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As a coincidence I have also started to learn about and implement stochastic vol models. So I first want to say that I am no expert in the field. However, I just struggled through implementations of calibration and simulation of the Heston model. For the calibration I used the article "Full and fast calibration of ...
VWAP is a ratio of the cumulative value (price ($p$) times volume ($q$)) traded in a stock, divided by total traded volume during the day: $$\frac{\sum{q \times p}}{\sum{q}}$$ Does the volume component ($q$) that appears in the numerator and denominator distinguish between buy and sell trades somehow? (negative $...
Does VWAP distinguish between buy and sell trading volume?
I want to apply the knowledge of this paper (Bayesian estimation of probabilities of default for low default portfolios, by Dirk Tasche) in R, but I can't find the right bayesian package and functions to use. Can you help me?
There is an exercise I struggle to solve. I hope you can give me a hint. Let X be random variable taking values in $I\subset \mathbb{R}$. I have to show that the Value at Risk is invariant under any increasing and continuous function $f:I \rightarrow \mathbb{R}$ for each $\alpha \in (0,1)$, i.e., $VaR_{\alpha}(f(X)...
What kind of entities use exotic derivatives, and do they serve any purpose other than hedging risk?
I have a problem in which I have been given data for two periods over fictional business partners. Each dataset consists of the following fields: ID (specific business partner), rating, PD, LGD, Original Exposure (on- and off-balance sheet exposures), EAD, RWA, Capital requirement, Expected Loss (EL= EADxPDxLGD)...
If I understand correctly, you're looking for an implementation of Monte Carlo simulations in this paper? If you can't find anything on CRAN or elsewhere on the internet, I think your best bets, in order are: 1. Contact the author 2. Implement it yourself
As @skoestlmeier and @noob2 commented there's much research going on about the profitability anomaly. Firstly, there are different ways of measuring profitability. Novy-Marx (2013, JFE) uses gross profitability, Fama and French (2015, JFE) total profitability and Hou et al. (2015, RFS) return on equity. The $q$-theo...
The dollar bars certainly allow for a partial recovery of normality through a price sampling process subordinated to a volume, tick, dollar clock. It is well known that returns are assumed to be normally distributed but in reality they have a high kurtosis and fat tails, they are leptokurtic. Dr de Prado posits i...
I am trying to compute Implied Volatilities from Heston OTM Call and OTM Put prices for short maturities ($\tau=$ 1 week). I am using the Carr & Madan representation, the little Heston trap formulation of the characteristic function, the Lord & Kahl algorithm for the optimal damping factor and the Kahl & Jäckel transfo...
In his book - Advances in Financial Machine Learning, Marcos Lopez de Prado familiarises the reader with a number of ways of normalizing our financial time series data. Below I provide a couple of examples (source: [here][1]) for the purpose of this question: 1. **Time Bars**: Bars indexed by time intervals, minute...
In finance, alternative risk measures such as value-at-risk (VaR) and GARCH are introduced as replacements to standard deviation volatility. Is there any application or value where several risk estimators or two are considered simultaneously? As an example, a ratio that describes the divergence between one another?...
You say: > At this point I don't really get any further, as I am unsure about > which "cross section" is being talked about here. Since I have created > 25 portfolios, I can only have all in all 25 values in the cross > section, right? Isn't that far too little for a sufficient regression? > Or do I have to run ...
I think your best shot is to share with us your 3,000 stocks. How far can that be from FF sample? As a quick check I took the 25 book-to-market portfolios and the Fama-French 3 factor model and run the standard fama macbeth regressions. 1) First stage results: [![enter image description here][1]][1] Only 6...
Consider a stopping time $\tau$ that represents the point in time when the first credit event (e.g. default) occurs on a compact interval $[0,T]$. Consider the expectation of the indicator function, $\mathbf{1}_{\{\tau\leq T\}}$, under a well-defined filtered probability space,$(\Omega,\{F_t\}_{t\geq0},P)$: $$E_P...
I am learning about market making and have been reading papers by Avellaneda-Stoikov, Bayraktar-Ludvkosli and L-Guéant-Fernandez. All of them attempt to maximize utility function but are based on a single order book. Are there any papers that propose a market making algorithm when two limit order books are perfectly co...
I have a hard time understanding how high IV effects a put backspread. Is it via the angle on the payoff or via the ratio one gets? or simply just that one has bigger numbers in the credit and debit for the position?
How does high IV effect a put backspread?
As @skoestlmeier and @noob2 commented there's much research going on about the profitability anomaly. Firstly, there are different ways of measuring profitability. Novy-Marx (2013, JFE) uses gross profitability, Fama and French (2015, JFE) total profitability and Hou et al. (2015, RFS) return on equity. The $q$-theo...
As @skoestlmeier and @noob2 commented there's much research going on about the profitability anomaly. Firstly, there are different ways of measuring profitability. Novy-Marx (2013, JFE) uses gross profitability, Fama and French (2015, JFE) total profitability and Hou et al. (2015, RFS) return on equity. The $q$-theo...
Let me try to answer. The generic CVA formula can be written as below ($Df(t)$ is the discount factor from $t_0$ to $t$, $V(t)$ is the portfolio value at time $t$): $$ CVA(t |\mathbb{F_{t_0}}) = \mathbb{E_Q} \left[ \int_{s=t_0}^{s=t} Df(s)* I_{(V_s>0)} * I_{(default_s)} *LGD* V(s) ds \right] = \\ = LGD* \mathbb{E_Q...
Let me try to answer. The term you mention in your question frequently appears in CVA (Credit Valuation Adjustment) calculations. The generic CVA formula can be written as below ($Df(t)$ is the discount factor from $t_0$ to $t$, $V(t)$ is the portfolio value at time $t$): $$ CVA(t |\mathbb{F_{t_0}}) = \mathbb{E_Q} \...
Let me try to answer. The term you mention in your question frequently appears in CVA (Credit Valuation Adjustment) calculations. In the context of CVA, the stopping time referring to a credit event is the point in time when a counterparty defaults (by "counterparty" I mean some financial or corporate institution which...
Let me try to answer. The term you mention in your question frequently appears in CVA (Credit Valuation Adjustment) calculations. In the context of CVA, the stopping time referring to a credit event is the point in time when a counterparty defaults (by "counterparty" I mean some financial or corporate institution which...
How does one compute the alpha of a global portfolio. Let's say we are using the Fama French 3 factor model and we have a portfolio of 50% US stocks and 50% German stocks. 1.Should the regression be applied on every country by using the country specific factors? And then reconstruct the portfolio? (Alpha Us * 50% +...
There is this huge growth in passive investing. If investors are passive, this shouldn't have any implications for the cross-section/time-series of asset prices, i.e. no real economic effects. Any references?
Is there any evidence that passive investing has real implications for asset prices?
The bisection method, Brent's method, and other algorithms should work well. But here is a very recent paper that gives an explicit representation of IV in terms of call prices through (Dirac) delta sequences: [Cui et al. (2020) - A closed-form model-free implied volatility formula through delta sequences][1] ...
I have a hard time understanding how high IV effects a put backspread. Is it via the angle on the payoff or via the ratio one gets i.e number of OTMs one can buy? or simply just that one has bigger numbers in the credit and debit for the position?
I noticed when I regress the return of a portfolio on the Fama French 3 factor model that the value and the statistical significance of the coefficients vary when I use daily versus monthly portfolio returns. I would like to know what is the best frequency to estimate these coefficients? For my purpose I do need to ...
Should a portfolio coefficients be calculated with daily or monthly returns?
How does one compute the alpha of a global portfolio. Let's say we are using the Fama French 3 factor model and we have a portfolio of 50% US stocks and 50% German stocks. 1. Should the regression be applied on every country by using the country specific factors? And then reconstruct the portfolio? (Alpha Us * 50%...
Suppose I am pricing a multi asset at the expiry payoff. Theoretically I define their joint distributions in the risk neutral measure, and price using expectation. However, how do I know that the model has accounted for cost of vega hedging? Cost of delta hedging is baked into the marginal distributions, but how to acc...
How does a pricing model 'understand' the cost of hedging?
So in Hull (2012) the main point is that $\Delta x^2 = b^2 \epsilon ^2 \Delta t + $higher order terms$ $ has a term of order $\Delta t$ and can not be ignored as the Brownian motion exhibits the quadratic variation of $\Delta t$. My question is now what does $\epsilon ^2$ correspond to. Cochrane (2005) notes that $dz^2...
I get the first part of the regression, basically it is a time series regression of returns on the proposed factors. So, I need to run the regression of monthly return on the monthly time series of the factors from 2000 to 2005. So from this, we get the alpha and beta for this stock from the period 2000 to 2005. Now...
There is an influential paper by Gormley et al. on this topic called ["Passive investors, not passive owners"](https://www.sciencedirect.com/science/article/pii/S0304405X16300319) Also a new working paper by Heath et al. ["Do index funds monitor?"](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3259433), which ...
I have a hard time understanding how high IV effects the amount of gamma obtained via a put backspread. Is it via the angle on the payoff or via the ratio one gets i.e number of OTMs one can buy? or simply just that one has bigger numbers in the credit and debit for the position?
I think the question also brings up a common confusion with notation. I think it is incredibly unfortunate to use notation such as $dW(t)$ (unless it's part of an integral), and I am really upset when I see it being used in textbooks. The definition of Brownian Motion is implicit and goes like this: (i) $W(t=0) ...
I think the question also brings up a common confusion with notation. I think it is incredibly unfortunate to use notation such as $dW(t)$ (unless it's part of an integral), and I get upset when I see it being used in textbooks. The definition of Brownian Motion is implicit and goes like this: (i) $W(t=0) = 0$ ...
It's hard for me to understand exactly what you are asking, but I will try to answer. If my answer misses the mark please clarify exactly it is what you don't understand and I will try again. We have \begin{aligned} P(\tau \leq t + dt \vert \tau > t) &= \frac{P(t < \tau \leq t+dt)}{P(\tau > t)} \\ &= ...
I think the question also brings up a common confusion with notation. I think it is incredibly unfortunate to use notation such as $dW(t)$ (unless it's part of a stochastic integral), and I get upset when I see it being used in textbooks. The definition of Brownian Motion is implicit and goes like this: (i) $W(t...
Yes, the second step of the Fama MacBeth procedure requires you to run a cross-sectional regression of the monthly returns of each stock against their betas for each month. This regression gives you a return for each factor for each period. The average factor return is the risk premium for the factor - see https://quan...
In general most of the forex market closes by 10:00 pm london time on Friday and opens back on Monday morning APAC time , somewhere around Sunday evening London time. I am interested in knowing if there are some reliable forex spot ( may be even forwards or future ) trade happening on Saturday or during first half of S...
I have a problem in which I have been given data for two periods over fictional customers. Each set consists of the fields: ID (specific customer), rating, PD, LGD, Original Exposure (on- and off-balance sheet exposures), EAD, RWA, Capital requirement, Expected Loss (EL= EADxPDxLGD), EADxPD, EADxLGD. PD = Prob...
You say: > At this point I don't really get any further, as I am unsure about > which "cross section" is being talked about here. Since I have created > 25 portfolios, I can only have all in all 25 values in the cross > section, right? Isn't that far too little for a sufficient regression? > Or do I have to run ...
I only have a very general theory-based knowledge on Jensen's Alpha. I'm very curious about Capital Asset Pricing Model with intercept at 0. May I know what is the intuition behind this? What does it indicates actually, and why it needs to be 0?
What is the intuition of CAPM model with Intercept at 0?
If we have two Ito (correlated) semimartingales over $[0,T]$: $$d X_t^i=a^i_tdt+\sigma_t^idW_t^i,\quad i=1,2$$ What is the relationship between $$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \rangle_t, $$ where $\langle \rangle_t$ denotes the quadratic variation? Assuming the correlation coeffi...
If we have two Ito semimartingales over $[0,T]$: $$d X_t^i=a^i_tdt+\sigma_t^idW_t^i,\quad i=1,2$$ What is the relationship between $$\langle X^1,X^2 \rangle_t \quad \text{and} \quad \langle W^1,W^2 \rangle_t, $$ where $\langle \rangle_t$ denotes the quadratic variation? Assuming the correlation coefficient $\rho$ ...
How can I simulate the Yield Curve?
My EuroDollar futures have monday of third week expiration and settlement date from CME exchange how ever Python Quantlib doesn't seem to like it. Anyway I can get away with IMM check? futures = { ql.Date(14,9,2020): 99.700, ql.Date(14,12,2020): 99.67, ql.Date(15,3,2021): 99.76 } futures...
How to make futuresHelpers in Quantlib work with monday settlement day not IMM?
I'm starting to experiment with covered call strategies, and I'm trying to find the right strike price to sell for my covered calls such that I can maximize premium while being generally "confident" that the strike price won't hit and I can just bank the premium. "Confidence" would hopefully be an adjustable parameter...
Asset returns are obtained by log-differencing prices. Standardizing or normalizing/scaling asset returns can be carried out by de-meaning the returns and dividing them by their standard deviation, causing their mean to change to 0 and standard deviation to be 1. Can we also expect the asset returns' skewness and kurto...
Does standardizing/normalizing asset returns change their skewness and kurtosis?
I think the question also brings up a common confusion with notation. I think it is incredibly unfortunate to use notation such as $dW(t)$ (unless it's part of a stochastic integral), and I get upset when I see it being used in textbooks. The definition of Brownian Motion is implicit and goes like this: (i) $W(t...
In market microstructure/high frequency finance, it is well known that the trading day has high activity in the opening hours and closing hours compared to mid-day. Can the same be said about the trading week, in that markets on Mondays and Fridays are more active than Wednesday as people open/close their positions at ...
More/less trading on Mondays and Fridays?
I want to maximize the return of a $n$-asset portfolio under known risk: $$max_{\{w \in \mathbb{R}^{n}|w_{1}+...+w_{n}=1\}} \; \mathbb{E}[\sum_{i=1}^{n}w_{i}R_{i}]$$ under the constraint $$ES(\sum_{i=1}^{n}w_{i}R_{i}) \le r$$ where $ES$ is the expected shortfall (at some level $\alpha$) and $r$ is the desired level...
I want to maximize the return of a $n$-asset portfolio under known risk: $$\max_{\{w \in \mathbb{R}^{n}|w_{1}+...+w_{n}=1\}} \; \mathbb{E}\left[\sum_{i=1}^{n}w_{i}R_{i}\right]$$ under the constraint $$ES\left(\sum_{i=1}^{n}w_{i}R_{i}\right) \le r$$ where $ES$ is the expected shortfall (at some level $\alpha$) and $...
I want to maximize the return of a $n$-asset portfolio under known risk: $$\max_{\{w \in \mathbb{R}^{n}|w_{1}+...+w_{n}=1\}} \; \mathbb{E}\left[\sum_{i=1}^{n}w_{i}R_{i}\right]$$ under the constraint $$ES\left(\sum_{i=1}^{n}w_{i}R_{i}\right) \le r$$ where $ES$ is the expected shortfall (at some level $\alpha$) and $...
I am trying to value an FRA in quantlib Python using the below code: import QuantLib as ql calendar = ql.UnitedStates() todaysDate = ql.Date(7, ql.May, 2017) # ql.Settings.instance().evaluationDate = todaysDate spotDates = [ql.Date(7, 5, 2017)+ql.Period(i*6, ql.Months) for i in range(0, 10)] ...
I am trying to value an FRA in quantlib Python using the below code: import QuantLib as ql calendar = ql.UnitedStates() todaysDate = ql.Date(7, ql.May, 2017) # ql.Settings.instance().evaluationDate = todaysDate spotDates = [ql.Date(7, 5, 2017)+ql.Period(i*6, ql.Mo...
I discuss the books I mentioned in the comments. They all deal with standard (theoretical) asset pricing (starting with one period utility maximisation and then branch off). Other books like [Björk][1] or [Shreve][2] focus more on time continuous models and derivatives pricing and I do not discuss them here. ## Coch...
I want to maximize the return of a $n$-asset portfolio under known risk: $$\max_{\{w \in \mathbb{R}^{n}|w_{1}+...+w_{n}=1\}} \; \mathbb{E}\left[\sum_{i=1}^{n}w_{i}R_{i}\right]$$ under the constraint $$ES\left(\sum_{i=1}^{n}w_{i}R_{i}\right) \le r$$ where $ES$ is the expected shortfall (at some level $\alpha$) and $...
Hi I am given two tables of two tables showing data about fictional corporate business partners and relevant credit risk data – ID, rating, probability of default (PD, defined by rating), loss given default % (LGD), original exposure (on- and off-balance sheet exposures), exposure at default (EAD), risk-weighted asset...
I suggest you take a look at this topic: [What are the quantitative finance books that we should all have in our shelves?][1] Other than that I strongly recomment Campbell's book mentioned above. Perfect for a 1st year PhD introduction into asset pricing. [1]: https://quant.stackexchange.com/questions/3886...
I want to maximize the return of a $n$-asset portfolio under known risk: $$\max_{\{w \in \mathbb{R}^{n}|w_{1}+...+w_{n}=1\}} \; \mathbb{E}\left[\sum_{i=1}^{n}w_{i}R_{i}\right]$$ under the constraint $$ES\left(\sum_{i=1}^{n}w_{i}R_{i}\right) \le r$$ where $ES$ is the expected shortfall (at some level $\alpha$) and $...
I am not really sure I understand the meaning of Risk weight defined as RW = RWA/EAD, where EAD = Exposure at Default RWA = Risk-Weighted assets From a bank's perspective: If I have two portfolios, over two time periods, of customers and each customer is given an EAD and RWA. Does it make sense compari...
I'm not sure I understand the question, but I'll give it a try anyway. The mean and variance specified for the terminal distribution $S_T$ are dependent on current asset price, $S_0$, and implied volatility, $\sigma_i$ (which needs to come from the market via hopefully same pricer that one uses). The expectation...
I have a problem in which I have been given data for two periods over a set of customers. Each set consists of the fields: ID, rating, PD, LGD, Exposure (on- and off-balance sheet exposures), EAD, RWA, Required Capital, Expected Loss (EL= EADxPDxLGD). PD = Probability of Default LGD = Loss Given Default ...
I'm not sure I understand the question, but I'll give it a try anyway. The mean and variance specified for the terminal distribution $S_T$ are dependent on current asset price, $S_0$, and implied volatility, $\sigma_i$ (which needs to come from the market via hopefully same pricer that one uses). The expectation...
I wonder if there are any websites/resources/sample codes on how to convert the American options to European options (when all else are equal). i.e. if given same underlying asset, same expiration date, same exercise price (but these are unknown). Now additionally we are given the price of an American option, are there...
With $V$ American option value, $H$ holding (aka continuation) value, and $B$ bank account value, we have: $$V_N(S_N) = (K-S_N)^+$$ and for $i$ backwards from $N-1$ down to 0, we have: $$ H_i(S_i) = \mathbf{E}\left[B_{i}B_{i+1}^{-1}V_{i+1}(S_{i+1})|S_i\right]$$ $$ V_i(S_i) = \max (K-S_i, H_i(S_i)) $$ The...
Assume we are working in the framework of modern portfolio theory. Now, let's say we have two securities (they could also be portfolios themselves) A and B. Portfolio A has expected return 10% and standard deviation 10% while portfolio B has expected return of 5% and a standard deviation of 5%. Is there a way to com...
Is it possible to make a portfolio with higher expected return and lower standard deviation than constituent securities?
Does it make sense to add more factors such as Quality Minus Junk (QMJ) and Betting Against Beta (BAB) in the Fama-French-Carhart model? Also, if anyone can point me to an article it would be appreciated. r=α+Rf+βm(Rm−Rf)+βs⋅SMB+βv⋅HML+βmom⋅UMD Would add QMJ and BAB to the regression.
This is not possible because if you want a portfolio with an expected return higher than 10% you need an asset with an expected return higher than 10%. However both asset A and B are lower or equal to 10%. So you can’t create the portfolio, or is leveraging allowed?
Specifically, the US govt bond market is segmented and the shape is difficult to model in a structural model, because recently there is a maturity gap from 12 to 20 year, and the front and back ends are so different. Nelson Siegel family clearly has difficulties at long end, even more complex/time-series models do not ...
There is a very popular parametrisation of the Implied Volatility (IV) Smile in terms of Total Implied Variance (see [Gatheral & Jaquier](https://arxiv.org/pdf/1204.0646.pdf)). A condition for an Implied Volatility Surface (IVS) to be free of calendar spread arbitrage is also derived in terms of Total Implied Variance....
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 ...
Does it make sense to add more factors such as Quality Minus Junk (QMJ) and Betting Against Beta (BAB) in the Fama-French-Carhart model? Also, if anyone can point me to an article it would be appreciated. r=α+Rf+βm(Rm−Rf)+βs⋅SMB+βv⋅HML+βumd⋅UMD Would add QMJ and BAB to the regression.
I'm assuming that ["limit offset"](https://investor.pershing.com/web/ada-investor/disclosures/stop-order-risk) just means ["limit price"](https://old.reddit.com/r/investing/comments/1ezbg5/questrade_trailing_stop_limittrailing_stop/ca5fivq/). [Help & How-to | Questrade](https://questrade-support.secure.force.com/mylear...
Why does Canada allow merely 9% spread between the trailing stop price and its limit offset?
On [Feb 11 2018 on Quora](https://www.quora.com/What-is-the-secret-behind-Renaissance-Technologies-Can-mathematics-really-be-used-to-bring-high-returns-to-investors/answer/Vladimir-Novakovski/comment/54182963), Vladimir Novakovski commented >Terry is Tier One in the RenTec world. His coauthor on compressed sensing d...
Which of Terry Tao's co-authors on compressed sensing consulted for Renaissance Technologies?
I want to maximize the return of a $n$-asset portfolio under known risk: $$\max_{\{w \in \mathbb{R}^{n}|w_{1}+...+w_{n}=1\}} \; \mathbb{E}\left[\sum_{i=1}^{n}w_{i}R_{i}\right]$$ under the constraint $$ES\left(\sum_{i=1}^{n}w_{i}R_{i}\right) \le r$$ where $ES$ is the expected shortfall (at some level $\alpha$) and $...
Are there any papers doing as broad as possible comparison of various portfolio theories performance? For example, compare specific theories: 1. MPT 2. Risk parity 3. Black-Litterman 4. etc or compare models: 1. Mean variance optimization 2. Risk allocation 3. Bayesian probabilities 4. etc I unders...
Studies on performance comparison of portfolio theories?
I want to maximize the return of a $n$-asset portfolio under known risk: $$\max_{\{w \in \mathbb{R}^{n}|w_{1}+...+w_{n}=1\}} \; \mathbb{E}\left[\sum_{i=1}^{n}w_{i}R_{i}\right]$$ under the constraint $$ES\left(\sum_{i=1}^{n}w_{i}R_{i}\right) \le r$$ where $ES$ is the expected shortfall (at some level $\alpha$) and $...
A popular solution technique are finite difference. By the Feynman-Kac theorem, partial differential equations are fundamentally related to (discounted, risk-neutral) conditional expectations. I understand how to derive finite difference approximations using Taylor's theorem and how to implement the final iteration....
> I know kalman filters and have ... If this knowledge extends to Unscented filters, UKF, you can think of the UKF being a sparse particle filter. With a UKF you have a few sigma points which are propagated forward via your model function and then after the measurement update these sigma points are updated via cova...
The NASDAQ open trading time is 09:30 AM which currently is 13:30 UTC. After DST is over does it stay the same? meaning does it become 14:30 UTC ?
Are NASDAQ open and close time consistent throughout the year?
Does it make sense to add more factors such as Quality Minus Junk (QMJ) and Betting Against Beta (BAB) in the Fama-French-Carhart model? Also, if anyone can point me to an article it would be appreciated. $$r=α+R_f+β_m(R_m−R_f)+β_s⋅SMB+β_v⋅HML+β_{umd}⋅UMD$$ Would add QMJ and BAB to the regression.
I am currently interested in doing some research on short term/intraday momentum trading strategies (highly liquid large cap stocks), especially to test if there are profitable strategies on higher frequency data (e.g. 5 minutes) using technical indicators like RSI or MACD. Do you have good studies/overview studies/...
I am a little bit stuck with my dissertation thesis, so help will be greatly appreciated. I am trying to analyze the performance of different mutual funds, which I have classified according to investment style using a holdings-based analysis. My sample period of reference ranges from 2007 to 2018, with quarterly observ...
I have written a simple script for modelling stock prices using Brownian motion. 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 the mu (average returns) and sigma (variance) using a linear or *stan...