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Simulate stock prices with Brownian motion with mu and signa based on 'normal' or continuous compounding? |
I wonder if there are any websites/resources/sample codes/papers on how to convert the American options to European options (when all else are equal). i.e. if given same underlying asset, same expiration date, same exercise price (but these are unknown). Now additionally we are given the price of an American option, ar... |
The drift in your code is:
drift = (mu - 0.5 * sigma**2) * delta_t
So I assume you are using the Geometric Brownian Motion to simulate your stock price, not just plain Brownian motion. Therefore your model is Lognormal, not Normal. Also, I assume that your time series you're downloading is **daily** closing p... |
The drift in your code is:
drift = (mu - 0.5 * sigma**2) * delta_t
So I assume you are using the Geometric Brownian Motion to simulate your stock price, not just plain Brownian motion. Therefore your model is Lognormal, not Normal. Also, I assume that your time series you're downloading is **daily** closing p... |
I understand how to derive finite difference approximations using Taylor's theorem and how to implement the final iteration. **I wonder how to improve a standard FD scheme?** For example, when solving the standard Black-Scholes equation, the following steps are often suggested
- The transformation $x_t=\ln(S_t)$ tu... |
The drift in your code is:
drift = (mu - 0.5 * sigma**2) * delta_t
So I assume you are using the Geometric Brownian Motion to simulate your stock price, not just plain Brownian motion. Therefore your model is Lognormal, not Normal. Also, I assume that your time series you're downloading is **daily** closing p... |
The drift in your code is:
drift = (mu - 0.5 * sigma**2) * delta_t
So I assume you are using the Geometric Brownian Motion to simulate your stock price, not just plain Brownian motion. Therefore your model is Lognormal, not Normal. Also, I assume that the time series that you're downloading is **daily** closi... |
I understand how to derive and implement standard finite difference schemes. **I wonder how to improve such a standard FD scheme?** For example, when solving the standard Black-Scholes equation, the following steps are often suggested
- The transformation $x_t=\ln(S_t)$ turns the Black-Scholes PDE into a PDE with c... |
I have written a simple script for modelling stock prices using Geometric Brownian Motion. The time series I am downloading are daily closing prices. My aim is to be able to change the prediction period and all other variables.
However, I am trying to understand the price prediction differences between calculating ... |
I have written a simple script for modelling stock prices using Geometric Brownian Motion. The time series I am downloading are daily adjusted closing prices. My aim is to be able to change the prediction period and all other variables.
However, I am trying to understand the price prediction differences between cal... |
Simulate stock prices with Geometric Brownian Motion motion with mu and signa based on 'normal' or continuous compounding? |
I have the following SDE (GARCH(1,1)) for the instantaneous variance:
$$ d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \psi \sigma_t^2 dW_t $$
I would like to find an expression for $IV_t = E[\int_{t-1}^t \sigma_{\tau}^2 d\tau]$.
Here is my reasoning:
$E[\int_{t-1}^t \sigma_{\tau}^2 d\tau] = \int_{t-1}^t ... |
So in the place I work, one of the traders is dealing with a cross-currency swap within a country that has really no market for that kind of product.
He wants to estimate a theoretical bid/ask, and thought perhaps of using proxys (some options were):
a) looking at ccy swaps in other countries
b) looking perhaps ... |
Another old question on this site (https://quant.stackexchange.com/questions/4589/how-to-simulate-stock-prices-with-a-geometric-brownian-motion) inspired me to ask the following question: **if we assume that regular returns could be normally distributed, doesn't that entirely invalidate the idea behind the GBM model?**... |
> I know kalman filters and have ...
If this knowledge extends to Unscented filters, UKF, you can think of the UKF being a sparse particle filter. With a UKF you have a few sigma points which are propagated forward via your model function and then after the measurement update these sigma points are updated via cova... |
Another old question on this site (https://quant.stackexchange.com/questions/4589/how-to-simulate-stock-prices-with-a-geometric-brownian-motion) inspired me to ask the following question: **if we assume that regular returns could be normally distributed, doesn't that entirely invalidate the idea behind the GBM model?**... |
# Don't solve the Black-Scholes PDE, solve the heat equation
One of the major result of mathematical finance is showing that the Black-Scholes PDE can be mapped to the heat equation. The heat equation is both mathematically nicer to handle, analyse, and computationally has much better solvers than other generic PDE ... |
# Don't solve the Black-Scholes PDE, solve the heat equation
One of the major results of mathematical finance is showing that the Black-Scholes PDE can be mapped to the heat equation. The heat equation is both mathematically nicer to handle, analyse, and computationally has much better solvers than other generic PDE... |
I am trying to use Brownian motion to predict option prices and compare the outcomes to Black and Scholes. For this purpose, I would like to calculate the average returns (mu) and volatility (sigma) of the underlying asset based on continuous compounding - hence I use log functions.
However, I think that there coul... |
How to best predict option prices using Brownian motion and compare it to the Black and Scholes model? |
Another old question on this site (https://quant.stackexchange.com/questions/4589/how-to-simulate-stock-prices-with-a-geometric-brownian-motion) inspired me to ask the following question: **if we assume that regular returns could be normally distributed, doesn't that entirely invalidate the idea behind the GBM model?**... |
I have a portofolio with 30 indexes and I want to calculate the annulised returns and volatility because I want to compare it with another portofolio with different number of indexes (but same time period)
My data are time series with 3 month frequency from 2009-12-31 to 2020-3-31.
I know that the general formula i... |
What is the Risk Neutral Measure?
I dont believe this has been answered on the internet well, with all the parts connecting.
So:
1) What is the risk neutral measure/pricing?
2) Why do we need it?
3) How we calculate the risk neutral measure or probabilities in practice?
4) What connection has risk neu... |
What is the Risk Neutral Measure? |
I am trying to use Brownian motion to predict option prices and compare the outcomes to Black and Scholes. For this purpose, I would like to calculate the average returns (mu) and volatility (sigma) of the underlying asset based on continuous compounding - hence I use log functions.
However, I think that there coul... |
I'm studying this [paper][1]. In the formulation of the theoretical setup they state:
> Our goal is to explain the differences in the cross-section of returns
$R$ for individual stocks. Let $R_{t+1, i}$ denote the return of asset
$i$ at time $t+1 .$ The fundamental no-arbitrage assumption is
equivalent to the exi... |
I have a porfolio of indexes and I built up a python model based on spearman correlation (I used a spearman and not a pearson because, after running some test on outliers and normality checks, I have some outliers and normality checks failed) to eliminate all the indexes with correlation coefficient higher th... |
You are consider two contracts: a Eurodlloar futures contract with six months to maturity, selling at 5%, settled on three-month LIBOR, marked to market every day; and a Eurodollar forward contract with contract with six months to maturity, selling at 5%, settled on three-month LIBOR
Which contract do you prefer?
... |
Disney, Ticker : DIS is triggering me, it seems that is has two different CIK's
If you search disney 10-K forms here:
https://www.sec.gov/cgi-bin/srch-edgar?text=company-name%3D%28DISNEY%29+and+form-type%3D%2810-K%29&first=2010&last=2020
It shows:
1000 Walt Disney Co [text] [html] 10-K 11/20/2019 ... |
Can someone explain me why Disney has two different CIK's? |
Another old question on this site (https://quant.stackexchange.com/questions/4589/how-to-simulate-stock-prices-with-a-geometric-brownian-motion) inspired me to ask the following question: **if we assume that regular returns could be normally distributed, doesn't that entirely invalidate the idea behind the GBM model?**... |
<sub>This question will serve as the [definitive community wiki of career anecdotes](https://quant.meta.stackexchange.com/q/1640/35). Future career questions can be pointed here.</sub>
There are as many career paths as there are people. How did you get started in the industry?
|
How did you become a quant? |
I first found-out about quantitative finance during my PhD program. Like most grad students, I was looking for an excuse to procrastinate from my dissertation. I began looking at the course catalog of different departments until I saw a course entitled "The Mathematics of Finance". The syllabus listed terminology I had... |
## Life Without Risk-Neutral Measure ##
How would we price assets without the measure $\mathbb Q$? Well, we would start with some version of the Euler equation $P_t=\mathbb{E}_t[M_{t+1}P_{t+1}]$, where $M$ is the stochastic discount factor (SDF). This equation holds under very weak assumptions (law of one price) and... |
Suppose I believe it would be profitable to build an investment portfolio by investing, say, USD 30,000 in stocks in the following ratio: 30% in shares of CompanyA, 30% in CompanyB and 40% in CompanyC. Say that my analysis shows that over the past 10 years, these stocks have performed well as a portfolio, and have a Sh... |
Is there a sound logic for buying a portfolio of call options in the same ratio as you would buy the underlying shares? |
You are consider two contracts: a Eurodlloar futures contract with six months to maturity, selling at 5%, settled on three-month LIBOR, marked to market every day; and a Eurodollar forward contract with six months to maturity, selling at 5%, settled on three-month LIBOR
Which contract do you prefer?
I dont under... |
The $k$ period log return is defined as $$r_{t}(k)=log(S_{t}/S_{t-k}),$$ Where $S_{t}$ is the stock closing price at time $t$. For argument sake, assume that by time I mean a stock trading day and historical stock closing prices are known and denoted by $S_{1}, S_{2}, S_{3}, ...., S_{T-1}$. Here are my questions:
1... |
you should be buying and selling over whatever the prediction time interval was. So, if you're return prediction was positive and was defined as being over the period wednesday from 10:00 am to 4:00 pm , then you should be buying on wednesday at 10:00 am and selling at 4:00 pm.
The prediction time interval is defin... |
I am trying to understand, in its simplest form, how the collateralized loan obligations (CLO) work.
I refer to an [article][1] in the Atlantic for those who are interested in learning about CLOs.
The way I understand how it works is similar to the CDOs as they are both types of structured credit:
(1) There ar... |
How does the Collateral in Collateralized Loan Obligations (CLOs) Work? |
## Life Without a Risk-Neutral Measure ##
How would we price assets without the measure $\mathbb Q$? Well, we would start with some version of the Euler equation $P_t=\mathbb{E}_t[M_{t+1}P_{t+1}]$, where $M$ is the [stochastic discount factor][1] (SDF). This equation holds under very weak assumptions ([law of one pr... |
In a recent CNBC interview, Black Swan author Nassim Nicholas Taleb gave a categorical advice about investing in the Corona period. “It is very unwise to do any form of investment without some form of tail risk hedge.”
I understand the concept of hedging tail risk. But how is it done in practice?
Suppose I have 1... |
How do traders hedge against “tail side risk” in practice? |
With difficulty and at costs, or secretively. Maybe even both. The costly but straightforward approach would be to buy equity index puts. However, I don't think anyone here can or will explain how you can do it significantly more cheaply and effectively. To support this negative answer, I offer this:
In May 2020, th... |
I wonder if the option chain data that is published everyday by websites like Yahoo finance can be used to quickly price options that are not yet traded? So in other words, to estimate option prices in the future?
For example, suppose I have the following information about a 12-month call option and I want to estima... |
Can option chain data be used as a quick and dirty substitute for proper pricing calculations using simulations, to price non-traded options? |
With difficulty and high costs and secretively. Successful ones are the ones that are able to do it more cheaply. This is also the reason for their secretiveness: prices would go up.
The costly but straightforward approach would be to buy equity index puts. However, I don't think anyone here can or will explain how ... |
I wonder if the option chain data that is published everyday by websites like Yahoo finance can be used to quickly price options that are not yet traded? So in other words, to estimate option prices in the future?
For example, can I take the price of a call that is USD 5 in the money today and expires in 3 months, a... |
I am trying to understand, in its simplest form, how the collateralized loan obligations (CLO) work.
I refer to an [article][1] in *The Atlantic* for those who are interested in learning about CLOs.
The way I understand how it works is similar to the CDOs as they are both types of structured credit:
1. There ... |
So I understand that the Fama-French factor model relates a stock's excess return with its beta, market cap, and book to price. How does one use the model in practice? Do people assume that market cap and book to price are fixed (use today's values), and then just supply an estimate for broad market return and basica... |
How would you in practice use factor models (like Fama French) to make decisions? |
So I understand that the Fama-French factor model relates a stock's excess return with its beta, market cap, and book to price. How does one use the model in practice? Do people assume that market cap and book to price are fixed (use today's values), and then just supply an estimate for next period broad market retur... |
I am looking for historical SPX 1minute data containing volume.
Anyone knows where to get them from?
Thank you, |
I'm not sure I understand the question, but I'll give it a try anyway.
The mean and variance specified for the terminal distribution $S_T$ are dependent on current asset price, $S_0$, and implied volatility, $\sigma_i$ (which needs to come from the market via hopefully same pricer that one uses).
The expectation... |
I wonder if the option chain data that is published everyday by websites like Yahoo finance can be used to quickly price options that are not yet traded? So in other words, to estimate option prices in the future?
**For example, can I take the price of a call that is USD 5 in the money today and expires in 3 months,... |
Can option chain data be used as a quick and dirty substitute to price non-traded options for proper pricing calculations? |
I wonder if the option chain data that is published everyday by websites like Yahoo finance can be used to quickly price options that are not yet traded? So in other words, to estimate option prices in the future?
**For example, can I take the price of a call that is USD 5 in the money *today* and expires in 3 month... |
Can option chain data be used as a quick and dirty substitute for proper pricing calculations for non-traded options? |
Prado's paper is really just an algorithm for solving some inverse problem. Has anyone seen a clear statement of that inverse problem? Or do you know how to write it simply?
The first step is just a linkage matrix (see scipy.linkage) so that is pretty clear and can probably be posed as a constrained DAG optimization... |
Is there a clear mathematical statement of what problem Hierarchical Risk Parity is solving? |
I have tick-by-tick data of an asset X denominated in EUR and minute-by-minute data on EURUSD. If I wanted to convert my tick-by-tick data to USD would it make sense to just consider every bucket of one minute in X and convert it using EURUSD? That is, every tick within a one-minute bucket would be converted at the sam... |
How it works has been described clearly by dm63. I commend that answer.
I would like to add a few words about "collateral", what does it refer to?
Obviously the buyers of the debt tranches issued by the SPV would not do buy if they thought the SPV was just an empty worthless box. They agree to lend their money be... |
How it works has been described clearly by dm63. I commend that answer.
I would like to add a few words about "collateral", what does it refer to?
Obviously the buyers of the debt tranches issued by the SPV would not do buy if they thought the SPV was just an empty worthless box. They agree to lend their money be... |
How it works has been described clearly by dm63. I commend that answer.
I would like to add a few words about "collateral", what does it refer to?
Obviously the buyers of the debt tranches issued by the SPV would not do buy if they thought the SPV was just an empty worthless box. They agree to lend their money be... |
The Quantitative Finance Stack Exchange is intended for **professionals** and **academics** involved in quant modeling or trading.
**Basically, your question must be one that someone who is**
- **earning a living** at this, or
- is **studying this specifically in graduate school**
**would confront in the co... |
The Markowitz mean-variance model is known to suffer from estimation error due to financial returns not meeting the assumptions of a normal distribution, providing portfolio weights that underperform out-of-sample.
Does this mean that if asset returns that have a mean of 0, standard deviation of 1, skewness of 0 an... |
Do normal returns make the mean-variance portfolio model perform properly? |
I have just finished my thesis on a heterogeneous dividend expectations model applied to the COVID-19 crisis. However after receiving some feedback there is one last issue I want to resolve. I'm using a *dynamic Gordon model* where estimated AR(1) coefficients (rho and tau) determine variable price-dividend ratios inst... |
I'm testing certain asset pricing factor models (e.g. Fama and French 3 factor model) and want to check if the alphas of my time series regressions are jointly zero.
Most papers use the Gibbons, Ross, Shanken (1987) Test to do so.
How do I get the GRS F statistic in R? How do I correct it for heteroscedasticity an... |
The Markowitz mean-variance model is known to suffer from estimation error due to financial returns not meeting the assumptions of a normal distribution, providing portfolio weights that underperform out-of-sample.
Does this mean that if asset returns that have a:
1. Mean of 0,
2. standard deviation of 1,
... |
I'm testing certain asset pricing factor models (e.g. Fama and French 3 factor model) and want to check if the alphas of my time series regressions are jointly zero.
Most papers use the Gibbons, Ross, Shanken (1989) Test to do so.
How do I get the GRS F statistic in R? This is not the F statistic from `Summary()`,... |
I'm testing certain asset pricing factor models (e.g. Fama and French 3 factor model) and want to check if the alphas of my time series regressions are jointly zero.
Most papers use the Gibbons, Ross, Shanken (1989) Test to do so.
How do I get the GRS F statistic in R? This is not the F statistic from `Summary()`,... |
Since both of them are used to swap between fixed rate payment and float rate payment? |
Can we think of Overnight Index Swaps as short-term IRS? |
I'm studying this [paper][1]. In the formulation of the theoretical setup they state:
> Our goal is to explain the differences in the cross-section of returns
$R$ for individual stocks. Let $R_{t+1, i}$ denote the return of asset
$i$ at time $t+1 .$ The fundamental no-arbitrage assumption is
equivalent to the exi... |
In the following course:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3257420
in the proof of variance reduction for a bagged estimator at page 8, I do not understand why we have the following equality:
$$ \sum_{j \neq i}^{N} \bar{\sigma}^2 \bar{\rho} = \sum_{j \neq i}^{N} \sigma_i \sigma_j \rho_i \rho_j
... |
The Future cost of carry model models the future price of a commodity.
$F = Se^{((r + s - c)t)}$
Where
$F$ = the future price of the commodity
$S$ = the spot price of the commodity
$r$ = the risk-free interest rate
$s$ = the storage cost expressed as a percentage of the spot price
$c$ = the conveni... |
Is the future cost of carry a upper bound for the future price? |
Many traditional finance models assume linear relationships between variables and features. Aren't linear correlations/covariances unable to capture financial processes empirically since they actually are more likely to possess non-linear correlations?
If I am trying to forecast volatility with 3 different algorit... |
Is non-linear correlation problematic in financial time series prediction? |
that is probably a rather simple question but I got confused and would be very thankful for help.
Imagine we are in 2015 and have an option that expires in either 2016, 2017, 2018, 2019, 2020 or 2021.
We are given the volatilies for t_1 = 31,84%, t_2 = 27,45%, t_3 = 26, 64%, t_4 = 26,27%, t_5 = 26,16% and t_6 = 26,25... |
That is probably a rather simple question but I got confused and would be very thankful for help.
Imagine we are in 2015 and have an option that expires in either 2016, 2017, 2018, 2019, 2020 or 2021.
We are given the volatilies for t_1 = 31,84%, t_2 = 27,45%, t_3 = 26, 64%, t_4 = 26,27%, t_5 = 26,16% and t_6 = 26,25... |
I am pretty sure the following is true
$$
\text{EAD} = \text{Drawn Amount} + \text{Undrawn Amount} \times CCF
$$
where $\text{CCF}$ is the credit conversion factor. It means if an overdraft line is drawn to 500 EUR & its limit is 1000 EUR with CCF = 0.5, EAD is 750 EUR.
Unfortunately, I have a really hard time... |
EAD = Drawn amount + Undrawn amount * CCF? |
I know that there is already a very similar post about this. Unfortunately it does not correspond exactly to my question, so I will try it again myself. I am trying to calculate the MRM of a Category 3 PRIIP. After I was unsure about my result I wanted to recalculate the example shown in the flow chart.
[EU Flow ch... |
In the following course:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3257420
in the proof of variance reduction for a bagged estimator at page 8, I do not understand why we have the following equality:
$$ \sum_{j \neq i}^{N} \bar{\sigma}^2 \bar{\rho} = \sum_{j \neq i}^{N} \sigma_i \sigma_j \rho_{i,j}
$$... |
In Lecture 4 of the following course:
Advances in Financial Machine Learning: 10 Lectures
by Marcos Lopez de Prado
[link][1]
in the proof of variance reduction for a bagged estimator at page 8, I do not understand why we have the following equality:
$$ \sum_{j \neq i}^{N} \bar{\sigma}^2 \bar{\rho} = ... |
I am attempting to create a volatility surface for a US electricity market that has a liquid futures market but nearly non-existent options market (<5 trades per month across all strikes and expiration dates). Options in the natural gas market for the region are not particularly liquid either -- and, even so, natural g... |
I guess we are familiar with the discussion, whether companies especially tech companies are paying their due in taxes. There was the huge discussion of Amazon paying nearly no taxes, using loopholes and so forth. I will stick with Amazon in this question as the most popular example but the question is general:
What... |
I'm having a problem to calculate the gap option greeks since there are 2 different exercise prices K1 and K2. Do you know the answer or where can I read about these particular greeks? |
What are the formulas to compute the greeks of a gap option? |
I understand the meanings of contango and backwardation, but I'm trying to better understand the theory behind what creates each. For future readers of this question, here are the examples from the CME website:
**Contango:**
[![enter image description here][1]][1]
**Backwardation:**
[![enter image descri... |
I understand the meanings of contango and backwardation, but I'm trying to better understand the theory behind what creates each. For future readers of this question, here are the examples from the CME website:
**Contango:**
[![enter image description here][1]][1]
**Backwardation:**
[![enter image descri... |
> Since contracts on physical goods have associated costs, it makes
> sense that the term structure curve would be upward sloping. Since
> there is no cost associated with delivery for the VIX and contango is
> considered to exist in healthy markets, is the upward slope simply
> accounting for the greater potential... |
Is there a way to formulate a Martingale series that will never explode? |
I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion.
My python script is below, but it contains an error because of which the end result is always zero ```0.0```. I cannot figure out where the error is.
The characteristics of this option are as follows... |
How to price a down-and-out leveraged barrier call option using Brownian motion? |
I was trying to compute the turnover when you have the initial monthly weights. Is not so easy to figure out the logic in the package used in the previous answer.
Here I show how to compute the DeMiguel and Nogales (Portfolio Selection with Robust Estimation, Operations Research 57(3), pp. 560–577, 2009) turnover.
... |
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