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Risk-free rate is that you get for letting someone else use your money in a riskless manner. Suppose we live in a world where there is no risk whatsoever. In particular, if you lend someone \$100 there is 100% certainty that he will pay you back in a year. Before the pay date, he can do whatever he wants with your$100, while you have no access to it. Even ...

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Your equations are for cum-dividend prices, i.e. the price plus dividend today. The paper refers to ex-dividend prices. The correct two equations for investor group $a=1$ are \begin{align} p^1(0) =&\ \frac{3}{4} \left(\frac{1}{2}p^1(0) + \frac{1}{2}(1+p^1(1))\right) \\ p^1(1) =&\ \frac{3}{4} \left(\frac{2}{3}p^1(0) + \frac{1}{3}(1+p^1(1))\right) ...

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If you want to learn more about price pressure, you should look after market impact of metaorders, which is a more adequate term. Because of the microstructure (i.e. the mix of orderbboks dynamics, trading rules, participants behaviours and habits, etc), the more you buy or sell, the more you influence the price an unfavorable way (for your trades). Just ...

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Beware, oversimplification ahead! (This means that the following is technically not correct, in fact it is false! But: It gives an intuition what is going on!) If you toss a coin and calculate heads as $-1$ and tails as $1$ you get a mean of $0$ with a variance of $1$. When you add up multiple coin tosses, i.e. create a random process $dz(t)$, the mean ...

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The risk free rate is important and the reason for the inclusion and consideration of the risk free rate is that investors do not get compensated for not taking on risk. Now, we can argue whether the risk free rate truly provides risk free returns (we all should know that it does not, but ...) but it is important in the context of pricing risky assets that ...

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Portfolio returns are analyzed to account for risk factors only to determine what the risk factor contributed to the returns, was it the underlying assets or the skill of the portfolio manager. Fama French model explains the returns in terms of principal component such SMB and HML besides the market related returns from CAPM. These links have more detais ...

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There are a few reasons to use factor models. Most importantly, stocks tend to move together. Stated alternately, the first principal component of the securities in a domestic market tends to explain a large share of the variance. If you're concerned with multiple securities (as in portfolio optimization), then you have to account for this or you will ...

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I can understand your concerns, but I think you are expecting too much from these theories. We cannot explain aggregate behavior from first principle based on a sound theory of individual decisions under uncertainty and I personally doubt that there will ever be such a Grand Unification in economics. Consumption-based asset pricing models are more related ...

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Another important reason for using risk-adjusted returns is to disentangle "skill" from "risk-taking". Think of a equation for a fund's performance like: $r_{i,t}-r_f=\alpha_i+\epsilon_{i,t}$ where $\alpha_i$ gives you the average excess return of fund $i$. Alpha is often interpreted as measure of a managers' skill in timing the market and selecting ...

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Let's consider a random process $X$. If $X$ is an adapted process, then we know, without any uncertainty, what its value is at the present time. This idea is formalized with measure theory. For $X$ to be a martingale, it needs to have the following property: at any given time, our best estimate of the value at some point in the future (i.e. forecast), is ...

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In my opinion, risk free rate is not necessarily positive and not so important to pricing theory. It happened to be positive in most cases, but imagine a planet using Uranium-235 instead of gold as the money and unknowingly suffers from a shrinking population, likely the risk free rate is negative. Below are what I regard as important in pricing theory, ...

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My non-rigorous answer: The future is uncertain. Even if there is no financial risk to investing in the "risk free" asset there is personal risk. For example, I could get hit by a car and die. Even if I survive till the moment that I liquidate my investment I will have less time left in my life to enjoy it. I need to be compensated for giving up this ...

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Pretty good explanation is in Schweser CFA Study Notes for CFA level III. Books 3 and 5, at least from 2009, if I remember right. See also Tsay R.S. Analysis of Financial Time Series (Wiley Series in Probability and Statistics). // 2010. - good example with implementation in R.

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I suggest to distinguish three things: Why risk-free rate? What defines its magnitude and sign? And what is its role in specific economic models? "Why risk-free rate" is very simple to answer: Because you want to compare cash at different points in time. One assumes that the difference between cash at time $t_1$ i.e. $C(t_1)$ and at time $t_2$ is $C(t_1) - ... 1 Edited: The risk free rate is positive because the factors of production, and the perception of time (from an individual's perspective) are limited. Limited supply of desirable goods such as houses (or limited capacity to make them from land, labour and capital), gives them a positive value in society (versus say air, which is in almost unlimited supply, ... 1 A risk free rate is the return rate from investing in an asset that has the lowest risk found in the market. It is a naming convention. The least risky of all returns is labelled as 'risk free' for the purpose of various models and resulting discussions. Another parallel answer is that you must understand what financial risk is in the first place. It is, ... 1 Ashwath Damodaran explains risk free rate in his 2008 paper, "What is risk free rate? A Search for the Basic Building Block". He has explained risk free rate from the perspective of investment. If we invest in an risk free asset then we expect guaranteed return. "An investment that delivers the same return, no matter what the scenario, should be ... 1 It is a decision problem, it is always a decision problem... The most basic decision problem is a lottery that gives you X (X > 0) for a chance of %Y and nothing for %(100-Y). Someone comes and offers you to buy your ticket for Z (0 < Z < X, otherwise it is trivial). What would you do? If Z is close to X, you need to be more risk-willing (or seeking) ... 1 That is true. Utility would not be concave anymore under prospect theory (only for gains), but convex for losses, which is evidence against CAPM. CAPM is valid either : -if the utility function is quadratic (which is nonsense in terms of economic interpretation, and in general, Von Neumann- Morgenstern utility describes poorly reality and should be ... 1 The general problem of the investor is: $$\max_{w\in[0,1]^n} U(\mu_p(w),\sigma_p(w))\quad s.t. \sum_{i=1}^n w_i=1$$ where$w$being the portfolio weights, and$U$utility function of portfolio risk$\sigma_p$and return$\mu_p$. CAPM assumes investors with concave utility function$U=\mu_p-\frac{1}{2}\sigma_p^2$, from which then follows that all ... 1 You can use the "Merton Jump Diffusion Model" to price European Options with jumps. The other points of your question are rather of practical relevance only. The negative drift of the underlying is usually not important, because the pricing goes under the riskneutral measure$Q$. 1 Intuitively, because of the central limit theorem: wiener process is a limit of a random walk, and after n steps a random walk moves away from the origin by ~$\sqrt{n}$Edit: here is a complete answer. First the formula for the sum. The trick is the following simple observation: if$X_1,.. X_n$are independent zero mean, then$E(\sum X_i)^2 = ...

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For a stochastic process $\left(X_{t}\right)$ to be adopted to a filtration $\left(\mathcal{F}_{t}\right)_{t\in T}$ the random variable $X_{t}$ must be $\mathcal{F}_{t}$-measurable for each $t\in T$. A stochastic process is a collection of random variables $X_{t}$, indexed by some set $T$. Each random variable is a mapping from a probability space into a ...

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A random process that is adapted to a filtration is measurable (ie X_t is F_t-measurable) but not necessarily a martingale. X_t is a martingale if E(X_t | F_s) = X_s for s < t.

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According to the literature in market microstructure, the price pressure is defined as "the change in price when large quantities of a security are traded". Here you can find an example of how price pressure influences the bond market and in which the authors provide a complete definition of the phaenomenon and the relative problem of the information ...

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Note that, for a smooth function and constant a $$f(S_t) = f(a) + f'(a) (S_t-a) + \int_a^{\infty}(S_t-x)^+f^{''}(x)dx + \int_{0}^a(x - S_t)^+f^{''}(x)dx.$$ Then, the payoff $1/S_t$ can be approximately hedged by call and put options: \frac{1}{S_t} = \frac{1}{a} -\frac{1}{a^2}(S_t-a)+ 2\bigg[\int_a^{\infty}\frac{(S_t-x)^+}{x^3}dx + \int_{0}^a\frac{(x - ...

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Consumption-based asset pricing theories are about representative agents, not necessarily about traders and investors in financial institutions. The agents are assumed to follow behaviors based on how people generally would decide whether to invest and how much to invest. The idea is that the average person in the economy does not invest for the joy of ...

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