Not sure if this is a relevant question for site, but I am looking to move to www.eoddata.com as my data source. If anyone has used it, can you tell me how the data quality is ? I am currently parsing yahoo for my prices but its not a very efficient process. What are the other sources of equity data out there ( affordable for retail investors like me) ? I am willing to pay \$250-\$300 per year for it.

Yahoo rounds the adjusted price to 2 decimals even though dividend amounts often have 3 decimal places. Since they apply the adjustment formula to adjusted prices, if you go far enough back in time, the value they give for Adjusted Price will be different than it would be if there were no rounding. edit: For example, for C (Citigroup), on January 2, 1990, Yahoo gives a close value of 29.37 and an Adjusted value of 1.50. Using the dividend data that Yahoo supplies, if they didn't round to cents on every adjustment, the adjusted value would be 1.677.

From my experience, EODData is pretty much that you get what you pay for. Its not a very sophisticated product. They email you the files you subscribe to, and thats that. I have had an issue before of where the emails didn't go through anymore and I never heard anything from them. On Quality, I can't make a claim on its accuracy. It seems good. I find issue with three things about the data: There is no way to signal an exchange change It doesn't give you an easy way to grab the symbol info [Names, etc] If there is...


Did Citigroup really fail three times in the last 40 years?

I watched a speech by Simon Johnson at UCSB and, at one point, he claimed that Citigroup has failed three times since the 1980s. For example, he claims that Citigroup failed and was saved by the government in 1982 because of "bad loans made in emerging markets". The second such failure is at the end of the 1980s because of "bad loans to commercial real estate". The third is of course the 2008/2009 financial crisis. What strikes me as odd is that the first two alleged failures are not mentioned in Wikipedia on Citigroup ! He doesn't appear to me as a crackpot as he is the former chief economist of IMF and effective whitewashing of the US's biggest bank holding company on Wikipedia is as unlikely. Does anyone know what affairs is he referencing?

(I worked there for 23 years.) Simon Johnson is correct. Citi (or its predecessors) was insolvent on those 3 occasions, and would have gone into liquidation without the bailouts by U.S. taxpayers.

I have worked on this topic extensively (pricing and calculating IV in production) and believe can offer an informed opinion. First of all Mathworks - the company that creates Matlab is not a trading firm so you should probably not rely on their advice so much. There are few closed form options pricing models, and all have practical shortcomings. Barone-Adesi and Whaley (please correct my spelling of last names as I'm typing from memory) model is simple approximation for American options but is unfortunately not very accurate, and does not deal with dividends. Roll-Geske-Whaley deals with dividends, but not very...

This isn't really an answer, but it's too long to add as a comment. I've always had a real problem with the correlation/covariance of price . To me, it means nothing. I realize that it gets used (abused) in many contexts, but I just don't get anything out of it (over time, price has to generally go up, go down, or go sideways, so aren't all prices "correlated"?). On the flip side, correlation/covariance of returns makes sense. You're dealing with random series, not integrated random series. For example, below is the code required to generate two price series that have...

It's an interesting question. I particularly agree with the $\mathbb{Q}-\mathbb{P}$ dichotomy mentioned by many. I would add to the other answers that, come to think of it, the Black-Scholes postulated Geometric Brownian Motion could be interpreted as an AR(1) process on the logarithm of the stock price as you discretise the SDE from which it is a solution, which is exactly what you do when running Monte-Carlo simulations (same thing for the Ornstein-Uhlenbeck process as explained here and noted by @Richard). Actually, when taking the continuous-time limit, many more econometric models can be shown to correspond to stochastic processes frequently...


What is the "delta" option quoting convention about?

At my work I often see option prices or vols quoted against deltas rather than against strikes. For example for March 2013 Zinc options I might see 5 quotes available for deltas as follows: ZINC MARCH 2013 OPTION DELTA -10 POINTS ZINC MARCH 2013 OPTION DELTA -25 POINTS ZINC MARCH 2013 OPTION DELTA +10 POINTS ZINC MARCH 2013 OPTION DELTA +25 POINTS ZINC MARCH 2013 OPTION DELTA +50 POINTS The same 5 values -10, -25, +10, +25, + 50 are always used. What are these deltas? What are the units? Why are those 5 values chosen?

Delta is the partial derivative of the value of the option with respect to the value of the underlying asset. An option with a delta of 0.5 (here listed as +50 points) goes up \$0.50 if the underlying asset goes up \$1.00. Or goes down \$0.50 if the underlying asset goes down \$1.00. Keep in mind that delta is an instantaneous derivative, so the value will change both in time (charm is the change in delta with time) and with changes in value of the underlying asset (gamma is the change in delta with the underlying asset, which is also...

Scott Mixon argues in What Does Implied Volatility Skew Measure that among all measures of implied volatility skew, the (25 delta put volatility - 25 delta call volatility)/50 delta volatility is the most descriptive and least redundant (volatility is Black-Scholes implied volatility). His paper, recently published in the Journal of Derivatives , gives a number of both theoretical and empirical arguments in favor of this measure. He distinguishes between "skew", which is a measure of the slope of the implied volatility curve for a given expiration date, and "skewness", which is the skewness of an option implied, risk neutral probability...

The volatiltiy surface is just a representation of European option prices as a function of strike and maturity in a different "unit" - namely implied volatility (while the term implied volatility has to be made precise by the model used to convert prices (quotes) into implied volatilities - for example: we may consider log-normal vols and normal vols). Volatility is often preferred over prices, e.g., when considering interpolations of European option prices (although this may introduce difficulties like arbitrage violations, see, e.g., http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1964634 ). A local volatility model can generate a perfect fit to the implied volatility surface via Dupire's...

VIX is a measure of implied volatility , specifically, model-free implied volatility , a concept originally developed by Demeterfi et. al. at Goldman Sachs in the 1990s . One of my recent questions, How to extrapolate implied volatility for out of the money options? , addressed some aspects of MFIV, and the papers mentioned in the question and answers will give you some more background on the concept. Also check out this JPMorgan research piece . The VIX is the square root of the market-implied variance swap rate, calculated based on the exchange-traded options portfolio typically used to hedge an...


How do you evaluate a covariance forecast?

Suppose you have two sources of covariance forecasts on a fixed set of $n$ assets, method A and method B (you can think of them as black box forecasts, from two vendors, say), which are known to be based on data available at a given point in time. Suppose you also observe the returns on those $n$ assets for a following period (a year's worth, say). What metrics would you use to evaluate the quality of these two covariance forecasts? What statistical tests? For background, the use of the covariances would be in a vanilla mean-variance optimization framework, but one can assume little is known about the source of alpha. edit : forecasting a covariance matrix is a bit different, I think, than other forecasting tasks. There are some applications where getting a good forecast of the eigenvectors of the covariance would be helpful, but the eigenvalues are not as important. (I am thinking of the case where one's portfolio is $\Sigma^{-1}\mu$, rescaled, where $\Sigma$ is the forecast covariance, and $\mu$ is the forecast returns.) In that case, the metric for forecasting quality should be invariant with respect to scale of the forecast. For some cases, it seems like forecasting the first eigenvector is more important (using it like beta), etc . This is why I was looking for methods specifically for covariance forecasting for use in quant finance.

This isn't really an answer, but it's too long to add as a comment. I've always had a real problem with the correlation/covariance of price . To me, it means nothing. I realize that it gets used (abused) in many contexts, but I just don't get anything out of it (over time, price has to generally go up, go down, or go sideways, so aren't all prices "correlated"?). On the flip side, correlation/covariance of returns makes sense. You're dealing with random series, not integrated random series. For example, below is the code required to generate two price series that have...

Garabedian, Typically, the "swap curve" refers to an x-y chart of par swap rates plotted against their time to maturity. This is typically called the "par swap curve." Your second question, "how it relates to the zero curve," is very complex in the post-crisis world. I think it's helpful to start the discussion with a government bond yield curve to clarify some concepts and terminologies. Consider the US Treasury market, using the outstanding Treasury notes and bonds (nearly 300 of them...), we can either use bootstrapping or more sophisticated spline models to construct a "fitted curve." Since this yield curve...

A general model (with continuous paths) can be written $$ \frac{dS_t}{S_t} = r_t dt + \sigma_t dW_t^S $$ where the short rate $r_t$ and spot volatility $\sigma_t$ are stochastic processes. In the Black-Scholes model both $r$ and $\sigma$ are deterministic functions of time (even constant in the original model). This produces a flat smile for any expiry $T$. And we have the closed form formula for option prices $$ C(t,S;T,K) = BS(S,T-t,K;\Sigma(T,K)) $$ where $BS$ is the BS formula and $\Sigma(T,K) = \sqrt{\frac{1}{T-t}\int_t^T \sigma(s)^2 ds}$. This is not consistent with the smile observed on the market. In order to match...

You are correct: evaluating volatility forecasts is quite different from evaluating forecasts in general, and it is a very active area of research. Methods can be classified in several ways. One criterion is to consider evaluation methods for single forecasts (e.g., for the time series of returns of a specific portfolio) vs multiple simultaneous forecasts (e.g., for an investable universe). Another criterion is to separate direct evaluation methodsfrom indirect evaluation methods (more on this later). Focusing on single-asset methods: historically the most commonly used approach by practitioners, and the one advocated by Barra is the "bias" statistics. If you have...


When should you build your own equity risk model?

Commercial risk models (e.g., Barra , Axioma , Barclays , Northfield ) have evolved to a very high level of sophistication. However, all of these models attempt to solve a very broad set of problems. The optimal risk model for, say, risk attribution in a fundamental portfolio may differ substantially from the optimal risk model for downside risk estimation of an optimized quantitative strategy or for hedging unwanted exposures in a pure relative value play. Suppose that one already subscribes to a decent risk model provider, so that cost is not an issue. For what applications is it most appropriate to build your own equity risk model? What are the main benefits of a customized risk model? When is it worth the time and effort to replicate the increasingly sophisticated data cleaning/analysis and statistical methods to reap these benefits?

1. Determine Factors Economically, the use of factor models can be either motivated using the ICAPM or the APT . Although there are some theoretical differences between the model, for empirical and practical work these differences are irrelevant. In the end, both models stipulate that returns and expected returns are linear functions of the factors: $$ r_{i,t} = \alpha_i + \sum_j \beta_{i,j} F_{j,t} + \epsilon_{i,t} \quad (1)$$ $$ \mathbb{E}[ r_{i,t}] = \lambda_o + \sum_j \beta_{i,j} \lambda_j \quad\quad\quad(2)$$ where $F_{j,t}$ is the factor surprise of factor $j$ at time $t$ and $\lambda_j $ is the factor risk premium of factor $j$....

Here couple pointers that may make it clearer: Drift can be replaced by the risk-free rate through a mathematical construct called risk-neutral probability pricing. Why can we get away with that without introducing errors? The reason lies in the ability to setup a hedge portfolio, thus the market will not compensate us for the drift above and beyond the risk free rate under risk-neutral probability pricing. As long as such hedge exists and couple other conditions are met (please look up Girsanov's Theorem) we can introduce a risk-neutral measure so that when applying it to the differential equation and through...

I think you're overlooking a third explanation: Nobody that found a successful technique to generate alpha has published it. I can think of the following causes: If you're an academic, why share your brilliant idea? These techniques require a lot of data and financial data can be expensive, researches that work at firms that have access to this data don't share their findings with the public. Academics did find a lot of signals already the old fashioned way. Despite this, fancy techniques such as AAD and Reinforcement Learning are discussed publicly. These methods don't generate any alpha however.

Great question. We would expect 3rd party risk providers to have specialized expertise (robust regression techniques, factor research, data cleansing etc.). We might grant them these advantages but still find weakness in the product design. Let's start off with the different uses of risk models and the procedure or metric which is maximized to solve for that use case. What we will see is that solving for a particular objective diminishes our ability to achieve other objectives. Portfolio construction = If you want to construct a minimum variance portfolio, for example, then the key here is developing a covariance matrix...


How to combine multiple trading algorithms?

Is it possible to combine different algorithms so as to improve trading performance? In particular, I have read that social media sentiment tracking, digital signal processing and neural networks all can be used for trading algorithms. Would it be possible to create a trading algorithm that combines elements from these three areas or are these methods mutually exclusive in that they are incompatible with each other? If you commit to one, can you use the other?

trading algorithmic-trading quant-trading-strategies performance

In order to answer your question (for you) you would need something to compare to . You would need numbers to know if it is slower/faster, how much, and if it will impact your system overall. Also knowing your performance goals could narrow down the options. My advice is to take a look at your overall architecture of the sytem you have or intend to build. To just look at QuickFIX is rather meaningless without the whole chain involved in processing information and reacting to it . As an example, say QuickFIX is 100 times faster than some part (in...

The way you do it in the first place is a discretization of the Geometric Brownian Motion (GBM) process. This method is most useful when you want to compute the path between $S_0$ and $S_t$, i.e. you want to know all the intermediary points $S_i$ for $0 \leq i \leq t$. The second equation is a closed form solution for the GBM given $S_0$. A simple mathematical proof showed that, if you know the initial point $S_0$ (which is $a$ in your equation), then the value of the process at time $t$ is given by your equation (which contains $W_t$,...

Yes. First, it is much easier to proceed if you standardize the output of your forecast so they are in the same units (returns, for example, or probabilities of an event/condition occurring). After you have done this, there are 3 general approaches: Signal weighting: Then you need to define a weighting scheme for your factors. Richard Grinold has an one answer to this question in his paper " Signal Weighting ". Note there are quite a few methods to weight signals (optimization, meta-models, forecast pooling, Bayesian model averaging, weighing based on out-of-sample performance, etc.). The general problem of "Signal Weighting"...


Returns possess non-zero skewness and excess kurtosis. If these assets are temporally aggregated both will disappear due to the law of large numbers. To be exact, if we assume IID returns skewness scales with $\frac{1}{\sqrt{n}}$ and kurtosis with $\frac{1}{n}$. I'm interested in a concise, clear and openly accessible proof for the above statement preferably for all higher moments. This question is inspired by this question by Richard which deals with, among other things, the behaviour of the higher moments of returns under temporal aggregation. I know about two papers that answer this question. Hawawini (1980) is wrong and Hon-Shiang and Wingender (1989) is behind a paywall and a bit inscrutable.

statistics returns volatility-skew kurtosis

Just to be painfully clear, it only seems to make sense to consider the logarithm of returns, i.e. $X=\log (1+\frac r{100})$ for a simple return of $r\%$ in an arbitrary period because this is what sums when returns are temporally aggregated. A basic property of cumulants is that cumulants of all orders are additive under convolution, for which a proof can be found here here . So if $X_1$, $X_2$, ... $X_n$ are i.i.d. , then all the cumulants of $$Y_n = \sum_{i=1}^nX_i$$ scale linearly with $n$, i.e. $$\kappa_k(Y_n)=n\kappa_k(Y_1).$$ However, I suspect that you are normalizing this sum so that...

There are some cases where you can blend your portfolios using weights directly. One case involves corner portfolios . In this case a linear combination of weights is also efficient. Another case is where you can treat the two separate weights you have produced each as distinct portfolio under the assumption that the correlation between these portfolios is relatively stable. In this scenario, the problem reduces to a two-asset portfolio optimization problem (each asset is simply the linear combination of weights produced via your two methods). The other class of methods involves blending via the expected returns. If you arrived...

The term has a different meaning to different people. to econometricians, microstructure noise is a disturbance that makes high frequency estimates of some parameters (e.g. realized volatility) very unstable. Generally this strand of the literature professes agnosticism as to the its origin; to market microstructure researchers, microstructure noise is a deviation from fundamental value that is induced by the characteristics of the market under consideration, e.g. bid-ask bounce, the discreteness of price change, latency, and asymmetric information of traders. The last example is frequently cited but I don't think it is accurate. Asymmetric information does not have to be a...

There seems to be a basic fallacy that someone can come along and learn some machine learning or AI algorithms, set them up as a black box, hit go, and sit back while they retire. My advice to you: Learn statistics and machine learning first, then worry about how to apply them to a given problem. There is no free lunch here. Data analysis is hard work . Read "The Elements of Statistical Learning" (the pdf is available for free on the website), and don't start trying to build a model until you understand at least the first 8 chapters....


Portfolio optimisation with VaR or CVaR constraints using linear programming

I would like to optimize a portfolio allocation (maximizing the exposure or the expected return), but with VaR or CVaR contraints. (some parts of my portfolio cannot exceed a certain VaR) How can I achieve that? Is there a way to turn the problem into a linear programming problem? or to approximate the results? Any links or ideas are welcome.

portfolio-management portfolio-optimization optimization value-at-risk

In my experience, a VaR or CVaR portfolio optimization problem is usually best specified as minimizing the VaR or CVaR and then using a constraint for the expected return. As noted by Alexey, it is much better to use CVaR than VaR. The main benefit of a CVaR optimization is that it can be implemented as a linear programming problem. Another option I have tried is the technique in this paper: http://www.math.uwaterloo.ca/~tfcolema/articles/bank_article.pdf Another option is the two-step heuristic where one first finds the mean-variance efficient frontier and then you could calculate whatever are the relevant portfolio statistics on only the...

I've worked at a hedge fund that allowed GA-derived strategies. For safety, it required that all models be submitted long before production to make sure that they still worked in the backtests. So there could be a delay of up to several months before a model would be allowed to run. It's also helpful to separate the sample universe; use a random half of the possible stocks for GA analysis and the other half for confirmation backtests.

This question goes to whether the historical returns to factors represent: Spurious results, overfitting, data mining... Mispricing Unexploitable effects Compensation for risk Case 1: Spurious results etc... If someone constructs a "stock tickers that begin with AAP or GOO" factor, the highly above average returns would almost certainly reflect a fishing expedition (or conditioning on future information) and would not be reproducible going forward. Under a null of no above average returns, you're going to get portfolios that have above average historical returns with t-stats over 2. Beware. For something like the Fama-French factor $\mathit{HML}$, this seems far less likely...


What is the most stable, non-trivial dependence structure in finance?

The highest rated answer to the question on What concepts are the most dangerous ones in quantitative finance work? is this one: Correlation Correlations are notoriously unstable in financial time series [...] My question My question is a little bit broader than just about linear dependence, it is: What is the most stable, non-trivial dependence structure in financial data? With non-trivial I mean that I don't want answers that are about direct connections, e.g. between derivative and underlying. The dependence structure can be either cross-sectional or through time with univariate time series, it can also be non-linear . The context of my question is that I am preparing the documentation for a new machine learning R package I wrote and I am looking for a good showcase in the financial sphere. Now this is not a trivial feat given that correlations are notoriously... see above ;-)

It is hard to find a stable non-trivial dependence structure in financial data. Usually when such is found it is hard to rationalize. One of my favorite (although I am sure there are others) is the so called "Presidential Puzzle". This is an old finding by Santa-Clara and Valkanov (2003) They find that " Excess return in the stock market is higher under Democratic than Republican presidencies: 9 percent for the value‐weighted and 16 percent for the equal‐weighted portfolio. At the time the finding was very robust and did not seem to be explained by anything else. What is more...

I would consider a motion chart that plots the eigenvalues of the covariance matrix over time. For a static view you can create a table: rows represent dates, and columns represent eigenvectors. The entries of the table represent changes in the angle of the eigenvector from the previous row. This will show how stable your covariance structure is. You can also create a second table this time with eigenvalues as the columns sorted from high to low (and the corresponding values below for each date). This shows the variance described by each eigenvector so you can see whether correlation as...

In general there are two basic ways to make money out of your option pricing models: Sell side (market maker, risk neutral): You use these models to calculate your greeks to hedge your portfolio, so that you live on the spread. Buy side (market/risk taker): You use your model to find mispriced options in the market and buy/sell accordingly. (A third possibility would be to write fancy books and papers about these models and get rich and/or tenure this way ;-)

I have worked on this topic extensively (pricing and calculating IV in production) and believe can offer an informed opinion. First of all Mathworks - the company that creates Matlab is not a trading firm so you should probably not rely on their advice so much. There are few closed form options pricing models, and all have practical shortcomings. Barone-Adesi and Whaley (please correct my spelling of last names as I'm typing from memory) model is simple approximation for American options but is unfortunately not very accurate, and does not deal with dividends. Roll-Geske-Whaley deals with dividends, but not very...


What are reasons not to do factor investing in equity markets?

Factor investing in equity markets is one of the hot topics of these days. Many manufacturers of investment products offer exposure to small cap, momentum, minvol, value and other pure factors or factor blends. Many of them beating the cap-weighted index at relatively low cost. I think of various reasons but I woud like to discuss: What are the reasons not everybody just invests in factor portfolios? What could be limitations, regulations, fears or other reasons not to invest in factor portfolios?

This question goes to whether the historical returns to factors represent: Spurious results, overfitting, data mining... Mispricing Unexploitable effects Compensation for risk Case 1: Spurious results etc... If someone constructs a "stock tickers that begin with AAP or GOO" factor, the highly above average returns would almost certainly reflect a fishing expedition (or conditioning on future information) and would not be reproducible going forward. Under a null of no above average returns, you're going to get portfolios that have above average historical returns with t-stats over 2. Beware. For something like the Fama-French factor $\mathit{HML}$, this seems far less likely...

Have a look at this classic paper: Honey, I Shrunk the Sample Covariance Matrix by O. Ledoit and M. Wolf The abstract answers your question already: The central message of this article is that no one should use the sample covariance matrix for portfolio optimization. It is subject to estimation error of the kind most likely to perturb a mean-variance optimizer. Instead, a matrix can be obtained from the sample covariance matrix through a transformation called shrinkage. This tends to pull the most extreme coefficients toward more central values, systematically reducing estimation error when it matters most. Statistically, the challenge...

From what I remember, there is no real relation between Markov and Martingale, and my intuition was confirmed by this post . Basically, it says that you can say neither of the following: If A is Markov, then A is a martingale. If A is a martingale, then A is Markov. further down the post, you can find two counter examples: $dX_t = a dt + \sigma dW_t$ is Markov but not a martingale and $dX_t = (\int_0^t X_s ds) dW_t$ is a Martingale but is not Markov. As for the assumption of these properties being true, I think it...

I know that I have seen things like this in the past. Wasn't there something recently that used Twitter? Here are a few recent papers as examples, although I will be brutally honest that I don't know if they speak to your decent quality requirement: "Trading Strategies to Exploit Blog and News Sentiment" (Zhang, Skiena 2010) "The Predictive Power of Financial Blogs" (Frisbee 2010) "An analysis of verbs in financial news articles and their impact on stock price" (Schumaker 2010)


Why does Gamma Pnl have exposure to realised volatility, but Vega Pnl only has exposure to implied volatility? I am confused as to why gamma pnl is affected (more) by IV and why vega pnl isnt affected (more) by RV? Essentially how do you show what gamma pnl will be mathematically and how do you show what vega pnl will be? I believe that gamma pnl is spot x (vega x IV - RV) Also does gamma pnl usually dominate (in $ terms) the vega pnl of an options, as most literature is on gamma pnl?

options implied-volatility option-strategies gamma vega

The risk-neutral measure $\mathbb{Q}$ is a mathematical construct which stems from the law of one price , also known as the principle of no riskless arbitrage and which you may already have heard of in the following terms: "there is no free lunch in financial markets". This law is at the heart of securities' relative valuation , see this very nice paper by Emmanuel Derman ("Metaphors, Models & Theories", 2011) and some part of this discussion. In what follows, assume for the sake of simplicity existence of a risk-free asset ; deterministic and constant rates, with risk-free rate $r$ ;...

The volatiltiy surface is just a representation of European option prices as a function of strike and maturity in a different "unit" - namely implied volatility (while the term implied volatility has to be made precise by the model used to convert prices (quotes) into implied volatilities - for example: we may consider log-normal vols and normal vols). Volatility is often preferred over prices, e.g., when considering interpolations of European option prices (although this may introduce difficulties like arbitrage violations, see, e.g., http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1964634 ). A local volatility model can generate a perfect fit to the implied volatility surface via Dupire's...

For an option with price $C$ , the P $\&$ L, with respect to changes of the underlying asset price $S$ and volatility $\sigma$ , is given by \begin{align*} P\&L = \delta \Delta S + \frac{1}{2}\gamma (\Delta S)^2 + \nu \Delta \sigma, \end{align*} where $\delta$ , $\gamma$ , and $\nu$ are respectively the delta, gamma, and vega hedge ratios. Then it is clear the vega P $\&$ L has exposure to the change of the implied volatility $\sigma$ . Note that, for the gamma P $\&$ L, \begin{align*} \frac{1}{2}\gamma (\Delta S)^2 = \frac{1}{2}\gamma S^2 \frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2\Delta t, \end{align*} where...

In general there are two basic ways to make money out of your option pricing models: Sell side (market maker, risk neutral): You use these models to calculate your greeks to hedge your portfolio, so that you live on the spread. Buy side (market/risk taker): You use your model to find mispriced options in the market and buy/sell accordingly. (A third possibility would be to write fancy books and papers about these models and get rich and/or tenure this way ;-)


How are distributions for tail risk measures estimated in practice?

Let's say you want to calculate a VaR for a portfolio of 1000 stocks. You're really only interested in the left tail, so do you use the whole set of returns to estimate mean, variance, skew, and shape (let's also assume a skewed generalized error distribution - SGED)? Or would you just use the left tail (let's say the bottom 10% of returns)? For some reason, using the whole set of returns seems more correct to me (by using only the left 10% or returns you'd really be approaching a non-parametric VaR). But using the whole set of returns would likely cause some distortion in the left tail in order to get a better fit elsewhere. How are the pros doing this? Thanks!

risk probability estimation value-at-risk expected-return

Perhaps you may want to consider article by D. Levine - Modeling Tail Behavior with Extreme Value Theory who gives practicale example on how EVT can be used to calculate probabilities on returns in tails with use of the Pickands-Balkema-de Haan Theorem and generalized Pareto distribution. It also contains some criterias and points on other methods that can be used to determine threshold value for PBH theorem: Contrary to this notion is the fact that the PBH theorem states a result based on the assumption of threshold values approaching the right endpoint of the distribution F. This implies that better...

Great question. We would expect 3rd party risk providers to have specialized expertise (robust regression techniques, factor research, data cleansing etc.). We might grant them these advantages but still find weakness in the product design. Let's start off with the different uses of risk models and the procedure or metric which is maximized to solve for that use case. What we will see is that solving for a particular objective diminishes our ability to achieve other objectives. Portfolio construction = If you want to construct a minimum variance portfolio, for example, then the key here is developing a covariance matrix...

One starts with the Black-Scholes equation $$\frac{\partial C}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2}+ rS\frac{\partial C}{\partial S}-rC=0,\qquad\qquad\qquad\qquad\qquad(1)$$ supplemented with the terminal and boundary conditions (in the case of a European call) $$C(S,T)=\max(S-K,0),\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad(2)$$ $$C(0,t)=0,\qquad C(S,t)\sim S\ \mbox{ as } S\to\infty.\qquad\qquad\qquad\qquad\qquad\qquad$$ The option value $C(S,t)$ is defined over the domain $0 Step 1. The equation can be rewritten in the equivalent form $$\frac{\partial C}{\partial t}+\frac{1}{2}\sigma^2\left(S\frac{\partial }{\partial S}\right)^2C+\left(r-\frac{1}{2}\sigma^2\right)S\frac{\partial C}{\partial S}-rC=0.$$ The change of independent variables $$S=e^y,\qquad t=T-\tau$$ results in $$S\frac{\partial }{\partial S}\to\frac{\partial}{\partial y},\qquad \frac{\partial}{\partial t}\to - \frac{\partial}{\partial \tau},$$ so one gets the constant coefficient equation $$\frac{\partial C}{\partial \tau}-\frac{1}{2}\sigma^2\frac{\partial^2 C}{\partial y^2}-\left(r-\frac{1}{2}\sigma^2\right)\frac{\partial C}{\partial y}+rC=0.\qquad\qquad\qquad(3)$$ Step 2. If we...

Nick Higham's specialty is algorithms to find the nearest correlation matrix. His older work involved increased performance (in order-of-convergence terms) of techniques that successively projected a nearly-positive-semi-definite matrix onto the positive semidefinite space. Perhaps even more interesting, from the practitioner point of view, is his extension to the case of correlation matrices with factor model structures. The best place to look for this work is probably the PhD thesis paper by his doctoral student Ruediger Borsdorf. Higham's blog entry covers his work up to 2013 pretty well.