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### EM for conditional Gaussian model

Let $$X_1\sim N(\mu_{X_1},\sigma_{X_2}^2)$$ $$X_2\sim N(\mu_{X_2}, \sigma_{X_2}^2)$$ where $\mu_{X_2}=c+aX_1$. Also, I have data $D$ (with missing values on $X_1,X_2$). How can I update/estimate the ...
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### $\mathbb{P}$ and $\mathbb{Q}$ probability measure/distribution interpretations

I'm trying to understand probability distributions implied from market prices and was reading through this reference explaining the interpretation of $N(d_1)$ and $N(d_2)$ in the log-normal vol Black-...
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### Correlation of a lognormal asset and a normal asset

So if i want to calcualte the correlation between a pair of assets, my intuition is that i should calculate whatever correlation i plan on using; When we look at correlation, it's normally the ...
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### Is it possible that under Black-Scholes: $\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$

I have a slide on which there is written that under Black-Scholes model: $$\ln S_{T} \sim N \left ( \ln S_t - \frac{1}{2}\sigma^2(T-t), \sigma^2(T-t) \right )$$ Now, here there is a good explanation ...
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I got little bit lost in the formulas. Assume to have two random variables distributed exponentially $X_i \sim Exp(\lambda_i)$ and $X_j \sim Exp(\lambda_j)$. Thus, the distribution functions are $... 0answers 18 views ### Correlation coefficient for different functions Can somebody explain the correlation coefficient values for the following set of functions$x and $y? -Independent Functions -Asymptotically Dependent Functions -Marginal Functions -Normally ... 0answers 56 views ### Normal Black&Schole model for swaptions isn't working properly I just wrote two functions in Matlab which calculates the swaption prices based on the Lognormal model and on the Normal model, although I have the idea that the Normal model is wrong because the ... 1answer 94 views ### Distribution of proportional bid-ask-spreads I already asked this yesterday at "Economics Stack Exchange" but think this question might be better suited here. In the meantime i really tried to solve it by myself, but couldn't find anything what ... 1answer 41 views ### Creating the histogram for the distribution of the portfolio returns Given log returns for some stocks$A$and$B$, which are the constituents of our hypothetical portfolio in equal weights, how does one actually come up with a distribution of the log returns of the ... 2answers 63 views ### Problem with obtaining densities For my research I need to obtain a series of densities, however, I am encountering some problems. The first problem is perhaps very simple, but the answer eludes me. Let's say I have an observation ... 5answers 299 views ### Kurtosis in asset logarithmic returns Assets such as stocks usually display kurtosis in their logarithmic returns. However, their logarithmic returns in a time interval$n$are the sum of smaller logarithmic returns in$1/ntime ... 1answer 205 views ### Why does Bloomberg's HRH test the simple returns for normality? On a Bloomberg terminal, it is possible to use the HRH (Historical Return Histogram) function on individual assets. It basically generates a histogram of the (simple) returns and overlays them with a ... 1answer 51 views ### Should earnings be modelled normally or lognormally? I am having difficulty deciding whether a company's earnings should be modelled normally or lognormally. If we consider two arguments: (i) The earnings of a company are the returns on the assets of ... 2answers 136 views ### The Distribution of Future Stock Price In Hull, we are presented that $$\frac{\Delta S}{S_{0}}=\mu \Delta t+\sigma\sqrt{\Delta t}\cdot \varepsilon.$$ Following some algebra, \begin{align*} \frac{\Delta S}{S_{0}} &=\mu \Delta t+\... 2answers 390 views ### Confidence Intervals of Stock Following a Geometric Brownian Motion In preparation for my Options, Future's and Risk Management examination next week, I have been presented with a series of questions and their answers. Unfortunately, my lecturer, one of the less ... 1answer 96 views ### What is the distribution assumption of the black scholes model As per wikipedia the Black Scholes assumption is: (... 0answers 199 views ### Monte Carlo simulation returns not normal distributed I am generating 100,000 paths of SPX out to 1 year using Euler discretization. I look at how S is distributed for 100,000 paths at the 1 year point and I find it is lognormally distributed. I look at ... 1answer 277 views ### Portfolio choice problem of a CARA investor with n risky assets Ok, I am working on a problem that consists of the following: I am looking to solve the portfolio choice optimization problem (maximizing utility with a known utility function) in the case where all ... 1answer 185 views ### Probability distribution and Stock Price Movement [closed] How can we use normal distribution for finding the probability of a stock price offer where current price offer depends upon the last price offer. The price offer on some day can go 10% above (at the ... 1answer 76 views ### Is the value also log-normally distributed? My book assumes many times that log(1+R) is normally distributed, so R is log-normal. But does this also mean that the value process is log-normal? Since V=V_0(1+R)\rightarrow V/V_0=1+R, and since ... 1answer 72 views ### Variability in the Expected Shortfall estimator Are there any results for calculating the variability in the Expected Shortfall measure. I am looking for Large sample confidence intervals under Normality for Expected Shortfall or calculation of ... 1answer 159 views ### Can Standardized unexpected earnings be considered a Z-score According to this wikipedia: http://en.wikipedia.org/wiki/Earnings_surprise, the SUE score is a "standardized" difference between reported earnings and expected earnings. Therefore, can the SUE score ... 2answers 366 views ### Normally Distributed Returns Become Leptokurtic Due to Compounding I was running a bunch of simple simulations in excel the other day in excel. Using the NORM.INV(RAND(),0,1) to simulate daily stock returns I noticed that the more compounded the returns, ie, the more ... 2answers 2k views ### An alternative to the Gaussian distribution to describe/fit market stock returns After the financial crisis in 2008, many people (including me) don't really believe that stock returns can be described in terms of the normal distribution (Gaussian distribution). But besides the ... 0answers 97 views ### Stock Price Question Can anyone show me how to answer this please? A stock has beta of 2.0 and stock specific daily volatility of 0.04. Suppose that yesterday’s closing price was 95 and today the market goes up by 3%. ... 0answers 138 views ### BS Implied Volatility under Normal returns If I use theoretical prices under a normal valuation model, and I estimate their implied volatility using BLACK SCHOLES implied volatility, do I'll get corresponding log normal volatility? 1answer 378 views ### Interpretation of cross-correlation matrix when one sample distribution is not normal I am looking at the variance of (log) price changes in securities vs. the amount of social media discussion about them. I'm not interested in building a model. I'm just looking to see if there is a ... 1answer 326 views ### What is the correct Stutzer index and Sharpe ratio relation, assuming a normal returns distribution? Assuming the returns distribution is normal, then there is a relation between Stutzer index and Sharpe ratio. However, I found in the following paper 2 different equation: Paper I (page 10-11)‎ ... 1answer 296 views ### How to design back-testing (validation) for such modified Vasicek model? Consider a classical Black Scholes model ,\frac{dS}{S} = \mu dt + \sigma dW$, where$dW$is a Brownian motion, that$W(t_1) - W(t_0) \sim N(0, t_1 - t_0)$. The back-testing strategy is straight-... 1answer 79 views ### Creditworthiness indicator for copula one-factor model In this paper in equation 15 on page 261 dealing with one factor copula model, one is using creditworthiness indicator as one of a variables. It is defined as Y_c = \sqrt{\rho_c} Z +... 2answers 340 views ### Transformation to reduce standard deviation without changing median Consider some negative skew and high kurtosis return time-series$X_t$. I do not know the functional form of the pdf of$X_t$and have about 150,000 data points. Suppose that I was to create an ... 5answers 619 views ### In Black-Scholes, why is$\log{\frac{S_{t+\triangle t}}{S_t}} \sim \phi{((\mu - \frac{1}{2}\sigma^2)\triangle t, \sigma^2 \triangle t)}$? Namely, I dont understand why the mean is$(\mu - \frac{1}{2}\sigma^2)\triangle t$and not just$\mu \triangle t\$. I am aware that it is supposed to represent a lognormal distribution, but I guess I'm ...
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Why does dividing daily returns by daily range eliminates fat tails and results in an (almost) gaussian distribution? And how could that distribution be exploited to enter trades?
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### normalized accumulation distribution

I am looking for a way to take an accumulation/distribution indicator and normalize it so I can compare a bunch of stocks with stock prices that have no relationship with each other. EDIT: This ...
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### a simpler test for normality given skewness, kurtosis and autocorrelation and size of time series

I typically do a JB (Jarque Bera) test and DW (Durbin Watson) tests for check for normality given skewness, kurtosis and autocorrelation of the data. However this requires a CHI distribution table ...
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### Monte carlo portfolio risk simulation

My objective is to show the distribution of a portfolio's expected utilities via random sampling. The utility function has two random components. The first component is an expected return vector ...
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### Whats the equation to calculate the area under the curve of a normal distribution, given an upper and lower standard deviation?

Lets say I want to find out the area under the graph of normal distribution curve, between X1=standard deviation of -0.5 and X2 = standard deviation of 0.5. Is there a formula for this? Case study: ...