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Financial Mathematics - Science topic

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Questions related to Financial Mathematics
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I intended to focus my research more to Financial Mathematics/Actuarial Science, I am open to collaborating & learning new things, I am looking for someone good in the area to mentor me through. Please if you can help contact me at @aminu.nass@fud.edu.ng
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My research covers Financial Mathematics/Actuarial Science. Most specifically, my research covers the technical aspects of finance (Asset Pricing, Derivatives, Risk Management, and Financial Engineering). I invite you to read several of my publications posted on my ResearchGate author page at:
If this is the type of research you are interested in, then email me at adoff21@yahoo.com to establish a working contact. Q.E.D.
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I am working on portfolio optimization using lower partial moment of order 1, can someone help me how to implement LPM-1 in excel sheet using "tau" as my threshold value as 0.00% and order (n) as 1.
Thank you all in advance for your contributions to my question.
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How will the model stability problem be addressed? Are these tests necessary to diagnose the model stability in the case of (ARDL)?
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yes absolutely, but the unit-root tests with breakpoints give graphics (in STATA) with the dates of breaks so you can judge if the test considers the appropriate breaks or not. For the chow test, it has to sequential test.
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It is required to submit an article in good journal
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You can visit web of science or scopus for updates jiurnals
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EMM used to be popular but there have not been so many papers lately. To get published now maybe you need more advanced methods like MCMC. But does that mean EMM has proven inadequate, or just not worth writing new papers about?
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I only use MCMC these days, but partly because it is very easy now with Stan software. See
for example. You might need to create a free account to get it.
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I am eager to study stochastic processes and their application in finance. as I am a student in economics the concepts are completely unfamiliar for me. Any help would be appreciated. Can anyone suggest me the introductory textbook?
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I think, It is very introductory book that everyone enjoy it:)
Thomas Mikosch
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I’m using 720 daily returns for each stock company (144 stocks) to calculate CARs and BHARs in monthly wise. For each stock at any period of time (36 months and each month has 20 trading days) the answers from both measures are different.
example: stock AA in first trading month CAR and BHAR is not same.
But in whole sample final answers at each month is almost similar in both methods. Is that possible or did I do any mistakes in the calculation steps?
  • ARR - first monthly average of each stock (144) separately and then sum of those average values in each month divided by nu of stocks. finally calculate the cumulative returns by adding monthly sample returns.
  • BHAR- simply followed the formula in daily basis and considered the values at each 20 days.
The formulas which I used is attached here. Please help me to solve this problem.
Thanks in advance.
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Dear Dilesha
You will generate different average returns the greater the variability in the annual returns. Your CAR formula is an arithmetic mean whereas your BHAR is a geometric mean. In addition, The BHAR implies your beginning portfolio of investments will not change over the holding period of your study as opposed to the CAR conceivably could have a different mix of investments across time.
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So I am new to actuarial science and trying to figure things out myself. After computing reserving estimates using those 3 methods : Chain ladder, link ratio and cadence method, how do we know or based on what do we decide to keep one and eliminate the others.
Thank you so much !
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Thank you very much Alexader & Fan. Your help is very appreciated.
Best regards
Sarah
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Option Pricing
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HI,
You may be interested by this recent paper:
Semi-analytical valuation for discrete barrier options under time-dependent Lévy processes
Journal of Banking and Finance, february 2017, Pages 167-183
Guanghua Lian, Song-Ping Zhu, Robert J. Elliott, Zhenyu Cui
Best regards
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I am writing a paper on bank stability in Africa. Z-score is used as a dependent variable. However, given the large variations in the z-score variable, I am tempted to log the variable to reduce scale bias.
I would like to know if it is possible to log the dependent variable (z-score) and if yes, what are the implications?
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The answers so far suggest yes and you need to deal with the issue of negative raw Z which can of course be scaled to achieve this.
Log transformations reduce right hand skewness and hopefully make the variable more normal.  Before rushing to this solution I suggest you review the distribution property of your variables and consider are they sufficiently normal to use in a linear regression context.  If not then rather than looking for transformations that may make the distribution normal a nonparametric regression might be better. 
Without knowing whether your data are time series, cross- section, panel (balanced or not) it is not sensible to suggest a particular approach.  If it is Panel and it satisfies a poolability test (Chow) then you have lots of possibilities.  There may be fixed effects present due to central bank regulations.
We have been working on Nepla data and found that the panel of bank data was poolable but for micro-finance institutions over the same time period it wasn't.  In the latter case we obviouslt had to use a cross-sectional approach and found quantile refgression appropriate.
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I'm modelling "deviations of interbank rate from policy rate (Yt)"  in function of a liquidity variable (Xt). The idea is that interbank will deviate more depending on the level (threshold) of Xt. Which econometric models can I use, and if you have some related literature I will be great. 
Thanks
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of course you can use either  Markov switching models, and smooth transition threshold models, However If I were you I will use STTM  as the change in your breaks ( Liquidity levels happens slowly)
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I'm trying to derive something and I got something but I wanted to check it.
I was wondering the following: 
Suppose  X_t is MA(1) with no intercept and MA parameter theta and variance say sigma squared.  is there an interesting relation between say X_t+n - X_t ? I showed on scrap that the difference process must be MA(1) with no intercept but does the MA parameter change to p^(n) and the variance  change to n sigmasquared. thanks for any help or confirmation.
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Hi Francesco: Yes,  I agree. Thanks so much for your beautiful derivations.
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... if you are not an expert in data mining (or similar) techniques?
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Excel is best  for an ease of use, any way if the nature of data is categorical or qualitative then SPSS is also an easy solution.
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Especially I am interesting in the topic: time value of money.
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The following books can be useful for students: 
2/ Paul Wilmos, Sam Howison and Jeff Dewwynne, The Mathematics for Financial Derivatives, A Student Introduction, Cambridge Univ. Press.
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I am looking for a data set to test a local volatility algorithm pricer for spread or crack options on commodity (WTI). I would need 
- Future data quote markets
- Options data (call / put) quote markets considered on spread option maturity
- Ideally : spread or crack quote markets,
- Ideally : corresponding Kirk / Bjerksund or Monte Carlo reference prices.
To your knowledge, is there any standard already considered data set to test performance and precision of the method ?
Thanks for any contribution !
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Hi,
As far as I'm aware it would be hard to create a standard data set due to at least two reasons - market conditions are changing and so are options and option-like products. Thus such data set would need to be modified frequently and that would render it very hard to maintain and use.
I would have taken a look at CBOE Livevol at:
Best regards,
 Stan
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My research is on the relationship between stock return volatility and trading volume. so i included both trading volume and forecasted volatility in a mean equation of GACH (1,1), however my model keeps failing. how do i solve this problem??
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In short, yes. Although usually in some kind of rolling window basis to generate the time variation 
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I would like someone to discuss the following hypothesis:
The Black-Scholes formula is not a valid optionpricing model.
When backtesting S&P stock options using B&S and the real volatility (= standard deviation) ex post the costs exceed the payoffs by 4 percent, using the VIX (=Volatility of the S&P500) by 26 percent. 
The method is: buy fictitious call options day by day over 15 years  at the money and at the price of fair value - compare the sum with the cumulated payoffs. The rationale:
The payoffs should somehow match the amounted procurement costs at least.
(For puts it's even worse - 18 / 46 percent overpricing.)
Any comment appreciated.
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Added a working paper that contains more detailed experiments applying the BS-formula to S&P and VIX data. 2 questions:
o Does the revers application of B&S induce a systematic error with a tendency to increase the index volatility?
o The payoffs of options on a broad portfolio like S&P don't match their fair value at emission time. Is the B&S formula still applicable - even for financial reports and balance sheets? 
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Xavier Vives defined fully revealing and partially revealing  rational expectations equilibriums in the pages 80-81 of its book, Information and Learning in Markets: The Impact of Market Microstructure.
I understood the literrary definitions but how mathematically we can prove if a REE is fully or partially revealing?
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Thank you Sergei and Amit for your answer.
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I am trying to find possible correlations between classical investment tools like equities, commodities, forex etc with alternative investments like art indices. Could you suggest me an advaced statistical, econometrical methodology?
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For each of the asset groups you need a return series.  In the case of equities you may choose a stock market index. If you a finer grouping there may be an appropriate sectoral index ot otherwiseconstruct a portfolio of the required form. The other asset groups likee forex can be similarly calculateCd from an aggregate index or portfolio.
Art as a portfolio can be tracked using vaeiuos art investment fund returns. These are not dissimilar to property trusts or real estate investment trusts.You might be surp4ised to find the number of coll3ctible investment funds there are.  There was/is a barbie doll index.
Once you have the portfolios you can calculate common period returns.  These timeseries can be correlated using a parametric or nonparametric approach.
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Hello
Different Sources states different ways for the lag order selection before running VAR or VECM in stata. Though stata manual notes the following "
To test for cointegration or fit cointegrating VECMs, we must specify how many lags to include.
Building on the work of Tsay (1984) and Paulsen (1984), Nielsen (2001) has shown that the Methods implemented in varsoc can be used to determine the lag order for a VAR model with I(1) variables.  The order of the corresponding VECM is always one less than the VAR. vec makes this adjustment automatically, so we will always refer to the order of the underlying VAR. "
Should I always choose first order differenced variables for VAR and Level variables for VECM ?
Thanks
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Rashad,
    The generally accepted protocol is to include enough autoregressive lags in the VAR or VECM to neutralize the bias that would result from failure to control for the autoregression.  The Schwartz Crtierion is often used as the information criterion to indicate the optimal number of lags to include in the model.  Helmut Lutekpohl discusses this in New Introduction to Multiple Time Series Analysis (2005) Sprinter in Chapter 4 on VAR order selection.  However, this is the order of autoregressive lags, not the order of covariance stationarity, to which you are referring when you speak of the order of VECM.   You will need to assure that you adjusted for the proper autoregressive lag first. Then you can test for the order of covariance stationarity.   You may encounter some variables that are I(2) and others that are I(1).  If there is a cointegrating vector, as indicated by a Johansen Trace or Maximum eigenvalue test, you may proceed to extract the eigenvectors containing the long-run dynamics from a proper equilibrium correction mechanism, properly adjusting for deterministic effects.  
       I hope this helps.
          Robert
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Hello 
Here I have attached daily Kazakhstan Stock Exchange Index from Jan 2007 to Jan 2015. There is no available monthly data, only daily basis. 
So, do you know an easy way (may be using marcoses) to transform it into monthly basis index data?
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Use of daily data or monthly data will usually depend upon the research you are undertaking. It is necessary to define the time period for your research context. In case you are considering a vast time period like many years, it may be difficult to work with voluminous data esp. if you take daily data. If that is the case, in a simple way, I would suggest you take data of the last day of the month and use it as monthly data of the time series. the variations within the month will of course not be captured in that case but in long term forecasting we are really not interested in day-to-day variations.
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By considering the Black-Scholes equation, how can I obtain the fractional Black-Scholes Equation and generalized fractional Black-Scholes Equation?
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Dear Khaliq, a study related the question in the attachment...
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There are various methods to measure stochastic volatility: from the basic model to GARCH. Which books/resources that give a good applied overview would you recommend?
Practically I am looking for resources that measure not only volatilty but the assumptions required to use particular models.
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It seems that there is some confusion in  this question when you are saying : "There are various methods to measure stochastic volatility: from the basic model to GARCH".  GARCH models and Stochastic Volatility  models  are two different approaches.
However, one of the book suggested by Dragan is the best one: Handbook of Volatility Models and Their Applications (Bauwens), particularly its Chapters 6, 7 & 8.
Regards
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The MCQ is a tool used to elicit individual intertemporal discount rates k, providing a set of alternative choices between lower more immediate amounts of money vs. higher delayed amounts of money. An estimate of the respondent’s discounting rate can be calculated as the geometric mean of the k at indifference between the two questions that reflect when the respondent changes between choosing the delayed reward versus the immediate reward. However sometimes the respondents are not consistent, i.e. they provide multiple switching points. How can you calculate an estimate of k in that cases? Many thanks
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I would exclude them. They are not systematically discounting, and might be using some strange choice process, e.g. the want the delayed reward at a certain time because of some life-event, but not before or after. If so, they are incomparable to other participants. However, if the switching points are close (within 20% of delayed reward value), than maybe okay to take average. See http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2765051/
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Agent Based Artificial Financial Markets are used as an alternative of real financial markets. How far this statement is correct?
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Agent based financial markets are simulation models of real financial markets, which are built from the bottom up by specifying simple behavioral rules of a plurality of agents. They can be used to analyze specific problems ( what would be the impact of an exogenpus shock) or to provide insights for more analitical models .
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I am interested in all of methods that relevant to fPDEs.
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Go through literature. Most of the numerical work in fPDE is by Haar, Chebychev, Legendre wavelets and by Bernstein polynomials.
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Details relevant to the question are in attachment. Thank you for your answers. 
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Dear Mantas Gabrielaitis,
Thank you very much for your answer to my question, I investigated your answer and this answer is enough for now, but associated with other versions of the FP equation will give you information, see you as soon as possible...
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In stata help for LSDVC model, it is explained that the results of estimation are saved in xtlsdvc saves in e():
Scalars
e(N) number of observations e(sigma) estimates of σ from the first stage  regression
e(Tbar) average number of time periods
e(N g) number of groups
Macros
e(cmd) xtlsdvc
e(depvar) name of dependent variable
Matrices
e(b) xtlsdvc estimates
e(V) var–cov matrix of the xtlsdvc estimator
e(b lsdv) xtreg,fe estimates
e(V lsdv) var–cov matrix of the xtreg,fe estimator
Functions
e(sample) marks estimation sample
but I do not know the command that show this result.
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After using -ereturn list- if you wanted to look at the results in more detail then the command -matlist- followed by e(*) for all the numerical e(results) will display them.  If you would like to display the string results, such as e(cmd), simply use -display- followed by relevent e(*).  Finally if you want to saves the results for later use you could use either -scalar- or -matrix- commands depending on the type of e(result).
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Say, to optimize buy/sell controls from a portfolio with transaction costs over 180 periods 
Theoretically, you use spectral decomposition, search for viscosity solutions ,
discretize, etc then solve backwards ... but run into curse of dimension or it yields too difficult to interpret and unstable solutions anyhow. Some adaptive online sampling methods seem to work sufficiently well (Q-learning, TD learning, NDP, SMC etc.) . Has anyone used them ? Thanks
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Thanks Sergei, in fact I was thinking about that : a robot that learns to trade based on charts should work best if it had a way to decide when, to switch between forward (new learning) and backward (consolidation of old maps) optimization. Maybe a simple measure like, the new data, how far is it ( > X times standard deviations) ? if it's new go forward, if not, go backwards.      
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I want to find a formula for calculating the NPV of the string of past values in a situations where the interest rates are changing annually rather than the constant interest rates.
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Dear Pezhman
Some clarification is needed. NPV of past values - must amount to a Future Value, FV, as seen from the beginning of the past string you consider.
It  is correct as Miguel says, that usually NPVs use a single (the appropriate risk adjusted e.g.) interest rate (discount rate) for the cash flow of a given project. However, we do in fact sometimes see, recommend and use different interest rates for different periods.
In some countries it is recommended by Government's economic offices that Social CBAs on long-term environmental projects are conducted with time varying interest rates, e.g. in Denmark the interest rate for the first 35 years is 5 %, then 4 % until year 70 and 3 % for any impacts further out than 70 years. Similar step schemes are found in UK and France.
Note that this is interpreted as and means that a payment arising in year 50 must be discounted by 4 % all the way back to year 35 (i.e. divided by 1.04^(50-35)) and then discounted with 5% for the last 35 years. You can see that it is straightforward to translate this to your case:
As you are essentially calculating a Future Value at time T_F = 0 (today) of a past cash flow stream of length T_Past with an annually, varying interest rate r_t and annual (perhaps also time varying) cashflow C_t, you can set it up like this (I hope it is written correctly as this is a bit odd for equations):
FV (T_F = 0 = today) = Sum from (t = 0) to (t=T_Past) [C_t] Product from (i = 0) to (i=t)[1 + r_i]
That should work - it just did in Excel
Bo
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1. in determining the no. of lags to use for the vecrank (trace test, maximum eigenvalue test), I use the information criteria to determine the no. of lags. However, when estimating the VAR and checking residuals, the residuals are not well-behaved. therefore, i ignore the information criterias, and keep increasing lags until residuals well-behaved in the corrgram, but I have to go up to 15 [and i am using monthly data]. should i just use 10 lags, or whatever the Mark processed information criteria recommends, even though residuals not well behaved?
2. if in pre-testing the different series, some of the series has a significant trend component [it is trend stationary], do I need to add a trend (rtrend) in the vecrank? [usually, I do trend(rconstant)]
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I prefer a method to obtain a close form solution of volterra integral equation of second kind with non linear kernel. But if such method doesn't exist, any method will be of help.
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Thanks every one. Prof. Patrick am looking at the paper presently, It seems the paper describes numerical method.
Prof. Shakouri, thanks for the idea.
Prof. Anton, I have not come across my kernel in any literature yet. Probably the close form method does not exist. Thanks so much.
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In islamic finance the rate interest must be nul, what must be the model of portfolio in this case?
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It can be used not as a Null but as floating, linking the inflation with a gradient. Similar to inflation indexed bonds. DCF should produce the current NPV of a project, else the selection process will be incomplete and theoretical or in other terms that gain could be termed as " possible Capital Gain". However the effect will remain the same.
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Dear all,
I need your help on the conversion of quarterly data to weekly via cubic interpolation.
I have weekly market data (stock prices) and quarterly balance sheet data (i.e. total assets). The paper that I read, refers that " The daily market data
is converted to a weekly frequency and matched with interpolated values of the quarterly balance sheet data. With that it is possible to generate weekly time series of growth rates of market-valued total assets..."
I will really appreciate your help.
Best regards.
Kostas
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I agree to Fernando  that the state-of-art approach is to use MIDAS or its State Space Equivalent. You may also refer to :
Bai, Jennie, Eric Ghysels, and Jonathan H. Wright (2013), "State space models and MIDAS regressions." Econometric Reviews 32.7 : 779-813.
But if you are still interested in conventional methods; the low frequency data can be converted into high frequency data using some kind of interpolation. Procedures are available in standard statistical software packages for that purpose. For example you can have a look at  "tempdisagg" package of  R.
However, you should be careful about "flow"and "stock" data while converting balance sheet items from quarterly to weekly frequency. For example if quarterly sale is $ 1.3 million, weekly sale will be around $ 0.1 million. But if quarter end balance of fixed assets is $1.3 million it will not become $0.1 million at weekly frequency, rather it will remain at the level of $1.3 million only.
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Using the historical var-cov matrix as an input in the optimizer leads to estimation errors. What other methods can be used in estimating the var-covar apart from shrinkage and diagonal methods?
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Implied Volatility as it is only the market's prediction of it from a Black- Scholes model, with quoted option prices , need not necessarily follow the statistical properties of a variance- co variance matrix .'Volatility smile' is an example for it
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Especially if this is a multi-factor model.
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Yes, I agree. Fama and MacBeth regression seems to tell us that we can get more accurate risk premium than original factor return. And thus we should use the cross section results as 'real' factor return instead of the original ones.
It is a pity that we cannot get the original factor returns from betas, which I have planned to do.
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Searching for the present value for a linear interest rate series.
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Dear Joao,
according to what I now, no way!
I suggest a semi-lexical distinction between "sum" and "series". They are different animals.and they must be kept distinct.
A sum is only a sum: the number of addenda is finite.
A series is a sum with an infinity of addenda and it is something else.
The widespread notion of "finite series" is simply pernicious.
Here are significant cases:
(1) - You have a sum, you have a closed form for it, you try to compute the limit of the sum for n -> + infinity. This is not the case.
(2) - You have a series (an infinity of addenda), but you are able to compute differently the sum(*), very well, but this, once again, this is not the case.
Of course I could be wrong. I'm Italian and hence Latin. A well known sentence in Latin, which is here relevant is "Onus probandi incumbit ei qui dicit".
I translate for people that don't practice Latin: "athe burden to prove an assertion concerns who asserts".
(*) - In the case of interest I can provide the indication of some interesting books (for instance H.G. GARNIR (1965): "Fonctions de variables réelles", Tome II, Librairie Universitaire - Louvain et Gauthier-Villars - Paris: it's in French, but nobody's perfect).
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In the paper by Hwa Kil Kim published in June 14, 2012. What is the meaning of (R^D) and the meaning of (J:(R^D)--->(R^D) is a matrix satisfying (
(J_v )_|_v) for all v in (R^D) ). The name of the paper is:Moreau-Yosida approximation and convergence of Hamiltonian systems on Wasserstein space, and it is on RG.
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It looks like D is a natural number and the condition on J means that v and Jv are orthogonal for each vector v, i.e. their scalar product is 0.
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Financial modeling
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Hi dear. On the Cochrane's personal page (one of the best in finance), you can find a lot of tinge:
Best, Matteo
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I am talking about the parameter that will make the non-arbitrage condition consistent with the model (as used and described by Hibbert, Mowbray & Turnbull, 2001, and Ahlgrim, D'Arcy & Gorvett, 2004). In a one-factor Vasicek model, some papers refer to it as the "lambda" and incorporate it in the equation to derive the price of any zero-coupon bond of maturity T. With a two-factor Vasicek model, i.e. one long term factor and one short term factor, I can not find any source with an equivalent parameter for the two-factor model and the calculus seems extremely complicated...
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Appendices in "Expectation Puzzles, Time-varying Risk Premia, and Dynamic Models of the Term Structure", Dai and Singleton
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Several books say that both the call and the put option get higher price when the volatility is higher.
I may understand it as , the stock price gets low when volatility is high , this will make the put option price higher, yet i cannot see  why it also makes the call option high.
Of course we may say, since the volatility means risk and thus high risk makes risky price high, yet this saying seems too naive.
We are in basic option stage and thus cannot apply BS pricing formula, just want a cheap and clear classroom answer, if it is possible.
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Typically the volatility measure does express only an estimate of the expected size of up or down moves, but not the direction.
In simplified terms, assume your one-day maturity option is at-the money, your daily expected move is either 10% up or down. Then your call payout is either 10% for an up move, or 0% for a down move, the equivalent put payout is either 0% for an up move or 10% for a down move. Thus in expectation call and put are both valued at the average of 5%.
Option pricing typically assumes a hedged position, meaning you own the option in combination with holding the offsetting numbers of shares. This eliminates your sensitivity with respect to the direction of the move of the underlying and leaves you mainly exposed to the expected size of the move, which you express in terms of volatility.