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 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...

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.

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.


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)

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...

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...

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...

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.

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...


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.

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 $j$....

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...


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...

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...

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...

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.

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...

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...

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 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...

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 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...

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...


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?

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 $g(t, x)$....

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.

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.

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...


Strategy of Renaissance Technologies Medallion fund: Holy Grail or next Madoff?

Renaissance Technologies Medallion fund is one of the most successful hedge funds - ever! Yet it is very secretive. Do you have information on the strategy used that is not yet mentioned in the Wikipedia article above? Is there really something fundamental going on (the Holy Grail of investing) - or will this be the next Madoff?

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 $r$ ;...

There are a some information about Renaissance Technologies available in The Quants from Patterson. Basically, and it's also what I heard in general, they are using intensively algorithmic trading, and from what I understood there are using Information Theory (they worked with Shannon if I remember well). I'd say it'd be harsh to say it's the next Madoff given the background they have, I can easily see them being simply better than the rest... It's just my opinion of course...


What are the best sources for equity quantitative research?

What are the best sources of quantitative finance research in equities? I will list a couple and note an asterisk if the research is available by request (i.e. non-clients) or online: BAC-Merrill Lynch - Savita Subramanian Deutsche Bank - Yin Luo* Credit Suisse - Pankaj Patel Barclay's - Matthew Rothman HSBC - Ely Klepfish* Goldman Sachs - Quantitative Strategies Neuberger-Berman - Wai Lee Nomura - Joseph Mezrich Perhaps there are also more un-conventional sources of quant research besides sell-side wall street: Hussman Funds - John Hussman* Emanuel Derman's collection of papers

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...

Sell Side Macquarie Quant - Venkat Eleswarapu Bernstein Research - Vadim Zlotnikov Nomura - Joe Mezrich JPMorgan Investment Strategies series Societe Generale - Alain Bokobza Independent CXO Advisory Empirical Finance Blog Russell Indexes: Research and Insights MSCI Research Papers Axioma Research Papers

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


Proof that you cannot beat a random walk

There is much speculation to what degree financial series are random (and what kind of randomness prevails). I want to turn the question on its head and ask: Is there a mathematical proof that whatever trading strategy you use you cannot beat a random walk (that is the expected value will always be 0 assuming no drift)? (I found this blog post where the author used the so called "75% rule" to purportedly beat a random walk but I think he got the distinction between prices and returns wrong. This method would only work if you had a range of allowed prices (e.g. a mean reverting series). See e.g. here for a discussion.)

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 can help you beat random walk 'in the way you want', i.e. the expected value $E[\$]$ will always be positive even assuming no drift. However, I have to warn people that $E[\$] > 0$ is NOT really an adequate condition for 'beating' in reality (at least to myself). Let's define some mathematical notations for derivation, and rephrase (simplify) vonjd's question without losing generality. Assume a trader plays a fair game, and his surplus $X(0), X(1), X(2), ... X(t)$ is a martingale. Q: Can the trader find a stopping time $s$ such that $E[X($s$)] > X(0)$? A proof supporting Bootvis'...

Sell Side Macquarie Quant - Venkat Eleswarapu Bernstein Research - Vadim Zlotnikov Nomura - Joe Mezrich JPMorgan Investment Strategies series Societe Generale - Alain Bokobza Independent CXO Advisory Empirical Finance Blog Russell Indexes: Research and Insights MSCI Research Papers Axioma Research Papers

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...


Is the stock price process a martingale or a Markov process?

Some people claim that the data-generating process for stocks is a "martingale" and that is has the "Markov property". Are they unrelated? Is it that the Markov property implies some sort of martingale property, or is it the other way around? How do you statistically test for such properties? How far from reality is it to assume such properties?

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)

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

I 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.


Why is an inverted yield curve a problem?

Immediately preceding the worst of the financial crisis, my professors all pointed out to me that the yield curve had inverted -- short-term yields were more risky than 20-year or 30-year Treasury securities. My professors all seemed nervous about this. Aside from such a situation being generally counter-intuitive, why was this viewed so negatively? What kind of effects does this have, in a more macro sense?

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...

Nano answer : declining yield curve is a symptom that the market expects the economy to worsen. Short answer : yield curve is built on expected future interest rates. Yield curve will be downward sloping if you expect interest rates to fall in the future. Interest rates fall because the central bank react to worsening economy. So if the yield curve is downward sloping it means the market expects the economy to worsen to the point that the central bank will have to intervene.


Why is the VIX futures market usually in a state of contango?

I'm a VIX newbie and I'm trying to understand why the VIX futures market is usually in a state of contango. All I can figure is that the sellers of VIX futures contracts demand high "prices" (because the seller is the holder of the short position and makes money when the price falls), and since there are willing buyers, namely ETNs and ETFs that are trying to track the S&P 500 VIX SHORT-TERM FUTURES INDEX (SPVIXSTR) through the purchase and sale of VIX futures, the contracts get sold. Also, the farther out the futures contract expires, the less certain the seller is about what the value of the VIX and the SPVIXSTR will be at the future's expiration date. The greater uncertainty over a longer term results in the seller demanding higher premiums over a longer term than he would demand over a shorter term. It seems like a prudent buyer of VIX futures wouldn't buy contracts that are priced so high. It seems like the VIX futures market should be in a state of contango about as frequently as it is in a state of backwardation. What causes contango in the VIX futures market, and why is it the usual state of the market?

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...

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...


We all know it does mean revert. The question is why. What's making volatility mean-revert? Is it some sort of cyclical behaviour of option traders? The way it's calculated? Why?

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...

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...


What does the VIX formula measure and how does it work?

I have read the CBOE's white paper on the VIX and a lot of other things, but I need to honestly say, I don't really get it, or I am missing something important. In semi-layman's terms, is the VIX simply the a measurement of the volume of protective options bought? That is, individual contract volatility increases when the option has been bid up by buyers due to desirability. One contract alone only marginally increases the overall market volatility, but if all contracts were bid up, then the entire VIX would be recalculated (on the fly) to reflect the higher value of protective options as a speculative/hedge solution. Do I understand this correctly, or do I really need to put more time into understanding that formula which comprises the VIX?

options volatility implied-volatility vix

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

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...

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...


Papers about backtesting option trading strategies

I am looking for all kinds of research concerning option trading strategies. With that I mean papers that publish results on different option trading strategies properly backtested with real-world data.

options backtesting quant-trading-strategies research option-strategies

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...

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 $W_t$,...

I did some digging and found the following papers - most of them offering quite a distinct perspective compared to classical option pricing theory! Stock Options as Lotteries by Brian H. Boyer et al. (2011) The Efficiency of the Buy-Write Strategy: Evidence from Australia by Tafadzwa Mugwagwa et al. (2010) The following is my favorite: You could do some backtests on your own with freely available data (using the VXO as volatility information) and with any spreadsheet - easy and elegant: How Students Can Backtest Madoff’s Claims by Michael J. Stutzer (2009) Loosening Your Collar: Alternative Implementations of QQQ Collars...

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...


How do I adjust a correlation matrix whose elements are generated from different market regimes?

Say I want to calculate a correlation matrix for 50 stocks using 3-year historical daily data. And there are some stocks that were recently listed for one year. This is not technically challenging because the correlation function in R can optionally ignore missing data when calculating pair-wise correlation. But honestly it worried me after deep thought: all the correlations for the short listed stock will bias to the regime it live in. For example, a stock that exists only in last few months will inevitably have higher correlation than the stocks that have full 3-year history. And I think it will cause bias and annoying problems in the following applications using this correlation matrix. So, my question is: how do I adjust my correlation matrix whose elements are generated from different market regimes? I feel a dilemma. In order to assure that the whole matrix live in the same regime, it seems that either I need to use least possible samples, or I have to throw that stock away. But either way looks a waste to me. I know the correlations for that stock got to be lower, but how much lower? Is there an approach or formula derived from solid arguments? I did some research and surprisingly found that all public research on correlation matrix assuming adequateness of data. How can it be possible in practice?! A mystery for me to see this seemingly common problem has not been publicly addressed.

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 $j$....

Quant Guy's list is really impressive! However, I am not sure they will readily solve your specific problem? I think there is one missing piece . Please note that imputing missing data is a very broad topic. There are many recipes to impute missings but that's for their specific 'assumptions' and purposes. They do not necessarily intend to well address your specific problem: the regime change. To best address your specific problem, you have to quantitatively define the market regime as a part of your adjusting formula. Otherwise, it wouldn't logically make sense that your model is aware of and...

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...

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...


Discrete returns versus log returns of assets

There have been similar posts here already but nevertheless I find the question worth posting: why do some people claim that log returns of assets are more suitable for statistics than discrete returns. E.g. in the ESMA CESR guidliens about SSRI log returns are used. I personally think that discrete returns are as good for means of risk management as continuous returns. Furthermore in portfolio context I can calculate the portfolio return by weighting the discrete returns of the assets which does not work with log returns. The time-aggregation of log returns is easier that's true. But people rather think in discrete returns. If my NAV drops from $100$ to $92$ then I have lost $8\%$ and that's it. Is there any study on this - any good reference? Anything that I can tell my regulator why I stick to discrete returns.

Arithmetic returns allow for easier cross-sectional aggregation and log returns allow for easier time-aggregation. The reason people use log returns (for equities) is that they are approximately invariant and hence easier to work with in estimating distributions. Meucci does better justice in describing invariance here . The basic idea (again, for equities) is that the distribution of security prices is log-normal, so the arithmetic returns will also be. However, making a log transformation results in approximately normal returns, which are easier to work with. Also, if you do assume them to be normally distributed, then there are convenient results for...

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...

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...


What are modern algorithms for trade classification?

When dealing with trade data, for example from TAQ, a common problem is that of determining whether a trade was a buy or a sell . The most commonly used classifier is the Lee-Ready algorithm ( Inferring trade direction from intraday data , 1991). Unfortunately, this method is known to be inaccurate: Lee and Radhakrishna ( Inferring investor behaviour: evidence from TORQ data , 2000) report that Lee-Ready incorrectly classifies 24% of the trades inside the spread. How to improve on Lee-Ready's recipie? What are the best algorithms for trade classification?

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.

There are a few approaches to use trade prices and quotes to classify the aggressor as "buy" or "sell." Also, many of these methods have historically had to deal with unsynchronized data streams. Classification of Approaches We can largely break trade classification methods into four types: Tick tests , advocated by Finucane (2000) , compare a trade price to the previous differing trade price with upticks (downticks) being evidence of a buy (sell); Midpoint tests , advocated by Lee and Ready (1991) , compare a trade price to a contemporaneous midpoint (average of bid and ask prices) with trades above...


I'm trying to understand why it's useful to shrink the covariance matrix for portfolio construction or in fact general. Think I missing something. I know if you have 5,000 stocks it's a lot of calculations but if we assume that computing power is not a problem.

statistics portfolio-management covariance portfolio-optimization

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

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...

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

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


I often hear people talking about the skew of the volatility surface, model, etc... but it appears to me that there isn't a clear standard definition unanimously used by practitioners. So here is my question: Does anyone have a clear and unifying definition that can be stated in mathematical terms of what skew is in the context of risk neutral pricing? If not, do you have a set of definitions of the skew with respect to a specific context?

options option-pricing implied-volatility volatility-skew

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...

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...

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 ;-)

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


How to quickly estimate a lower bound on correlation for a large number of stocks?

I would like to find stock pairs that exhibit low correlation. If the correlation between A and B is 0.9 and the correlation between A and C is 0.9 is there a minimum possible correlation for B and C? I'd like to save on search time so if I know that it is mathematically impossible for B and C to have a correlation below some arbitrary level based on A to B and A to C's correlations I obviously wouldn't have to waste time calculating the correlation of B and C. Is there such a "law"? If not, what are other methods of decreasing the search time?

Yes, there is such a rule and it is not too hard to grasp. Consider the 3-element correlation matrix $$\left(\begin{matrix} 1 & r & \rho \\ r & 1 & c \\ \rho & c & 1 \end{matrix}\right)$$ which must be positive semidefinite . In simpler terms, that means all its eigenvalues must be nonnegative. Assuming that $\rho$ and $r$ are known positive values, we find that the eigenvalues of this matrix go negative when \begin{equation} c Therefore the right hand side of this expression is the lower bound for the AC correlation $c$ that you seek, with $\rho$ being...

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


What is the necessary level of Econometrics-Know-How for a quant

It seems quants increasingly use econometric models at work. As someone who has sold his soul to probability theory and stochastical analysis I would like to catch up. What are the econometric tools a quant should be able to wield ? As I see it, the answer will be highly dependant on where one works. Thus perhaps it would make sense to distinguish: Buy side Sell side Fixed Income Equity Risk Management and Model Validation Book suggestions that cover the necessary knowledge will be appreciated. Also, if someone feels like it, a list of topics (e.g. ARCH, GARCH etc.) would also be very helpful.

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...


Why dynamics of local volatility is wrong?

In Dupire's local volatility model, the volatility is is a deterministic function of the underlying price and time, chosen to match observed European option prices. To be more specific, given a smooth surface $(K,T)\mapsto C(K,T)$ where K is the strike and T is time to maturity. Dupire equation implies that there exits an unique continuous function $\sigma_{loc}$ defined by $$\sigma_{loc}^{2}(K,T)=\frac{\partial_{T}C(K,T)+rK\partial_{K}C(K,T)}{\frac{1}{2}K^{2}\partial_{KK}C(K,T)}$$ for all $(K,T)\in(0,\infty)\times(0,\infty)$ such that the solution to the stochastic differential equation $dS_{t}/S_{t}=rdt+\sigma(t,S_{t})dW_{t}$ exactly generates the European call option prices. What do the dynamics of the local volatility mean? Are dynamics equivalent to the volatility surface? Why the dynamics of local volatility model is highly unrealistic?

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...

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...

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...


I am reading paper High-frequency trading in a limit order book by Marco Avellaneda and Sasha Stoikov. At the end of the paper they obtain a closed-form solution to the optimal market-maker quotes under diffusion without drift. They found that the optimal behaviour of the market-maker would be to set a bid/ask spread of size: $$ spread = \gamma\sigma^2(T-t) + \frac{2}{\gamma}ln(1+\frac{\gamma}{k}), $$ where $\gamma$ is a discount factor, $\sigma^2$ is the variance of the process, $k$ is the parameter corresponing to the intensity of arrival of market orders, $T$ is terminal time and $t$ is curent time, around a reservation price given by: $$ price = s - q\gamma\sigma^2(T-t), $$ where $q$ is the state of the inventory and $s$ is the current price. However, I do not see any specification of bounds for this reservation price and therefore I think there is no guarantee that ask prices computed by the market-maker will be higher or bid prices will be lower than the current price of the process. | How is this necessity of market makers' ask prices being higher and bid prices being lower than the actual price enforced in their model (e.g. in their simulations)? Edit : To be more concrete, I just specify, that in my opinion, it needs to hold that: $$ price + spread/2 - s > 0 $$ Lets denote $price$ by $p_{mm}$ and $spread/2$ by $s_{mm}$ . Then $$ p_{mm} + s_{mm} - s > 0, \\ s - q\gamma\sigma^2(T-t) + \frac{\gamma\sigma^2(T-t)}{2} + \frac{1}{\gamma}ln(1+\frac{\gamma}{k}) - s >0 \\ (...) \\ \frac{1}{2} + \frac{ln(1+\frac{\gamma}{k})}{\gamma^2\sigma^2(T-t)} > q $$ However, this situation does not need to happen, so there is no guarantee he will set prices compatible with current market prices.

The market-maker makes a bid-ask spread $\delta$ around the reservation price $r$. So at any time, the market-maker quotes the bid price $$ p_b = r - \delta/2, $$ and the ask price $$ p_a = r + \delta/2. $$ Bid price is hence always below the reservation price and ask price is always above the reservation price. The reservation price $$ r = s - q\gamma\sigma^2(T-t) $$ is the market price minus a term that depends on the inventory $q$ that the market-maker is holding. If $q$ is positive the reservation price moves lower (below the market price) and...

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...

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....


What is a coherent risk measure, and why do we care? Can you give a simple example of a coherent risk measure as opposed to a non-coherent one, and the problems that a coherent measure addresses in portfolio choice?

I'm just providing a global answer to the question, as I think it can be interesting for some beginners in quant finance. The properties given by TheBridge: Normalize $\rho (\emptyset)=0$ This means you have no risk in taking no position. Sub-addiitivity $\rho(A_1+A_2) \leq \rho(A_1)+\rho(A_2)$ Having a position in two different can only decrease the risk of the portfolio (diversification) Positive homogeneity $\rho(\lambda A) = \lambda \rho(A)$ Doubling a position in an asset A doubles your risk. And finally, Translation invariance $\rho(A + x) = \rho(A)-x$ That is, adding cash to a portfolio only diminishes the risk. So a risk-measure is...

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....

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

I 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...


Does mean-variance portfolio optimization provide a real edge to those who use it?

Mean-variance optimization (MVO) is a 50+ year concept, and perhaps the first seminal idea of quantitative finance. Still, as far as I know, less than 25% of AUM in the US is quantitatively managed. While a small minority of fundamental managers use MVO, that is counterbalanced by statistical arbitrage and HF strategies that often use optimization but not MVO, so the percentage of AUM not allocated using Markowitz' invention is surely not less than 70%. My questions: if MVO is such a great idea, why after all this time, so few people use it? if MVO was such a bad idea, how come companies like Axioma, Northfield and Barra still make money off it? is there there a rationale for the current mixed equilibrium of users and non-users? A few caveats on what I just said: i) perhaps the first and most important idea in finance is that of state-contingent assets, which is Arrow's; ii) I am focused on the buy side. I believe that optimization is widely used for hedging on the sell side.

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...

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...

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 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 ;-)


Why are options trades supposed to be delta-neutral?

I'm reading Natenberg's book, and he says that all options trades should be delta neutral. I understand that this prevents small changes in the underlying price from changing the price of the option, but couldn't there be a case where you would want that? I (think I) also understand that if you're betting against just volatilty, it would make sense, since you don't care what direction the underlying price moves, but I don't entirely understand why he says all options trades should be delta neutral.

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...

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

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...