Vertical Skew IV

Vertical Skew is the shape of the implied volatility (IV) term structure for a single options chain maturity. There is also something called a horizontal skew which is the IV across maturities. The movement of the vertical skew structure has been of interest to me recently when analyzing some of my option positions.

I had a long put butterfly position on with center short strikes 20 points below market. At trade initiation my greeks were as follows: Delta: -56, Gamma: -1.04, Theta: 117.9, Vega -343.8. This is generally what you’d expect for a short vol position. Delta is slightly negative due to the fly being bearishly positioned. My expectation was that for any decline in the markets you’d expect reduced pnl decline due to the fact that the delta will partially offset vega. As it turns out this is not the case, my position actually gained money on a price decline which was opposite to what my greeks are telling me. To understand we must look at IV Skew.

Below are the IVs for RUT September 2015 expiration put options. I specifically picked this period to illustrate the transition from high vol to low vol environment. If you look closely, you will notice that the green line is steeper than the red line. The second graph is the difference between the two line – its increasing. As we go from OTM options to ITM options, the rate of change of IV increases. Another words in our graph, ITM options (right side) will decline more in value (steepens) then OTM and ATM option when IV drops (vice versa for increases in IV – flatten). This phenomenon is not captured by the Black Scholes model as it assumes fixed volatility during the lifetime of the option.

IV

Diff

 

Now how does this help me understand what happened to my position? Well, my right wing (highest strike) put option within my fly is an ITM option. When I was analyzing my position, I assumed that the long put wings of my fly had equal pnl contribution but that’s not the case. My right wing benefited the most from a given IV increase which means the overall negative effect of vega was over-estimated. In fact it may (and I am not 100% sure) be that both delta and vega was working in my favor assuming that the pnl contribution of the long wings was greater than the losses incurred from the short center strikes of my fly. Armed with this information, I think people can incorporate this in to their adjustments and maybe create some ways to exploit this. Open to any ideas!

Pretty cool eh?

Automated Trading System – Internal Order Matching

Most automated trading systems (ATS) are built such that there are little to no interactions between component models. This is limiting. Here I am referring to a trading system as the overarching architecture that houses multiple individual models.

Without interactions, each model is operating within an environment that it is preconceived in. For example, mean reversion can happen in different time/event frequencies. A model that is parameterized to take advantage on certain frequency will not have knowledge of others.

One component within an ATS that is rather complicated to architect is the order management system. The OMS is the component that handles all order requests generated by the prediction models. It must always be aware of outstanding orders (limit/market, etc), partial fills, and proper handling of rejects,etc. Now the complexity is increased when a portfolio of prediction models all generate an order for a given tick. (Which should be processed first?)

The general rule of thumb in dealing with this is to aggregate all orders by asset to reduce transaction costs. If there are a mix of long/shorts, the net will be the final order quantity. When it is filled, simply dis-aggregate them back in to component fills to the respective models (internal matching). The annoying part, in my opinion, is when you introduce multiple order types. For example, Limit and Market orders. How would one architect the OMS to handle both? Hmm.. This goes back to the debate of the degree of coupling between the strategy and the OMS itself….

CRSM Code

The code below represents the CRSM algorithm. It is adopted to the SIT framework. I have refactored the code so that its is easier for the user to use and understand.

I would like to once again thank you David Varadi for his tireless effort in guiding me along this past year. My gratitude also goes out to Adam Butler and BPG Associates for their support  all along the way.

Download Code: here

Thanks,

Mike

Shiny Market Dashboard

I’ve been asked multiple times regarding code for the dashboard so thought I’d release it. I coded the whole thing in one night last year so it’s not the best and most efficient, buts it’s a good framework to get your own stuff up and running. It’s long, north of 700 lines of code.

https://www.dropbox.com/sh/rxc8l4xnct5bcci/AABvTD2iJjC6wicLed3q9Qr3a

On another note, I’ve recently graduated from University and am excited to be moving to Chicago in a month for work. I am looking forward to the exciting opportunity. Also I will be releasing my graduating thesis in the coming weeks on RSO optimization. David Varadi has been my thesis advisor and mentor and I want to thank him for that!

Mike

Natural Language Processing

I’ve recently updated some matlab code about machine learning on to github. Today I’ll be adding to that with some Natural Language Processing (NLP) Code. The main concepts we covered in class were ngram modelling which is a markovian process. This means that future states or values have a conditional dependence on the past values. In NLP this concept is utilized via training n gram probabilities models on given texts. For example, if we specify N to equal to 3, then each word in a given sentence depends on the last two words.

So the equation for conditional probability is given by:

CodeCogsEqn

Extending this to multiple sequential events, this is generalized to be (chain rule)

CodeCogsEqn (1)

 

This above equation is very useful for modelling sequential stuff like sentences. Extension to these concepts to finance are utilized heavily in hidden Markov models that attempts to model states in various markets.  I hope the interested reader comment below for other interesting applications.

The last topic we are covering is class is computer vision. As of now, topics like image noise reduction via Gaussian filtering, edge detection, segmentation are being covered. I will post more about them in the future.

Code Link

Cheers,

Mike

Artificial Intelligence

Artificial intelligence surrounds much of our lives. The aim of this branch of computer science is to build intelligent machines that is able to operate as individuals; much like humans. I am sure most of us have watched the Terminator movie series and questioned to what extent will our own society converge to that in the movie. While that may sound preposterous, much of what automated system developers do revolves around building adaptive systems that react to changes in markets. Inspired by a course I am taking at school right now, I would like to use this post as a general fundamental intro to AI.

If you ask people what intelligence is, most will initially find it hard to wrap words around the idea. We just know what it is and our gut tells us we, humans, are the pinnacle of what defines intelligence. But fact is, intelligence encompasses so much. According to the first sentence in wikipedia, there are ten different ways to define it.

“Intelligence has been defined in many different ways including logic, abstract thought, understanding, self-awareness, communication, learning, having emotional knowledge, retaining, planning, and problem solving.” -Wikipedia

Since it encompasses so much it is not easy to define it in a single sentence. What can be said is that intelligence relates to one’s ability to problem solve, reason, perceive relationships, and learn.

Now that I’ve offered a sense of what intelligence means, what, on the other hand, is artificial intelligence? Artificial intelligence is the field of designing machines that is capable of intelligent behavior; machines that is able to reason; machines that is able to learn. More precisely, the definition of AI can be organized in to four different sections:

  • Thinking Humanly
  • Thinking Rationally
  • Acting Humanly
  • Acting Rationally

The first two relates to thoughts processes while the last two relates to behavior. Thinking humanly revolves around whether the entity in question is able think and have a mind of its own. This is essentially making decision, learning and  problem solving. Acting humanly is whether a machine is able to process and communicate language, store information or knowledge and act based on what it knows and learn to adapt based on new information. These set of required traits are formulated based on the famous Turing Test which examines if a machine is able to act like a human through answering questions asked by anther human being. The machine passes the test if the person asking the question isn’t able to determine if its a machine or human. Thinking rationally closely incorporates the study of logic and logical reasoning. It was first introduced by Aristotle who attempted to provide a systematic way of inferring a proposition based on a given premise. An famous example would be “Socrates is a man; all man are mortal; therefore, Socrates is mortal.” Lastly, acting rationally is the idea of choosing the most suitable behavior that produces the best expected outcome. Another word one is rational if given all its knowledge and experience, selects the action that maximizes their own performance measure/utility.

Agents

When studying AI, the term agent is used to represent an entity/model that interacts with the environment. More precisely, an agent perceives the environment through its sensors and employ actions through actuators. Comparing this to humans, imagine sensors as eyes/ears and actuators as arms and legs. The sensors will at each time step take inputs, called percepts, which are than processed by the agent program. The agent program then passes the inputs in to an agent function. The agent function maps inputs to correct outputs (actions) which are then sent via the agent program to the actuators. This agent based framework closely relates to automated trading systems. The environment is the market and the changing prices at each time interval. The agent program would be our trading system which takes in daily price information and pipes it into the agent function, or the logic of the trading system. For example, todays new price is updated which is passed in to the trading logic. The logic specifies that if the current price is $10, it will sell. The sell action is passed back to the environment as a sell order.

The above example is a very basic type of agent known as the simple reflex agent. This type of agent only makes decision solely on the current percept (price). It doesn’t have a memory of the previous states. A more complex agent known as the model based reflex agent is one in which it has memory of the past, known as its own percept sequence. Also this agent has an internal understanding of how the environment works which is detailed in its own model. This model of the world takes inputs and identify the state it is in. Given the state, the model forecasts what the likely environment will be like in the next time step. Proper action is then recommended and executed via actuators. (Think of markov models)  So far, the agents I’ve introduced largely reflect that of a function that take input and spits out a output. To make things more humanly, the next agent I will introduce is called a goal based agent. This is similar to how given our current circumstances, we aim to maximize out objective function. The objective function can be money or anything that makes us happy. More concretely, the goal based agent is an extension of the model based reflex agent but it assigns a score for each recommended action. The agent will choose the one that maximizes its own objective function.

The reader will most likely ask how this knowledge helps them make money in the markets. What I can say is that finance is enter a brave new world where together with technology is transforming how money is being made in the markets. Having a understanding in finance and statistics in my opinion is not enough. Those are the areas where your competitors are already fishing (mostly). Knowledge in subject areas like AI, speech recognition, natural language processing, machine learning, and computer vision (just to name a few) will allow you to be more creative in design. I urge the curious minds to explore the unexplored!

Beyond Pairs

Writing rather prolifically this past week. Last week was the end of my midterms (for now!) and continuation of job search and preparing for my final 5 weeks of university.(!) I hope my readers are finding my posts interesting and enlightening.

As mentioned in an earlier post on statistical arbitrage, the interesting aspect of it comes when we consider multi leg portfolios. To construct a multi-leg portfolio, the traditional way to do it would be to employ a multivariate linear regression (factor model). The intuition behind this is that we are trying to estimate a fair value for an asset using various predictors or independent variables. For example, we know that the S&P 500 is composed of stocks from various sectors. Therefore, an intuitive way is to derive a fair value for S&P 500 using the 9 different sector Spdrs by the following equation:

CodeCogsEqn (3)

The residual return that is left over, “alpha”, is considered to be neutral (uncorrelated) against the industry sectors. With this framework, we now can essentially make ourselves neutral to any factors we want. For example, we have access to a wide variety of ETFs that mimic underlying asset class movements. If we want to be neutral to interest rates, credit risk, and volatility, we can employ ETFs: TLT, HYG, and VXX respectively. Below is a chart demonstrating this, showing the estimated fair value of SPY relative to the actual ETF:

Screen Shot 2014-03-02 at 11.33.55 AM

Below is the spread that can be traded via long short on each leg:

Screen Shot 2014-03-02 at 11.38.21 AM

The concept of being able to control the factors we are exposed to is very appealing as it allows us to potential shy away from turbulent events that transpire from specific assets. Not only that, these uncorrelated return streams when combined in to a portfolio allows significant risk reduction. As Dalio said, the ability to combine 15 uncorrelated return streams allows us to effectively reduce 80% of risk. (Chart below) Interestingly, from my understanding of what Bridgewater does, I am pretty confident they are employing spread trading too, but purely from a fundamental way. For example, how does a set of asset classes react to the movements of economic indicators? From there they construct synthetic spreads to trade off of these relationships.

Screen Shot 2014-03-02 at 12.06.52 PM

Below is the code that generated the data for this post:


spread.analysis<-function(data, y.symbol, x.symbol, lookback=250){
    y = data$prices[,y.symbol]
    x = data$prices[,x.symbol]
    lm.holder<-list()
    fv = NA * data$prices[,1]
    colnames(fv) = c('FairValue')

    for( i in (lookback+1):nrow(data$prices) ){
        cat(i,'\n')
        hist.y = y[(i-lookback):i, y.symbol]
        hist.x = x[(i-lookback):i, x.symbol]
        lm.r = lm(hist.y ~ hist.x)
        lm.holder[[i]] = lm.r
        fv[i,] = lm.r$coefficients[1] + sum(lm.r$coefficients[-1] * x[i,])
    }
    mat = merge( x,y,fv )
    return( list( mat = mat, fv = fv, reg.model = lm.holder ) )
}

Also here are some links I’ve found to be very informative.

The paper on high frequency statistical arbitrage is rather a relevant one as it relates to my previous blog posts on energy related pairs trading. Essentially, the author goes on to construct a meta algorithm for ranking pairs to trade. This meta-algorithm is composed of correlation coefficient, minimum square distance of normalized price series, and a co-integration test value. I don’t have intraday (paper used 15 min bars) equities data nor do I have the infrastructure to test it but the idea resonates with me from my research in top N momentum systems. A lot of ways to improve.