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9 votes
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On a time integral of Brownian motion up to the hitting time

Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\...
FoolAlex's user avatar
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9 votes
0 answers
3k views

Arbitrage free smoothing of volatility smile - cubic spline - implementation procedure

I am studying the paper Arbitrage-Free Smoothing of the Implied Volatility Surface, from Matthias R. Fengler (https://core.ac.uk/download/pdf/6978470.pdf). The problem I want to solve is much simpler ...
AnUser's user avatar
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5 votes
0 answers
1k views

Fitting Student t-distributions to log-returns

It seems that some tail-risk centric groups are bent on using Paretian and t-distributions to account for tail risk when fitting log-returns. It has been observed, however, that with and without ...
user avatar
4 votes
0 answers
158 views

Alternative Method for Determining Option-Implied pdf

As I am refining a pricing model to incorporate skew, and not just ATM volatilities, I need to create random realizations of the underlying consistent with the skew-implied pdf. When searching, one ...
ZRH's user avatar
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3 votes
0 answers
331 views

Large deviations theory in finance

In probability theory, the theory of large deviations concerns the asymptotic behavior of remote tails of sequences of probability distributions. A related post says: Large deviations theory is ...
develarist's user avatar
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3 votes
0 answers
341 views

Example how to model stock price with Pareto distribution according to Mandelbrot and Taleb

There's a paper by B. Mandelbrot and N. Taleb Mild vs Wild Randomness that says that Pareto distributions is a better fit for modelling price changes. ...
Alex Craft's user avatar
3 votes
0 answers
145 views

What is the relation between return volatility and return rank volatility, and how can I control the latter?

I have no experience in finance, but I've been playing around with a virtual portfolio. I'm trying to control the "rank volatility" distribution - that is, the volatility of a stock's daily rank in ...
Elliot JJ's user avatar
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2 votes
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79 views

non gaussian distributions with higher moments and time scaling properties?

If we assume a portfolio comprised of n asset classes, whose log returns can be modeled with a distribution. I am interested in finding a distribution that: incorporates higher moments (skewness and ...
torino's user avatar
  • 21
2 votes
0 answers
135 views

The distribution of the jump diffusion process

In the Merton jump diffusion model the process of the share price can be expressed as $$S_{t}=S_{0}\cdot\exp\left\{ X_{t}\right\} ,$$ where $$X_{t}=\mu t+\sigma W_{t}+\sum_{i=1}^{N_{t}}Y_{i}.$$ Here $...
Kapes Mate's user avatar
2 votes
0 answers
938 views

law of absolute of max of brownian motion

What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion? Any references for properties of this process?
user40929's user avatar
  • 131
2 votes
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Convolution of generalized hyperbolic distribution

I have a question concerning the convolution of generalized hyperbolic distributions. Proposition 6.13 of McNeil, Embrechts, Frey states the following: If $X$ has a $d$-dimensional generalized ...
Cettt's user avatar
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2 votes
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$\int_{0}^1W_x(t)dW_y(t)/(\int_{0}^1W_x^2(t)dt)^{1/2}$ normally-distributed?

I have came across the following stochastic integrals: $$\frac{\int_{0}^1W_x(t)dW_y(t)}{(\int_{0}^1W_x^2(t)dt)^{1/2}}$$ which was claimed to be standard normally distributed ($W_x$ and $W_y$ are ...
Tamas's user avatar
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2 votes
0 answers
152 views

Bivariate risk neutral distribution through copula

I want to build a bivariate risk-neutral distribution from two liquid assets (A and B) through the use of a copula. As A and B are liquid, I have the marginal distributions from the market. All I have ...
Pierre's user avatar
  • 143
2 votes
0 answers
221 views

Simulating t-distributed returns by calibrating degrees of freedom $\nu$ from variance or kurtosis

A slight twist (I hope) on the familiar problem of simulating log returns from a t-distribution. My two questions concern calibration to sample data. First, one can infer the degrees of freedom, $\nu$...
LukeG's user avatar
  • 21
2 votes
0 answers
159 views

Gaussian Copula with t margins

I am trying to fit a Gaussian Copula with t margins to my data (log returns of two stocks). It has already worked for a Gaussian Copula with normal margins with: normcopula_dist = mvdc(copula=...
mrsdalloway's user avatar
2 votes
0 answers
109 views

How to choose a window for curve fitting and prediction?

I am using Pareto distribution to fit a serie of survival rates (with least square). My ultimate goal is to use this fitting curve for prediction. Thus I would mainly focus on the tail of the ...
NewLong's user avatar
  • 121
1 vote
0 answers
45 views

Assymetric Rate Distribution

The pandemic has disavowed any notion of nominal rate distributions to being truncated at 0%. However, if Central Banks at Debtor nations are conflicted in that they are incented to suppress interest ...
AlRacoon's user avatar
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1 vote
0 answers
41 views

How can I find the distribution function of the following random variables?

Suppose that the random variables $Z_i$ are defined as follows: \begin{equation} Z_i = D(0, t_i)(R_{i-1} +c)\Delta N, \end{equation} where $D(0, t_i)= \exp\{-\int_{0}^{t_i} r_u du\}$ for which $r_u$ ...
user53249's user avatar
  • 419
1 vote
0 answers
103 views

Full Copula View using Meucci's Full Flexible View

I'm currently setting up an "Investment Framework" that should allow the following steps: Investment Committee (IC) has to decide on probabilities for 4 different market states. I have historical ...
R. Steigmeier's user avatar
1 vote
0 answers
49 views

Skewness and kurtosis measures when full distribution is not available

I have asked this question here, but did not get any answer. I was wondering if anybody knows a method of deriving skewness and kurtosis measures from different quantiles, mean, and/or variance. I do ...
AK88's user avatar
  • 1,850
1 vote
0 answers
55 views

relationship between option vol and option payoff

Has anyone thought of the relationship between the option vol and distribution of option payoff? for example, I have 1000 paths of simulated underlying prices, keeping all inputs the same but only ...
option_q's user avatar
1 vote
0 answers
150 views

Pricing call option on bond under CIR model by simulating noncentral chi square distribution

In the original paper of CIR model, there is a pricing formula about call option on bond $$ \begin{array}{l}{C(r, t, T ; s, K)} \\ {=P(r, t, s) \chi^{2}\left(2 r^{*}[\phi+\psi+B(T, s)] ; \frac{4 \...
Tak wa Ng's user avatar
1 vote
0 answers
34 views

Sample distribution of cross-sectional statistics of returns

Currently doing an application of VaR on sample of industry portfolios in the US. I have a matrix of $n$ industry portfolios with $m$ time-series observations. I calculate cross-sectionally (for each ...
alexbougias's user avatar
  • 1,426
1 vote
0 answers
42 views

Finding the distribution and moments of returns with GARCH models (in R if possible)

I understand the GARCH type models and I know how to fit a model to a time series. But, there is a paper which calculates the moments of the distribution of returns (Variance, Skewness, and Kurtosis) ...
Novic's user avatar
  • 155
1 vote
0 answers
106 views

Probability of outlier events for laplace distribution

I've read that the laplace distribution is better for forecasting purposes than the normal distribution due to it better accounting for fat tails. However, when I run the numbers in matlab, laplace ...
QFqs's user avatar
  • 125
1 vote
0 answers
53 views

Bootstrapping to Judge the Fit of a Sampled Return Distribution

Consider the following: I have sampled yearly stock returns from a specified distribution. What I want to do is compare how well my sampled distribution fits the empirical distribution of yearly ...
user33475's user avatar
  • 147
1 vote
0 answers
20 views

Question about Paul Kupiec's "concentrated Bond loss rate distribution"

I wonder if anyone here has read the following paper by Paul Kupiec in which he approximates a loss rate distribution for a portfolio composed of (possibly) concentrated bond positions. https://www....
RDA's user avatar
  • 36
1 vote
0 answers
342 views

Probability Integral Transform: Standardisation

I've been applying the probability integral transform as shown here to standardise date for input into a neural network: https://math.stackexchange.com/questions/592076/mapping-cdfs-to-each-other?...
Bazman's user avatar
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1 vote
0 answers
331 views

Expected shortfall of stable distribution by Stoyanov

I've been working on calculating parametric ES assuming the returns follow Paretian stable law. Given the four parameters - $\alpha, \beta,\sigma,\mu$- Stoyanov introduces closed form solution of the ...
Jan Sila's user avatar
  • 732
1 vote
0 answers
182 views

Interpretation of Skew and Kurtoisis - strategy backtesting

I am working on my dissertation and i would like to provide a nice interpretation of two tables which i will present below. I have 10 portfolio buckets which i sort on 6 different attributes. One of ...
Alex Bădoi's user avatar
1 vote
0 answers
309 views

student-t asset path

I am trying to simulate an asset path based on a t-distribution. I found a lot of ressources and the fact that it will be difficult to do a path. But now I changed my Geometric Brownian Motion ...
lechim's user avatar
  • 21
1 vote
0 answers
786 views

Large deviations theory and extreme value theory

I'll enter into details of both, sooner or later, but for the moment I'm concerned about the differences (and relationships, if any) between these two theories. Can someone give me a brief, but still ...
simmy's user avatar
  • 585
1 vote
0 answers
2k views

Skewed Generalized Error Distribution's (SGED) pdf

I want to use the SGED distribution of Theodossiou for GARCH estimation, however, I am struggling to understand which is the correct pdf function of the distribution. Let me just say that the ...
Masher's user avatar
  • 491
1 vote
0 answers
86 views

Cross-sectional moments

I got a seminar topic named Forecasting risk from cross sectional moments? Could at least someone tell me what should I write about and if there is any paper that I could read. Thank you very much in ...
user17539's user avatar
1 vote
1 answer
385 views

Distribution of realized volatility for stock prices from a GBM

If you generate random stock price paths according to a GBM with daily increments, what will be the distribution of the realized volatility? Assume that the realized volatility is measured over daily ...
dm63's user avatar
  • 17.2k
0 votes
0 answers
35 views

Taking skewness into account when determining daily expected ranges

I use a method to determine daily expected ranges by combining both daily IV and daily realized vol. with different weights to get the expected range, and it worked pretty accurately. However I want ...
c.m.'s user avatar
  • 1
0 votes
0 answers
12 views

constrains of return distribution and risk return trade off

Suppose we have a portfolio $V$, we are only allowed to invest in one stock $S$, its price movement follows the geometric brownian motion, i.e. $dS=S(\mu dt+\sigma dW)$. We are allowed to choose ...
Mango's user avatar
  • 31
0 votes
0 answers
48 views

Determing "fair" implied volatilities for SPX options

I'm trying to come up with a method to calculate fair IVs for SPX options based on historical data. I can't find much information on this so here's how I've thought to do it: Determine a metric for ...
SuperCodeBrah's user avatar
0 votes
0 answers
60 views

Taking a set of normally distributed random variables as the sample space to fitting an exponential distribution

Disclaimer, this is my first question/interaction in this forum. Let's assume I have random variables that are normally distributed. Then, say I take the observations that are greater than the mean, i....
ak10's user avatar
  • 1
0 votes
1 answer
106 views

Probability Theory: Maximizing the difference between distribution functions

Given a sample of observations $X$, by changing a parameter $p$ we can divide $X$ into two subsamples $X_1$ and $X_2$ (this division is done in a non-trivial way which is nonetheless irrelevant to ...
bond-pricer's user avatar
0 votes
0 answers
48 views

How to compute the combined probability of loss for 2 time series (consisting of historical stock prices)?

May I please ask the community's support with the following problem? I have 2 time series, with approximately 1000 observations each (same number of observations for both). They represent the daily ...
mihnea_11235's user avatar
0 votes
0 answers
102 views

basic numerical integration question related to case of high positive volatility skew

is the below equation true irrespective of if that 2nd derivative turns out to be negative or >1 , (ie even if theres an arbitrage) ? the reason i ask is that i am writing a single asset montecarlo ...
Randor's user avatar
  • 796
0 votes
0 answers
572 views

Probability Distribution at each Simulation Period using Geometric Brownian Motion

I am using the equation $S_t = S_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma\epsilon\sqrt{t}} $ to simulate a financial metric at each $t$, where $t=1$ and $T=5$. Stated in plain English, I am trying to ...
Dmitriy's user avatar
  • 75
0 votes
0 answers
549 views

What should degrees of freedom $\nu$ be set to when modeling financial returns that follow the t-distribution?

The closer the t-distribution degrees of freedom ($\nu$) is to 0, the more heavy are the tails, whereas high degrees of freedom recovers the normal distribution. In finance, what value is usually used ...
develarist's user avatar
  • 3,030
0 votes
0 answers
66 views

Density of a portfolio's returns is the weighted average of asset distributions?

The expected return of a portfolio can be formulated as a weighted average of the constituent assets' returns: $$r_p = w_1 r_1 + w_2 r_2 + \dots + w_N r_N + \epsilon$$ Does it also follow that the ...
develarist's user avatar
  • 3,030
0 votes
0 answers
52 views

Degree of freedom input for Monte Carlo simulation of asset returns with multivariate t distribution

How do I calculate or estimate the degrees of freedom in order to perform a Monte Carlo simulation of asset returns with multivariate t distribution using R functions? I am able to calculate the mean ...
sjedi's user avatar
  • 25
0 votes
2 answers
308 views

Estimating distribution of rate of return

Let $f[t]$ be the price of a stock at time $t$. We can calculate the rolling rate of return of the stock in a window of length $n$ by computing: $$r[t] = \frac{f[t] - f[t-n]}{f[t-n]}$$ $r[t]$ is ...
Vivek Subramanian's user avatar
-1 votes
1 answer
577 views

Intensity of Exponential Distribution

How do I show the following: Suppose $\lambda=-\frac{S'(x)}{S(x)}$, where $S(x)=1-F(x)$ is survival probability. Show that $\lambda$ is the intensity of the exponential distribution with cdf $F(x)=1-e^...
lrh09's user avatar
  • 155
-1 votes
1 answer
183 views

Why do I get this error using ghyp-distribution function?

I want to fit multivariate GH distribution on my data, and then generate simulations for that distribution. Using the instructions given in ghyp package, I wrote following lines of code in R. ...
HasnainMamdani's user avatar
-2 votes
1 answer
126 views

How to compute Pr(S>100) when S follows Geometric Brownian Motion?

I have been trying to resolve this problem, under (b), but I cannot find the correct answer. For i=1, my ultimate answer (P=1) deviates from the correct answer (P=0.7580). Please let me know whether ...
AltTabsen's user avatar