Factor models such as Fama-French or the other ones that are partially summarized here work on the cross-section of asset returns. How are the factors built, how are sensitivities/coefficients estimated? In this context Fama-MacBeth regressions are usually mentioned. How does this method work intuitively? Could anyone give a step-by-step manual? EDIT: Links to papers and manuals have been posted in the two answers - this is great. But can someone provide more intuition in the answer? Say we have a universe of stocks (say MSCI Europe) and we group them by value and size. How can we proceed? How do we construct the factors and how do we construct the sensitivities? Could someone please give a more direct explanation, without a link? thanks!

The standard story (also told by @vonjd) is of "The Drunk and Her Dog". This is based on "A Drunk and Her Dog: An Illustration of Cointegration and Error Correction" (1994). The story is itself based on the standard illustration for a random walk known as the "drunkard's walk". The Dickey-Fuller test is used to check for a unit root . It can be used as part of the general Engle-Granger two-step method (although it isn't the only option). In this case, while the two assets themselves are not stationary, you are able to test if the residuals between a...

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.

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 $j$....


Has high frequency trading (HFT) been a net benefit or cost to society?

Various studies have demonstrated the very large and growing influence of high frequency trading (HFT) on the markets. HFT firms are clearly making a great deal of money from somewhere, and it stands to reason that they are making this money at the expense of every other participant in the market. Defenders of HFT will argue that HFT firms provide an essential service to the economy in the form of greater liquidity. What research has been done on the benefits and costs of HFT? Has any study attempted to measure either the benefits or the costs? How would one attempt to measures these benefits and costs? What would be the effect of banning rapidly cancelled limit orders (see follow-up question ), e.g. via a minimum 1-second tick rule? Any references and professional opinions (backed by research) on this topic would be appreciated.

algorithmic-trading research high-frequency market-microstructure

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...

The lead paper in the January 2011 Journal of Finance ( Hendershott, Jones, and Menkveld ) addresses algorithmic trading (AT). In short, they find that AT improves liquidity as measured by bid-offer spreads. Taking the econometrics as correct (it is in the Journal of Finance) the next question is if bid-offer spreads are a sufficient statistic for measuring liquidity (or any other benefits). It is a difficult question to answer because, given current market structure, AT may improve liquidity (as measured by bid-offer spreads), but without data on other market structures, it is hard to say that we wouldn't better...

The minimum variance solution loads up on securities that have low variances and co-variances. Theoretically you are correct that this should have a low expected return profile. However, it turns out - in contradiction to modern portfolio theory - that securities that have low-volatility or low-beta experience higher returns than high-volatility or high-beta stocks. This is well-documented in the literature as the low-volatility anomaly . As a result, many funds and ETFs have been launched in recent years to exploit this phenomenon. There are a couple arguments as to why the anomaly exists. The paper I cite above argues that...

I can think of an application in options pricing. I came across the following paper a long time ago but think it explains FT very eloquently as applied to pricing options under BS: http://maxmatsuda.com/Papers/2004/Matsuda%20Intro%20FT%20Pricing.pdf The fun starts on page 112 but it relies on the 1998 paper by Madan and Carr. What I like about the paper is that it gives a thorough introduction to FT and only when the groundwork is set it applies it to option pricing. Not a bad approach vs many other papers which make a lot of assumption and assume the reader can jump right...


How to estimate real-world probabilities

In the world of finance, Risk-neutral pricing allow us to estimate the fair value of derivatives using the risk free rate as the expected return of the underlyings. However, the behavior of financial assets in the real-world might be substantially different to the evolution used in a risk-neutral context. For instance, if I want to estimate the real-world probability of an equity asset reaching certain thresholds, which models and calibration techniques could be used? In particular, some questions that may arise in the estimation of real-world probabilities are: Calibration : Should real-world probabilities be calibrated to current market prices or, alternative, historical data should be used for this type of estimation? No-arbitrage conditions : Could they be relaxed or they still play a role in the assessment of real-world probabilities? Expected returns : Assuming that I have already estimated the expected return of an asset $\mu$, how accurate would be a real-world estimation that combines a widely used evolution model (e.g. Geometric Brownian motion ), with the use of $\mu$ instead of the risk free rate $r$? Per comments, I understand that in order to estimate real-world probabilities: I should use expected returns instead of the risk-free rate. The asset evolution should still respect the no-arbitrage conditions (i.e: the real-world dynamics should still reproduce the current prices of vanilla options). However, if we just use $\mu$ instead of $r$, the underlying asset behavior might not be consistent with the observed option prices. For instance, if we just change $r$ by $\mu$ (with $\mu>r$) the underlying asset dynamics will lead to call prices above its current market price, and put prices below its market price. Therefore, in addition to use expected returns, which other adjustment might be needed in order to estimate real-world probabilities? Any papers or references regarding real-world estimation will be greatly appreciated.

derivatives risk-neutral-measure forecasting stochastic-discount real-world-measure

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 ;-)

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....

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 $r$ ;...

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...


In my firm we are beginning a new OMS (Order Management System) project and there is a debate whether we use Quickfix or we go for a professional fix engine? Because there is a common doubt that QuickFix is not enough fast and obviously we will not get any technical support. I heard that in BOVESPA it has been used for a while. They are changing it with a paid one now. Well that is enough for me. If they use it I can use it. Should I choose a professional one over QuickFix? Is it not good enough?

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...

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 $j$....

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...

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.


Why aren't econometric models used more in Quant Finance?

There is a big body of literature on econometric models like ARIMA , ARIMAX or VAR . Yet to the best of my knowledge practically nobody is making use of that in Quantitative Finance. Yes, there is a paper here and there and sometimes you find an example where stock prices are being used for illustrative purposes but this is far from the main stream. My question Is there a good reason for that? Is it just because of tradition and different schools of thought or is there a good technical explanation? (By the way I was pleased to find an arima tag here... but this is again a case in point: only 8 out of nearly 7,000 questions (~ 0.1% !) use this tag! ...ok, make this 9 now ;-)

modeling forecasting econometrics models arima

Because of: The (extreme) dominance of noise over signal The prevalence of non-repeating patterns (many of which we know are not going to repeat) A pathetic sample size for cross-validation Regime changes due to exogenous events. These are typically in the cross-val window which makes it even worse. (GFC, financial integration, trade law changes, interest rate adjustments by central banks, some idiot in a bank was hiding trades and loses 5 billions dollars, etc). It is well known that non-linear relationships are generally just artefacts of the in sample dataset There is also the following: Much price changes are driven...

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)

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 ;-)

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...


How should I calculate the implied volatility of an American option in a real-time production environment?

There are many models available for calculating the implied volatility of an American option. The most popular method, employed by OptionMetrics and others, is probably the Cox-Ross-Rubinstein model. However, since this method is numerical, it yields a computationally intensive algorithm which may not be feasible (at least for my level of hardware) for repeated re-calculation of implied volatility on a hundreds of option contracts and underlying instruments with ever-changing prices. I am looking for an efficient and accurate closed form algorithm for calculating implied volatility. Does anyone have any experience with this problem? The most popular closed-form approximation appears to be Bjerksund and Stensland (2002), which is recommended by Matlab as the top choice for American options, although I've also seen Ju and Zhong (1999) mentioned on Wilmott . I am interested in knowing which of these (or other) methods gives the most reasonable and accurate approximations in a real-world setting.

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 $W_t$,...

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...

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...

The standard story (also told by @vonjd) is of "The Drunk and Her Dog". This is based on "A Drunk and Her Dog: An Illustration of Cointegration and Error Correction" (1994). The story is itself based on the standard illustration for a random walk known as the "drunkard's walk". The Dickey-Fuller test is used to check for a unit root . It can be used as part of the general Engle-Granger two-step method (although it isn't the only option). In this case, while the two assets themselves are not stationary, you are able to test if the residuals between a...


Mapping symbols between tickers, Reuters RICs and Bloomberg tickers

Is there any known solution (preferably open source) to map between ticker symbols, Reuters and Bloomberg symbols. For example: Ticker: AAPL Reuters: AAPL.O (may be prefixed with RSF.ANY. dependent upon infrastructure) Bloomberg: AAPL US Equity Edit: by mapping I mean translating from one symbol naming convention to another. For example let's say we have RSF.ANY.AAPL.O and want to get Bloomber equivalent, which is "AAPL US Equity". Edit2: Fixed Bloomber mapping, it should be "AAPL US Equity" not "AAPL:US"

Here are some pointers. First of all: What you list as a Reuters RIC, RSF.ANY.AAPL.OQ , is not really a RIC, only the AAPL.OQ is. The initial part is some stuff which is essentially site specific and tells me that you are working on a site that has a legacy RTIC infrastructure (some Reuters/TIBCO technology which is quite old these days and for all practical purposes has been deprecated in favour of other distribution mechanisms, most notably the ADS). Ok, the AAPL.OQ is the RIC, and only that. The initial part, the RSF.ANY denotes the feed and that is because...

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...

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...


What type of investor is willing to be short gamma?

As far as I understand, most investors are willing to buy options (puts and calls) in order to limit their exposure to the market in case it moves against them. This is due to the fact that they are long gamma. Being short gamma would mean that the exposure to the underlying becomes more long as the underlying price drops and more short as the underlying price rises. Thus exposure gets higher with a P&L downturn and lower with a P&L upturn. Hence I wonder who is willing to be short gamma? Is it a bet on a low volatility? Also, for a market maker in the option market, writing (selling) an option means being short gamma, so if there is no counterparty willing to be short gamma, how are they going to hedge their gamma?

options hedging portfolio-management greeks investing

Great question! I think the most useful starting point is Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options by Bakshi, Kapadia and Madan (2003) . Their paper proposes a definition of model-free implied skewness (they originally called it risk-neutral skewness, but MFIS is more accurate), which they prove will have a P&L directly proportional to the realized skewness of the underlier. Subsequent papers (there are literally dozens) have thoroughly explored the properties of MFIS. In particular, Does Risk-Neutral Skewness Predict the Cross-Section of Equity Option Portfolio Returns? by Bali and Murray (2011) estimates the empirical...

The standard story (also told by @vonjd) is of "The Drunk and Her Dog". This is based on "A Drunk and Her Dog: An Illustration of Cointegration and Error Correction" (1994). The story is itself based on the standard illustration for a random walk known as the "drunkard's walk". The Dickey-Fuller test is used to check for a unit root . It can be used as part of the general Engle-Granger two-step method (although it isn't the only option). In this case, while the two assets themselves are not stationary, you are able to test if the residuals between a...

Being short gamma simply means that you are short options regardless of whether they are puts or calls. The most common type of investor that is willing to be short gamma is someone who sells options, also known as a premium collector. These investors commonly use strategies such as short puts, covered calls, iron condors, vertical credit spreads, and a few others. These strategies are typically referred to as income generation strategies. They offer the investor a return known in advance, in exchange for the risk of being short options. Frequently these types of income trades have have a probability...


What are the popular methodologies to minimize data snooping?

Are there common procedures prior or posterior backtesting to ensure that a quantitative trading strategy has real predictive power and is not just one of the thing that has worked in the past by pure luck? Surely if we search long enough for working strategies we will end up finding one. Even in a walk forward approach that doesn't tell us anything about the strategy in itself. Some people talk about white's reality check but there are no consensus in that matter.

The minimum variance solution loads up on securities that have low variances and co-variances. Theoretically you are correct that this should have a low expected return profile. However, it turns out - in contradiction to modern portfolio theory - that securities that have low-volatility or low-beta experience higher returns than high-volatility or high-beta stocks. This is well-documented in the literature as the low-volatility anomaly . As a result, many funds and ETFs have been launched in recent years to exploit this phenomenon. There are a couple arguments as to why the anomaly exists. The paper I cite above argues that...

Strictly speaking, data snooping is not the same as in-sample vs out-of-sample model selection and testing, but has to deal with sequential or multiple tests of hypothesis based on the same data set. To quote Halbert White: Data snooping occurs when a given set of data is used more than once for purposes of inference or model selection. When such data reuse occurs, there is always the possibility that any satisfactory results obtained may simply be due to chance rather than to any merit inherent in the methody yielding the results. Let me provide an example. Suppose that you have...

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...


There is a concept of trading or observing the market with signal processing originally created by John Ehler . He wrote three books about it. Cybernetic Analysis for Stocks and Futures Rocket Science for Traders MESA and Trading Market Cycles There are number of indicators and mathematical models that are widely accepted and used by some trading software (even MetaStock), like MAMA, Hilbert Transform, Fisher Transform (as substitutes of FFT), Homodyne Discriminator, Hilbert Sine Wave, Instant Trendline etc. invented by John Ehler. But that is it. I have never heard of anybody other than John Ehler studying in this area. Do you think that it is worth learning digital signal processing? After all, each transaction is a signal and bar charts are somewhat filtered form of these signals. Does it make sense?

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 $r$ ;...

I can think of an application in options pricing. I came across the following paper a long time ago but think it explains FT very eloquently as applied to pricing options under BS: http://maxmatsuda.com/Papers/2004/Matsuda%20Intro%20FT%20Pricing.pdf The fun starts on page 112 but it relies on the 1998 paper by Madan and Carr. What I like about the paper is that it gives a thorough introduction to FT and only when the groundwork is set it applies it to option pricing. Not a bad approach vs many other papers which make a lot of assumption and assume the reader can jump right...

Wavelets are just one form of "basis decomposition". Wavelets in particular decompose in both frequency and time and thus are more useful than fourier or other purely-frequency based decompositions. There are other time-freq decompositions (for instance the HHT) which should be explored as well. Decomposition of a price series is useful in understanding the primary movement within a series. In general with a decomposition, the original signal is the sum its basis components (potentially with some scaling multiplier). The components range from the lowest frequency (a straight-line through the sample) to the highest frequency, a curve that oscillates with a...


Skew arbitrage: How can you realize the skewness of the underlying?

It's not clear to me how to realize skewness. In other words, how do you implement skew arbitrage? There seems to be no well-known recipe like in volatility arbitrage . Volatility arbitrage (or vol arb) is a type of statistical arbitrage implemented by trading a delta neutral portfolio of an option and its underlier. The objective is to take advantage of differences between the implied volatility and a forecast of future realized volatility of the option's underlier. My hypothetical skew arbitrage definition: Skew arbitrage is a type of statistical arbitrage implemented by trading a delta and volatility neutral portfolio. The objective is to take advantage of differences between the implied skew and a forecast of future realized skew of the option's underlier. Is it possible to make such a skew-arb portfolio in practice? If I have great confidence in my skew forecast but not in my volatility forecast, I am tempted to engage in this type arbitrage. But again, this is just a hypothetical version of skew arbitrage. If you know a correct and more practical version, you are welcome to correct me! The same question but in a different voice: In practice, a skew bet is implemented through vertical spread, i.e. buying and selling options of different strikes. How do options traders hedge / realize the edge of the spread they trade that is indifferent to the underlying volatility?

options volatility trading hedging option-strategies

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...

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...


Is there an intuitive explanation for the Feynman-Kac-Theorem?

The Feynman-Kac theorem states that for an Ito-process of the form $$dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dW_t$$ there is a measurable function $g$ such that $$g_t(t,x) + g_x(t, x) \mu(t,x) + \frac{1}{2} g_{xx}(t,x)\sigma(t,x)^2 = 0$$ with an appropriate boundary condition $h$: $g(T,x) = h(x)$. We also know that $g(t,x)$ is of the form $$g(t,x)=\mathbb{E}\left[h(X_T) \big| X_t=x\right].$$ This means that I can price an option with payoff function $h(x)$ at $T$ by solving the differential equation without regard to the stochastic process. Is there an intuitive explanation how it is possible to model the stochastic behaviour of the Ito-process by a differential equation, even though the differential equation does not have a stochastic component?

Martingales + Markovian Here is the motivation. Conditional expectations are martingales by the tower property of conditional expectations (an easy exercise to show). Suppose $r=0$, by the risk neutral pricing theorem $E^\star\left[h(X_T)\bigg|\mathscr{F}_t,\,X_t=x\right]$ is the price of any derivative security with $X$ as the underlying asset and payoff function $h$ assuming for the moment that the underlying security and the derivative itself pay no intermediate cashflows. In a Markovian setting, it must be the case that the price of the derivative is a measurable function of the current asset price and the time to maturity only, say a function $g(t, x)$....

I can only talk about quantitative trading. As a rule of thumb, the lower frequency you work in, the more econometrics is important, whereas for a higher frequency, the more econometrics becomes useless . (I would still recommend a top econometrician for HFT since they have what it takes to succeed, it's just the models aren't out-of-the-box applicable.) But if I was interviewing someone who was educated in econometrics for a quantitative research position, I would hope for (given the relevance to financial time-series): I have tried to put in a legend, ^ is something you should learn later and...


How to identify technical analysis chart patterns algorithmically?

I'm working on a small application that will provide some charts and graphs to be used for technical analysis. I'm new to TA but I'm wondering if there is a way to algorithmically identify the formation of certain patterns. In most of the TA literature I've read the authors explain how to identify these patterns visually. Is there a way to algorithmically determine these patterns so that I could, for example, examine the prices in code and identify a possible Head and Shoulders pattern?

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.

I would offer the distinctions are i) pure statistical approach, ii) equilibrium based approach, and iii) empirical approach. The statistical approach includes data mining. Its techniques originate in statistics and machine learning. In its extreme there is no a priori theoretical structure imposed on asset returns. Factor structure might be identified thru Principal Components, for example. The goal here is to maximize predictive accuracy at the expense of intuition and explanatory power. This approach increasingly dominates at very short frequencies in modeling market microstructure, market making algorithms, volatility modeling, etc. However, even in high-frequency trading one can impose a factor...

As mentioned elsewhere on this site, Lo, Mamaysky, and Wang (2000) do exactly what you're talking about, namely algorithmic detection of head and shoulders patterns. Their definition: Head-and-shoulders (HS) and inverted head-and-shoulders (IHS) patterns are characterized by a sequence of five consecutive local extrema $E_1,...,E_5$ such that $$ HS \equiv \begin{cases} E_1 \text{ is a maximum} \\ E_3 > E_1, E_3 > E_5 \\ E_1\text{ and }E_5\text{ are within 1.5 percent of their average} \\ E_2\text{ and }E_4\text{ are within 1.5 percent of their average,} \end{cases} $$ $$ IHS \equiv \begin{cases} E_1\text{ is a minimum} \\ E_3

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...


How to simulate stock prices with a Geometric Brownian Motion?

I want to simulate stock price paths with different stochastic processes. I started with the famous geometric brownian motion. I simulated the values with the following formula: $$R_i=\frac{S_{i+1}-S_i}{S_i}=\mu \Delta t + \sigma \varphi \sqrt{\Delta t}$$ with: $\mu= $ sample mean $\sigma= $ sample volatility $\Delta t = $ 1 (1 day) $\varphi=$ normally distributed random number I used a short way of simulating: Simulate normally distributed random numbers with sample mean and sample standard deviation. Multiplicate this with the stock price, this gives the price increment. Calculate Sum of price increment and stock price and this gives the simulated stock price value. (This methodology can be found here ) So I thought I understood this, but now I found the following formula , which is also the geometric brownian motion: $$ S_t = S_0 \exp\left[\left(\mu - \frac{\sigma^2}{2}\right) t + \sigma W_t \right] $$ I do not understand the difference? What does the second formula says in comparison to the first? Should I have taken the second one? How should I simulate with the second formula?

equities simulations stochastic-processes brownian-motion

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)

I recently read "Modeling financial data with stable distributions" (Nolan 2005) which gives a survey of this area and might be of interest (I believe it was contained in "Handbook of Heavy Tailed Distributions in Finance" ). Another more recent reference is "Alpha-Stable Paradigm in Financial Markets" (2008). I'm not aware of anything covering "risk of fluctuations" and this is still certainly not at the center of the field (i.e. most theory still includes some version of Gaussian or mixture of Gaussians). Would also be interested in other references.

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 $W_t$,...


Trading a synthetic replication of the VIX index

One cannot directly buy and sell the VIX index. Theoretically, however, one could approximate the index by purchasing an at-the-money straddle on the SP500, then delta-hedging the straddle. Does anyone have experience with such a "synthetic" replication of the index? It might be very useful for betting on volatility or for spreads against the VIX futures (a sort of basis trade), but I can see potential problems if the replication is too inaccurate. (To anticipate your comments: I'm aware of the many VIX-related ETFs; but, no, I would not consider using them. I'm also aware that the VIX calculation uses other strikes beyond the ATM options; this proposed synthetic is admittedly an approximation.)

A synthetic model for the VIX would be quite useful. I just mention this since it has been covered elsewhere in the past, although I don't think that it's a real solution to your problem (for a number of reasons). Several blogs posted on the "William's VIX Fix" (WVF) in the past: marketsci , trading the odds , mindmoneymarkets . The WVF is intended to be a synthetic VIX calculation, derived by Larry Williams (see the original article here ), and is represented by the following formula: $wvf = \frac{Highest(Close, 22) - Low}{Highest(Close, 22)} * 100$ In R, this can...

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.

I haven't read Natenberg but it of course depends on your side in the trade: Are you a market maker or a risk taker? So do you live on the spread (first) or are trying to make money based on e.g. forecasts on direction (second). This is the great divide in QuantFinance! Only in the first case will all your option trades be delta neutral. There is a nice short paper which elaborates on both concepts (it calls the first one Q and the second P ): Meucci: 'P' Versus 'Q': Differences and Commonalities between the Two Areas of Quantitative...


What exactly is meant by "microstructure noise"?

I see that term tossed around a lot, in articles relating to HFT, and ultra high frequency data. It says at higher frequencies, smaller intervals, microstructure noise is very dominant. What is this microstructure noise that they refer to?

volatility statistics high-frequency market-microstructure high-frequency-estimators

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...

Volatility is mean reverting if the underlying security doesn't drop to zero. If the security has some underlying "value" then its price is co-integrated with that "value". The volatility is the uncertainty of that price as it tracks the security's "value". Edit 12/03/2011 ================================================= @pteetor, I may have missed something, but the question was " Why is volatility mean-reverting?". I realize that the standard answer is that the VIX (I'm assuming he's asking about the VIX) is related to the historical volatility of the S&P. A simple version of that relationship provides a reasonable R^2 (see Fig. 1). It relates...

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...

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 $r$ ;...


How do we use option price models (like Black-Scholes Model) to make money in practice?

In quantitative finance, we know we have a lot of option price models such as geometric Brownian motion model (Black-Scholes models), stochastic volatility model (Heston), jump diffusion models and so on, my question is how can we use these models to make money in practice? My comments: Because we can read option price from the market, by these models (Black-Scholes), we can get the implied volatility, then we may use this implied volatility to compute other exotic option price, then we can make money by selling/buying this exotic option as a market maker, is this the only way to make money? For stocks, we know that if we have a better model to predict future stock prices, then we can make money, but for option, it seems that we didn't use these models to predict the future option prices? so how can we make money with these models?

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...

Wavelets are just one form of "basis decomposition". Wavelets in particular decompose in both frequency and time and thus are more useful than fourier or other purely-frequency based decompositions. There are other time-freq decompositions (for instance the HHT) which should be explored as well. Decomposition of a price series is useful in understanding the primary movement within a series. In general with a decomposition, the original signal is the sum its basis components (potentially with some scaling multiplier). The components range from the lowest frequency (a straight-line through the sample) to the highest frequency, a curve that oscillates with a...

I can only talk about quantitative trading. As a rule of thumb, the lower frequency you work in, the more econometrics is important, whereas for a higher frequency, the more econometrics becomes useless . (I would still recommend a top econometrician for HFT since they have what it takes to succeed, it's just the models aren't out-of-the-box applicable.) But if I was interviewing someone who was educated in econometrics for a quantitative research position, I would hope for (given the relevance to financial time-series): I have tried to put in a legend, ^ is something you should learn later and...


What is the Risk Neutral Measure? I don't believe this has been answered on the internet well and with all the parts connecting. So: What is the risk neutral measure/pricing? Why do we need it? How we calculate the risk neutral measure or probabilities in practice? What connection has risk neutral pricing to the drift of a SDE? Does this help with 3)?

option-pricing stochastic-processes risk-neutral-measure pricing martingale

Intro: Great answer given by Kevin. I would like to contribute an additional perspective. My experience with and my understanding of the Risk Neutral measure is entirely based on "no arbitrage" and "replication / hedging" arguments. The way I would like to explain this view is via the following three-step construction : (i) First, I want to build the intuition with a one-period discrete model: only a single stock and a risk-free account, no derivatives . The aim is to show that even without trying to price derivatives, one can create a mathematical object called a "risk-neutral probability measure", just...

Because of: The (extreme) dominance of noise over signal The prevalence of non-repeating patterns (many of which we know are not going to repeat) A pathetic sample size for cross-validation Regime changes due to exogenous events. These are typically in the cross-val window which makes it even worse. (GFC, financial integration, trade law changes, interest rate adjustments by central banks, some idiot in a bank was hiding trades and loses 5 billions dollars, etc). It is well known that non-linear relationships are generally just artefacts of the in sample dataset There is also the following: Much price changes are driven...

Consider the standard error , and in particular the distance between the upper and lower limits: \begin{equation} \Delta = (\bar{x} + SE \cdot \alpha) - (\bar{x} - SE \cdot \alpha) = 2 \cdot SE \cdot \alpha \end{equation} Using the formula for standard error, we can solve for sample size: \begin{equation} n = \left(\frac{2 \cdot s \cdot \alpha}{\Delta}\right)^{2} \end{equation} where $s$ is the measured standard deviation, which you already have from your IR calculation. High-frequency Example I was testing a market-making model recently that was expected to return a couple basis points for each trade and I wanted to be confident...


What is the so-called Swap Curve, and how does it relate to the Zero Curve (or spot yield curve)? Does it only refer to a curve of swap rates versus maturities found in the market? Or is it a swap equivalent of a spot-yield curve constructed from bootstrapping a bond yield curve? The context of this question is set against a backdrop of a plethora of terminology (that seems to be used interchangeably). I am looking into how the so-called Zero Curve (or spot yield curve) is constructed in order to discount various IR derivatives (including swaps) when pricing them.

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...

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...


What is the role of stochastic calculus in day-to-day trading?

I work with practical, day-to-day trading: just making money. One of my small clients recently hired a smart, new MFE. We discussed potential trading strategies for a long time. Finally, he expressed surprise that I never mentioned (much less used) stochastic calculus, which he spent many long hours studying in his MFE program. I use the products of stochastic calculus (e.g., the Black-Scholes equation) but not the calculus itself. Now I am wondering, does stochastic calculus play a role in day-to-day trading strategies? Am I under-utilizing a potentially valuable tool? If this client was a Wall Street investment bank that was making markets in complicated derivatives, I'm sure their research department would use stochastic calculus for modeling. But they're not, so I'm not sure how we would use stochastic calculus. ( Full disclosure: I have Masters degrees but not a PhD. I'm an applied mathematician, not a theoretician.)

Martingales + Markovian Here is the motivation. Conditional expectations are martingales by the tower property of conditional expectations (an easy exercise to show). Suppose $r=0$, by the risk neutral pricing theorem $E^\star\left[h(X_T)\bigg|\mathscr{F}_t,\,X_t=x\right]$ is the price of any derivative security with $X$ as the underlying asset and payoff function $h$ assuming for the moment that the underlying security and the derivative itself pay no intermediate cashflows. In a Markovian setting, it must be the case that the price of the derivative is a measurable function of the current asset price and the time to maturity only, say a function $g(t, x)$....

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...

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...

This is pure speculation: MFE's are really tailored toward valuation models (how can we develop a model to price x swap, etc.). You don't entirely have to worry about those details in order to trade them: you're just quoted a price based on these models. But if you go in-house at a bank and are working as a product quant (structured products, etc.), then you really need to worry about these things. Alternatively, it could be relevant to a trading strategy if you think that the current model is mispricing things and there's an arbitrage opportunity. This is why banks...


What is the best data structure/implementation for representing a time series?

I was wondering what is best practice for representing elements in a time series, especially with large amounts of data. The focus/context is in a back testing engine and comparing multiple series. It seems there are two options: 1) Using an integer index, or 2) Using a date-based index At the moment I am using dates, but this impacts on performance & memory usage in that I am using a hash table rather than an array, and it requires some overhead in iteration (either forward or backwards) as I have to determine the next/previous valid date before I can access it. However, it does let me aggregate data on the fly (e.g. building the ohlc for the previous week when looking at daily bars) and most importantly for me allows me to compare different series with certainty I am looking at the same date/time. If I am looking at an equity issue relative to a broader index, and say the broader index is missing a few bars for whatever reason, using an integer indexed array would mean I'm looking at future data for the broad index vs present data for the given security. I don't see how you could handle these situations unless you're using date/times. Using integer indexes would be a lot easier code wise, so I was just wondering what others are doing or if there is best practice with this.

Representing time series (esp. tick data) using elaborate data structures may be not the best idea. You may want to try to use two arrays of the same length to store your time series. The first array stores values (e.g. price) and the second array stores time. Note that the second series is monotonically increasing (or at least non-decreasing), i.e. it's sorted. This property enables you to search it using the binary search algorithm. Once you get an index of a time of interest in the second array you also have the index of the relevant entry in the first...

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 $j$....

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...


Any known bugs with Yahoo Finance adjusted close data ?

Yahoo Finance allows you to download tables of their daily historical stock price data. The data includes an adjusted closing price that I thought I might use to calculate daily log returns as a first step to other kinds of analyses. To calculate the adj. close you need to know all the splits and dividends, and ex-div and ex-split dates. If someone gets this wrong it should create some anomalous returns on the false vs. true ex dates. Has anyone seen any major problems in the adj. closing price data on Yahoo Finance?

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.

Consider the standard error , and in particular the distance between the upper and lower limits: \begin{equation} \Delta = (\bar{x} + SE \cdot \alpha) - (\bar{x} - SE \cdot \alpha) = 2 \cdot SE \cdot \alpha \end{equation} Using the formula for standard error, we can solve for sample size: \begin{equation} n = \left(\frac{2 \cdot s \cdot \alpha}{\Delta}\right)^{2} \end{equation} where $s$ is the measured standard deviation, which you already have from your IR calculation. High-frequency Example I was testing a market-making model recently that was expected to return a couple basis points for each trade and I wanted to be confident...

The minimum variance solution loads up on securities that have low variances and co-variances. Theoretically you are correct that this should have a low expected return profile. However, it turns out - in contradiction to modern portfolio theory - that securities that have low-volatility or low-beta experience higher returns than high-volatility or high-beta stocks. This is well-documented in the literature as the low-volatility anomaly . As a result, many funds and ETFs have been launched in recent years to exploit this phenomenon. There are a couple arguments as to why the anomaly exists. The paper I cite above argues that...


What is the intuition behind cointegration? What does the Dickey-Fuller test do to test for it? Ideally, a non-technical explanation would be appreciated. Say you need to explain it to an investor and justify why your pairs trading strategy should make him rich!

time-series statistics cointegration intuition

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...

The standard story (also told by @vonjd) is of "The Drunk and Her Dog". This is based on "A Drunk and Her Dog: An Illustration of Cointegration and Error Correction" (1994). The story is itself based on the standard illustration for a random walk known as the "drunkard's walk". The Dickey-Fuller test is used to check for a unit root . It can be used as part of the general Engle-Granger two-step method (although it isn't the only option). In this case, while the two assets themselves are not stationary, you are able to test if the residuals between a...

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...

The minimum variance solution loads up on securities that have low variances and co-variances. Theoretically you are correct that this should have a low expected return profile. However, it turns out - in contradiction to modern portfolio theory - that securities that have low-volatility or low-beta experience higher returns than high-volatility or high-beta stocks. This is well-documented in the literature as the low-volatility anomaly . As a result, many funds and ETFs have been launched in recent years to exploit this phenomenon. There are a couple arguments as to why the anomaly exists. The paper I cite above argues that...


How do you mix quantitative asset allocation with qualitative views?

Usually in asset allocation you have a quantitative approach (which can be from example mean-variance), but you (or you and your firm) also have a more qualitative approach given market-conditions, economic outlooks, or tactical indicators. Hence, you will eventually come up with 2 allocations, the ones strictly dictated by the numbers $w^*$ which is the result of your quantitative algorithm and the one you have in mind from your personal expectations $\bar{w}$. What are common the ways $f$ to mix them together such that $w=f(w^*,\bar{w})$ is your "final" allocation?

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...

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.


What are the advantages/disadvantages of these approaches to deal with volatility surface?

I would like to know if someone could provide a summarized view of the advantages and disadvantages of the approaches on the volatility surface issues, such as: Local vol Stochastic Vol (Heston/SVI) Parametrization (Carr and Wu approach)

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 ;-)

Your questions about contango in VIX futures have close analogies in options too. The Black & Scholes model suggests that all time frames and all strikes should have the same implied volatility, but they don't. I think one of the reasons is that the B&S model assumes that stock returns are distributed in a normal (gaussian) distribution, but the actual returns don't match a gaussian distribution all that well. For example the actual occurrence of big crashes / gains is much more likely than a normal distribution would predict. Since crashes do occur people are willing to pay what the...

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...


What methods do you use to improve expected return estimates when constructing a portfolio in a mean-variance framework?

One of the main problems when trying to apply mean-variance portfolio optimization in practice is its high input sensitivity. As can be seen in ( Chopra , 1993) using historical values to estimate returns expected in the future is a no-go, as the whole process tends to become error maximization rather than portfolio optimization . The primary emphasis should be on obtaining superior estimates of means , followed by good estimates of variances. In that case, what techniques do you use to improve those estimates ? Numerous methods can be found in the literature, but I'm interested in what's more widely adopted from a practical standpoint . Are there some popular approaches being used in the industry other than Black-Litterman model? Reference: Chopra, V. K. & Ziemba, W. T. The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice . Journal of Portfolio Management , 19: 6-11, 1993.

modern-portfolio-theory mean-variance expected-return estimation

Wavelets are just one form of "basis decomposition". Wavelets in particular decompose in both frequency and time and thus are more useful than fourier or other purely-frequency based decompositions. There are other time-freq decompositions (for instance the HHT) which should be explored as well. Decomposition of a price series is useful in understanding the primary movement within a series. In general with a decomposition, the original signal is the sum its basis components (potentially with some scaling multiplier). The components range from the lowest frequency (a straight-line through the sample) to the highest frequency, a curve that oscillates with a...

Hey, it's early days yet. After all it is still called MODERN portfolio theory. I think there are two main issues and they are both really cultural: 1) specifying alphas 2) wild results Alphas I agree with Gappy that alphas are the key thing you need to have effectiveness (unless you are doing minimum variance). Having a vector of expected returns is quite a natural thing for quant managers. But it is something foreign to fundamental managers. They have to map their views into a number for each asseet in the universe. That is not necessarily an easy task, and...

Short of having a 'reasonable' predictive model for expected returns and the covariance matrix, there are a couple lines of attack. Shrinkage estimators (via Bayesian inference or Stein-class of estimators) Robust portfolio optimization Michaud's Resampled Efficient Frontier Imposing norm constraints on portfolio weights Naively, shrinkage methods 'shrink' (of course,no?) your estimates (arrived at using historical data), toward some global mean or some target. Within the mean-variance framework, you can use the shrinkage estimators, for both, the expected returns vector, as well as the covariance matrix. Jorion introduced application of a 'Bayes-Stein estimator' to portfolio analysis. Bradley & Efron have a...


Any research on how natural language processing can be used to forecast stocks?

Is there any published research of decent quality linking news or unstructured information to asset returns? I know that Thomson Reuters offers its Machine Readable news (MRN), so somebody must use it. But I can't find much in the public domain.

equities forecasting research prediction nlp

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...

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...

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)


Can I get some recommendations for a book on market microstructure? I'm not looking for some author's questionable methods for trading, I'm just looking for a book that provides me with facts about how order books, closing auctions, order execution, etc. really works. I'm also NOT looking for the type of depth that a HFT would be interested in. I know that whole books can be written about the various plumbing of a particular exchange. I'm simply looking for a general overview of how exchanges work.

Consider the standard error , and in particular the distance between the upper and lower limits: \begin{equation} \Delta = (\bar{x} + SE \cdot \alpha) - (\bar{x} - SE \cdot \alpha) = 2 \cdot SE \cdot \alpha \end{equation} Using the formula for standard error, we can solve for sample size: \begin{equation} n = \left(\frac{2 \cdot s \cdot \alpha}{\Delta}\right)^{2} \end{equation} where $s$ is the measured standard deviation, which you already have from your IR calculation. High-frequency Example I was testing a market-making model recently that was expected to return a couple basis points for each trade and I wanted to be confident...

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...

I've not yet read it, but Lehalle's recent book is bound to be a goldmine of good micro-structure bits and pieces. Market Microstructure in Practice EDIT: I'm reading the book now, so far it's quite good.


Separating the wheat from the chaff: What quant methods separate skillful managers from lucky ones?

Fund managers are acting in a highly stochastic environment. What methods do you know to systematically separate skillful fund managers from those that were just lucky? Every idea, reference, paper is welcome! Thank you!

Larry Harris has a chapter on performance evaluation in Trading and Exchanges . He states that over a long period of time, a skilled asset manager will consistently have excess returns whereas a lucky one will be expected to have random and unpredictable returns. Thus, we start with the portfolio's market-adjusted return standard deviation: \begin{equation} \sigma_{adj} = \sqrt{\sigma^2_{port} + \sigma^2_{mk} - 2\rho\sigma_{port}\sigma_{mk}} \end{equation} where $\rho$ is the correlation between the market and portfolio returns. For a sample size $n$ (generally number of years), the average excess returns, and the adjusted standard deviation from above, we have a t-statistic : \begin{equation}...

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.


Why do high frequency traders use rapidly cancelled limit orders?

In reading about the various practices and strategies of high frequency traders, one of the most mysterious to me is "fleeting orders," or orders that are cancelled almost immediately after they are sent (see Hasbrouck and Saar (2011) ). Why do HFTs use these orders? Some of those trying to explain the practice claim it gives HFTs an informational advantage. How? What information do they get and how do they use it? Update I am hoping for someone with actual experience in HFT to answer and verify some of the hypotheses laid out by academics. In particular, one hypothesis is that rapidly cancelled limit orders bring forth market orders on the other side, which are then executed at a less favorable price also placed by the HFT. In other words, HFTs are gaming the system to exploit sub-optimal behavior by slower traders. Is there any evidence of this still occurring, now that the practice of fleeting orders is widespread and well known? This explanation would also imply that it is relatively simple to avoid being taken advantage of, thus weakening the policy implications. Is there some reason that slower traders prefer sending market orders only after seeing a limit order meeting their price target?

The lead paper in the January 2011 Journal of Finance ( Hendershott, Jones, and Menkveld ) addresses algorithmic trading (AT). In short, they find that AT improves liquidity as measured by bid-offer spreads. Taking the econometrics as correct (it is in the Journal of Finance) the next question is if bid-offer spreads are a sufficient statistic for measuring liquidity (or any other benefits). It is a difficult question to answer because, given current market structure, AT may improve liquidity (as measured by bid-offer spreads), but without data on other market structures, it is hard to say that we wouldn't better...

All HFTs are event driven. In the most basic sense, they have some model that is a function of order book events. For every order book event the model calculates some micro price that is the HFTs perceived fair value. This is often a function of the current bid, ask, depth, last n trade prices, inventory, etc. Given the most up to date view of fair value, the HFT will be adjust orders in the market. As you can imagine, the rate of events for order book data is very high. This results in a very high scratch rate (cancel...


Applications of Fourier theory in trading

What are fashionable applications of Fourier analysis in trading? I have heard vague ideas of applications in High Frequency Trading but can somebody provide an example, maybe a reference? Just for clarification: The approach to split up a stock price in its cosines and to apply this for forecasts or anything similar seems theoretically not justified as we can not assume the stock price to be periodic (outside of the period of observation). So I don't really mean such applications. Put differently: are there useful, theoretically valid applications of Fourier theory in trading? I am curious for any comments, thank you! EDIT: I am aware of (theoretically $100\%$ valid) applications in option pricing and calculation of risk measures in the context of Lévy processes (see e.g. here p.11 and following and references therein). This is well established, I guess. What I mean are applications in time series analysis. Sorry for any confusions.


how to derive yield curve from interest rate swap?

According to some textbooks, to derive the yield curve, quote overnight to 1 week: rates from interbank money market deposit, 1 month to 1 year: LIBOR; 1 year to 7 years: Interest Rate Swap; 7 years above: government bond. I'm a bit lost here: how can an IRS rate be used to derive yield curve? Yield rate is the discount rate, if $ yield (5 years) = 4.1 \% $ , it means the NPV of 1 dollar 5 years later is $ NPV ( 1 dollar, 5 years) = 1/[(1+4.1\%)^5] = 0.818 $. While interest rate swap is a contract among to legs. Assume a 5 years' IRS contract is leg A pays fixed rate to B @ 8.5%, while A receives floating rate @ LIBOR +1.5% leg B pays floating rate to A @ LIBOR +1.5%, B receives fixed rate@ 8.5%. , how could this swap contract help deriving the 5 years' yield rate?

interest-rates swaps yield-curve interest-rate-swap irs

The standard story (also told by @vonjd) is of "The Drunk and Her Dog". This is based on "A Drunk and Her Dog: An Illustration of Cointegration and Error Correction" (1994). The story is itself based on the standard illustration for a random walk known as the "drunkard's walk". The Dickey-Fuller test is used to check for a unit root . It can be used as part of the general Engle-Granger two-step method (although it isn't the only option). In this case, while the two assets themselves are not stationary, you are able to test if the residuals between a...

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...

You should take a look at the example from Hull's book. Assume that the 6-month, 12-month, 18-month zero rates are 4%, 4.5%, and 4.8%, respectively. Suppose we know that the 2-year swap rate is 5%, which implies that a 2-year bond with a semiannual coupon of 5% per annum sells for par: $$2.5 e^{-0.04 \bullet 0.5} + 2.5 e^{-0.045 \bullet 1.0} + 2.5 e^{-0.048 \bullet 1.5} + 102.5 e^{-2 \bullet R} = 100 \; . $$ Solving for $R$ above gives a 2-year zero rate $R$ of 4.953%. We can keep going to compute the 3-year zero rates, etc.


Is there a standard model for market impact? I am interested in the case of high-volume equities sold in the US, during market hours, but would be interested in any pointers.

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...

There is a family of models that is so commonly used among practitioners that it can be almost regarded as standard. For a survey, check out Rob Almgren's entry in the Encyclopedia of Quantitative Finance. Check out also Barra, Axioma and Northfield's handbooks. In general, the impact term per unit traded currency is of the form $$MI \propto \sigma_n \cdot \text{(participation rate)}^\beta$$ where the exponent is somewhere between 1/2 and 1, depending on the model being used, and the participation rate is the percentage of total volume of the trade, during the trading interval itself. When including the total MI...

Consider the standard error , and in particular the distance between the upper and lower limits: \begin{equation} \Delta = (\bar{x} + SE \cdot \alpha) - (\bar{x} - SE \cdot \alpha) = 2 \cdot SE \cdot \alpha \end{equation} Using the formula for standard error, we can solve for sample size: \begin{equation} n = \left(\frac{2 \cdot s \cdot \alpha}{\Delta}\right)^{2} \end{equation} where $s$ is the measured standard deviation, which you already have from your IR calculation. High-frequency Example I was testing a market-making model recently that was expected to return a couple basis points for each trade and I wanted to be confident...


Transformation from the Black-Scholes differential equation to the diffusion equation - and back

I know the derivation of the Black-Scholes differential equation and I understand (most of) the solution of the diffusion equation. What I am missing is the transformation from the Black-Scholes differential equation to the diffusion equation (with all the conditions) and back to the original problem. All the transformations I have seen so far are not very clear or technically demanding (at least by my standards). My question: Could you provide me references for a very easily understood, step-by-step solution?

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...

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...


How are cryptography and speech recognition technology applied to forecasting financial markets?

One of the answers to my previous question regarding the strategy of Renaissance Technologies , there was a reference to The Quants: How a New Breed of Math Whizzes Conquered Wall Street and Nearly Destroyed It . After doing some browsing in the book I found that it states that Renaissance Technologies obviously very successfully employs cryptography and speech recognition technology for forecasting financial time series. Do you know of any good papers (or other references) where the use of either of these technologies in connection with finance is shown?

quant-trading-strategies research forecasting machine-learning

By "cryptography" you mean information theory. Information theory is useful for portfolio optimization and for optimally allocating capital between trading strategies (a problem which is not well addressed by other theoretical frameworks.) See: --- J. L. Kelly, Jr., "A New Interpretation of Information Rate," Bell System Technical Journal, Vol. 35, July 1956, pp. 917-26 --- E. T. Jaynes, Probability Theory: The Logic of Science http://amzn.to/dtcySD --- http://en.wikipedia.org/wiki/Gambling_and_information_theory http://en.wikipedia.org/wiki/Kelly_criterion In the simple case, you would use "The Kelly Rule". More complicated information theory based strategies for allocating capital between trading strategies take into account correlations between the performance of trading strategies...

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.

Wavelets are just one form of "basis decomposition". Wavelets in particular decompose in both frequency and time and thus are more useful than fourier or other purely-frequency based decompositions. There are other time-freq decompositions (for instance the HHT) which should be explored as well. Decomposition of a price series is useful in understanding the primary movement within a series. In general with a decomposition, the original signal is the sum its basis components (potentially with some scaling multiplier). The components range from the lowest frequency (a straight-line through the sample) to the highest frequency, a curve that oscillates with a...

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.


How to interpret the eigenmatrix from a Johansen cointegration test?

I ran a Johansen cointegration test on 3 instruments, A B and C. The results that I got are: R 36.7 18.9 21.1 25.8 r=1 --> 8.4 12.29 14.26 18.52 r=2 --> 0.21 2.7 3.8 6.6 EigenValues EigenMatrix 0.03 --> 0.25 | 0.512 |-0.79 0.007 --> -0.96 | -0.618 | 0.14 0.00017 --> 0.05 | 0.59 | 0.59 My question is how I interpret these results? How do I know there is a cointegration for the these instruments. How to build a portfolio using the eigen vector? Which eigen vector should I choose to build my portfolio?

I can only talk about quantitative trading. As a rule of thumb, the lower frequency you work in, the more econometrics is important, whereas for a higher frequency, the more econometrics becomes useless . (I would still recommend a top econometrician for HFT since they have what it takes to succeed, it's just the models aren't out-of-the-box applicable.) But if I was interviewing someone who was educated in econometrics for a quantitative research position, I would hope for (given the relevance to financial time-series): I have tried to put in a legend, ^ is something you should learn later and...

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 $j$....

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....

From remote memory, The first question is Yes/No question. Is there any stationary, i.e. I(0), time series for different levels of combination r? This question is answered by your first table. For example, if [r=2]'s test stat is say 7 while the critical value of 99% confidence is 6.6 like your example, then I have over 99% confidence to say that all instruments A, B, and C are stationary by themselves. You don't even need to build a co-integrated portfolio/combination. They are ready for mean-reversion strategy already. Obviously, in your example, your [r=2] stat is way much lower than even...


Why are there no papers about stock prediction with machine learning in leading financial journals?

I'm writing my master's thesis about stock price prediction using machine learning methods. During my literature review, I noticed that a lot of research produced on this topic is of poor quality, published in non-finance related journals or unpublished/peer reviewed alltogether. There is no paper to be found in leading journals like journal of finance or journal of financial economics on the topic. I'm curious as to why this is the case. Did the academic world move on, and simply accept that markets are generally efficient a long time ago? Or are the leading journals overlooking a key technique that could effectively forecast stock price?

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.

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 fascinating thing about volatility pumping (or optimal growth portfolio , see e.g. here ) is that here volatility is not the same as risk, rather it represents opportunity. Additionally it is a generic mechanical strategy that is independent of asset classes. My question: Do you know examples where volatility pumping is actually implemented? What are the results? What are the pitfalls?

The optimal growth portfolio is obtained by applying the Kelly criterion which is one of the pillars of the sound risk management. Ed Thorp's weekend forays to Las Vegas to play blackjack were one of the first historically documented cases of successful practical implementation of the Kelly strategy. Since then this method and its modifications have been systematically used by Thorp himself and other hedge fund managers as an important risk control tool.

Great question! I think the most useful starting point is Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options by Bakshi, Kapadia and Madan (2003) . Their paper proposes a definition of model-free implied skewness (they originally called it risk-neutral skewness, but MFIS is more accurate), which they prove will have a P&L directly proportional to the realized skewness of the underlier. Subsequent papers (there are literally dozens) have thoroughly explored the properties of MFIS. In particular, Does Risk-Neutral Skewness Predict the Cross-Section of Equity Option Portfolio Returns? by Bali and Murray (2011) estimates the empirical...

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...