utility ...
7
votes
1answer
1k views
Finding parameters of an utility function in a market making strategy to apply it in practice
I am reading this paper below about optimal bid-ask spread in a market making strategy.
It finds an approximation for optimal solution, but I cannot understand how it's practice to set the parameters ...
1
vote
0answers
94 views
How to derive Expected utility approximation with power function
My question concern how to derive the expected utility of a power function in the following model:
I have two normally distributed risky assets X and Y and a risk-free asset B, for which :
rA = 0.5y ...
1
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0answers
142 views
optimizing the expected utility
The market consist of one single stock and call options with different strike price based on the given stock.Suppose the market believes the stock follows the following GBM:$$dS_t=\mu S_tdt+\sigma ...
3
votes
1answer
217 views
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 ...
2
votes
0answers
93 views
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)...
3
votes
0answers
179 views
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 ...
3
votes
2answers
653 views
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-...
2
votes
1answer
106 views
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 ...
2
votes
1answer
881 views
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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0answers
716 views
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 ...
0
votes
1answer
137 views
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 ...
1
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1answer
121 views
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$$
for some constants $a&...
3
votes
1answer
212 views
List of risk-averse utility functions
In the context of optimal portfolio allocation, I am looking for a (possibly exhaustive) list of risk-averse utility functions verifying part of the so-called Inada conditions.
Essentially, I am ...
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1answer
1k views
Explain Four Basic Axioms of Maximising Expected Utility
I begin learn PRM , Someone help me understand Four Basic Axioms of Maximising Expected Utility most intuitive way .Thank you very much
2
votes
1answer
257 views
example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$
I have a market with safe rate r and risky asset S
$$ \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)$$
where W, B are Brownian Motions with ...
1
vote
1answer
89 views
optimal strategy problem (using Jensen's inequality)
I have a strategy in Samuelson model with zero safe rate defined as $$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 \...
7
votes
3answers
11k views
Why do we assume quadratic utility in portfolio theory?
In my text (Investments by BKM), the investor's mean-variance utility (given as $U = E[R] - \frac12A\sigma^2$) is stated to be the objective function we wish to maximize. Upon further digging, it ...
8
votes
2answers
635 views
Utility Theory and portfolio optimization - Proof of a lemma
I have a question on the following problem from chapter 9 of D. Luenberger, Investment Science, International Edition:
(Portfolio Optimization)
Suppose an investor has utility function $U$. There are ...
2
votes
1answer
2k views
Utility Theory - Certainty equivalent approximation formula derivation
I have a question on an exercise from chapter 9 of D. Luenberger, Investment Science, International Edition, where I suspect there may be a typo.
Exercise 8 (Certainty approximation)
There is a ...
2
votes
1answer
3k views
Utility Theory - How to show that this exponential utility function is wealth-independent?
I have a question on the following exercise from chapter 9 of D. Luenberger, Investment Science, International Edition.
Exercise 2 (Wealth Independence)
Suppose an investor has exponential utility ...
1
vote
2answers
180 views
How to arrive at expectation of negative utility function via Taylor series expansion
I'm attempting to follow an author's steps in an argument and having trouble seeing how Taylor series expansion can be applied to give the stated result. The scenario is as follows.
The mid price ...
