Linear Approximations

The shortest distance
Lines have two parts [y = mx+b]: they start somewhere [b] and with unwavering speed [m] they get somewhere. Pretty simple.

Since I am simple, lines describe me pretty nicely too. I have 50 dollars and each redbull vodka costs 10 dollars. 5 redbull vodkas later, I am numb to my net worth.

Relationships can suffer over distance
To estimate how much money you would make investing at 8% a year with 100 dollars, see that you make 8 dollars in your first year. The amount of money that you have changed by only 8% so far, so figuring that you will make about 8 dollars the next year on the same investment is only 8% off. After two years, I have made about $16.

After a greater number of years, the compounding of error would become a problem. After all, what is the difference after 10 years. ($100+$80) - [100*1.08^10]. The error gets worse and worse and we lose love for the line.

Over that line
A threshold is a point where everything changes. I was a jello crazed child. One thing was very clear to me: if there is anything better than one cup of jello, it is two cups of jello (rational by any economic sense). One way or another, I got enough money to satisfy my dreams. I ate an orgy of jello. Unlike the linear promise of joy, I threw up -- violently. The formula for maximizing pleasure can be tricky.

Call me
Some things can be described well linearly. Take for example how much you gain or lose on a stock when it goes up by a dollar, but what happens when you add a threshold to the payoff? A call is similar to holding a stock contract except that it only entitles you to money if the stock is beyond a certain threshold like 100 bucks (when the contract ends). Of course, you could sell your call contract before the end of the contracts life. Instead of getting dollar for dollar returns, you have to value your call with an equation that is much more difficult (at the top of the post). All due to a little threshold. After all, if I can hook up with an 18 yr old, why can't I hook up with a 17 yr old? Thresholding can be difficult to understand.

............................
Igor 'I only post when you make me' Schmertzler

Quotations

Just about everyone loves a quotation.

• A penny saved is a penny earned. (I just made a penny!)
  
• Follow your heart. (Yay, I love not going anywhere)
  
• Live for the moment. (I was alive every moment writing this blog)
  
• And a trading favorite, balls to the wall. (There are too many walls!)
  

''Keep it simple, stupid"
Imagine following your heart and living for the moment. You see a delicious apple and since you are hungry, you eat it. You see a beautiful girl and you walk up to talk to her. You feel like reading a book or writing a blog. And you rack up points.

"I'm kind of a big deal around here"
You rack up points -- big time -- because everything your heart desires not only helps you in the present, it makes your life better in the future. An apple a day, anyway. And maybe the girl wanted to seize not only the day with you. Sometimes, there is no trade-off between the present and the future. Sometimes, seizing the day is what it's all about. Big opportunities are captured when a person is trying to capture opportunities. That rarely happens when practicing long division.

More likely than not, saving pennies is not something your heart truly desires. Is that how Bill Gates made his money? Certainly, it's not how Trump made his money. It's better to be dollar smart and penny foolish. After all, the more you spend, the more you save.

"Opportunity is missed by most people because it is dressed in overalls and looks like work"
Opportunities can only be seized when they come. Fortunately for you, opportunity can only be seized by those who prepare. So figure out when your opportunity is coming and prepare for that time. It would be hard to do many good trades when the market is closed. Saving that penny in preparation for a once in a lifetime investment; learning how to put on makeup without looking like a prostitute and meeting the love of your life; practicing that jump shot for hours for the important three seconds you'll remember for the rest of your life -- that opportunity will come.

"In a minute there is time / for decisions and revisions which a minute will reverse"
All men are not created equal. In fact, some mothers smoke or drink while pregnant. There are fathers who did not want to create anything at all. The nature is different, the nurture is different, and the randomness is -- random. Minutes in my life were not created equally either. My time was worth about $7 / hour when I was 16. I was just as good at reading Shakespeare then as I am now, but I get paid slightly more these days to focus less on shakespeare and more on trading. Is this the most I will ever make? Are my minutes now deserving of thinking only about the short term?

"To perceive evil, where it exists, is a form of optimism"
My time right now is every bit as important but not every bit as useful ($ / hr) as it will be in the future. When a person thinks like that, he makes his thoughts a self-fulfilling prophecy. If you are in college, invest in yourself and take a harder class. Calculate your grocery bill in your head and add tax. Commit yourself to not borrowing from tomorrow. Debt, anorexia, not putting away the dishes, drugs, and laziness is borrowing from tomorrow. I am an optimist; I think your opportunity will come tomorrow.

"Equality of opportunity is freedom, but equality of outcome is repression"
They say, the gap between the haves and the havenots is growing. I am not sure if that is true on percentage terms, but I am sure that the gap grows between people who live for the moment and people who prepare for their moments. If you have spent your life preparing, we'd love to have you at Optiver trading.

You've worked hard, why not let your investments pay off. Email me.

More Posts!

Welcome Stephanie
In keeping with the intention of sliding away from Optiver specific material and instead sliding towards new ideas (and increasing post frequency) we officially have a new 'poster girl.' I am excited about her first post, which I think she might put up tonight.

As the name 'poster girl' implies, you will want to be like her and perhaps write for this blog as well. Feel free to contact me if you want posting abilities on this blog. Or you can just leave great comments like these.

Themes
As the blog gets more people posting, the need for separate themes becomes more apparent. Still, separation of finance, math, economics, 'quant,' is largely imaginary. Here are the themes as we have defined them.

• I will try to maintain my preference for writing about 'MathFinance,' (which is both math and finance and neither math nor finance at the same time). Also, I will be in charge of making up words.
  
• Stephanie has promised to write about 'FinanceEconomics.' She has also volunteered to be in charge of blog sarcasm :)
  

It may seem like in just those two themes we would not be restricted from covering almost everything! I can think of some things we might not cover. Here is a short list of the things that you could potentially write about:

1. Advances in computer science (algorithmic, software packages, and hardware)
   
2. Trading (technicals, new ideas, psychology)
   
3. Investments (putting a lot of capital to work)
4. Strategic theory
5. Mathematics (what has been proven, what is contested, how do you develop your own framework)
   
6. Quantitative Modelling (CAPM, APT, Black-Litterman, Back-testing, Signal Theory, ...)
   
7. Computational Finance
   
8. Econometrics
   
9. Financial Business Decisions
10. Financial Culture and Lifestyles
11. etc
    

Accumulated Topics
Over the past couple of months, I have promised to talk about all sorts of topics that I never got the chance to write about. Here is a list that is meant to serve as an apology:

1. What is dynamic replication? Can I arbitrage volatility?
2. Adverse selection
3. Volatility as the sensitivity of the market to information
4. Taylor expansion for quick calculations
5. Fourier series
6. Principal components analysis
7. Total least squares vs ordinary least squares (introduced in the last post)
   

Topics that others have asked for:

1. What is a broker, prof trade, wholesales trader?
2. How do I prepare for the Optiver interview?
3. What is a swap?
4. What is Beta?

I will do my best to get to all of these and perhaps others will volunteer to write great posts. In the meantime, thanks for reading.

Applied Finance

Investing and Trading - Similar but Different

In investing, one must find some security to hold for a relatively long duration. Trading, however, is a bit different. In trading, one must find something that is under- or over- priced for quick (often relative value) market corrections. In particular, traders try to find opportunities which are (1) hard for others to spot, (2) hard for others to exploit, or hard for others to exploit (3) as quickly as the traders can exploit the opportunities.

Speaking generally, in both investing and trading one uses tools to reason which investments / trades will act accordingly. In both cases, one also filters choices based on standardized descriptions. But in trading choices happen fast and relative value needs to be found more and more deeply every year. As implied above, mathematics is important because it:

1. is a language for common understanding
   
1. gives rise to standardized filtering which allows for very quick decision making
2. allows for standard methods of information discovery (so you can use the same tools for different products)
   
2. is a set of tools for reasoning (to see deeply)
   
3. is expertise to avoid misunderstanding and to avoid false conclusions (to see clearly)
   
4. is much more, but not relevant to the post
   

This blog starts with 3 and goes backwards.

Expertise to avoid misunderstanding

• Some math is considered so basic that we take it for granted that we can apply strict methods to it. 1+1 = 2 right? If that is true, then how many chewing gums do you get by adding one to another? How many empty sets do you have if you add 5 empty sets together? Can you show me half a piece of chalk? This math is basic enough that you would never make this mistake with these examples. The point is that there is an assumption made even with something as basic as counting.
  
• Some math is used so commonly that we forget the changes implicitly made when moving from fractions to decimals and back. Take for example averaging.
  
• Alice ate 1/2 an apple. Varun ate 1/4 of an apple. What is the average number of apples eaten?
  
• (.5 + .25) / 2 = 3/8 Simple.
  
• A baseball player had a hitting average of .5 in year one and .95 in year two.
  
• Simple method: (.5 + .95) / 2 = .725
• Actually, it turns out he hit 15 balls in year one and 95 balls in year two. So 15/30 + 95/100= 110/130, well over 80%. We forgot that the decimals no longer had the weighting information.
  

-----Examples for the general populace------------------

Risk free investments are risk free- You are promised a risk-free, 10% return after a year (there is no chance of default). You reason that you should be about 5% richer after half a year (or a little less if there is a high frequency of compounding). Unexpectedly you need the money so you decide that you want to take your profits in the middle of the year by selling your risk-free investment to someone else. The only problem is that the market is now offering 20% "risk-free" and you are only able to sell at a major loss. The assumption that you wouldn't have to pay for opportunity cost is unreasonable (thanks to Stephanie for pointing out the correct way to say this). Your investment is not as risk-free as you thought.

---------Examples for Finance majors (chosen because these are so accepted)-------

Diversification leads to higher returns - Why should diversification lead to higher returns? Why doesn't diversification just water down the skill you put into choosing your investments, as Buffet proposes? The answer is to most is simple: either you have no skill or your skill isn't better than the value of diversification. Who knew that the nice guys telling you to diversify were actually implying that you suck. In mathematics, one says geometric returns lag arithmetic returns. The more the lag (which happens with high volatility), the more to be had from diversifying.

Least Squares Regression for Hedge Ratio- Many people, sadly including myself, use "math" blindly. Think of the last time you ran a regression to determine a hedge ratio. What happened? Which security did you choose to be the independent variable? Based on what algorithm did the computer fit the best line?

What would happen if you chose the other security to be the independent variable? Ans: Using financial data, you would get a different hedge ratio. That should raise a major red flag. This anomaly happens because your standard linear regression assumes that you have no error in determining the value of the independent variable. You should be using an "orthogonal least squares." Check the assumptions here. [More in a later post]

What's the point?!
The point is that not only do we use math blindly, but also we use blind math often. Somewhat rationally, you might argue to yourself that you should specialize in finance. That means, of course, you cannot waste time learning theoretical math because you can just use tools developed by others.

Due to our preference for "specializing," you, me, and the next finance major are the people using math incredibly blindly and not even knowing how blind we are. It is a mistake to feel that the immense precision of math means that one is accurate to some nice decimal places (accuracy vs precision). Not only is there a difference between accuracy and precision (ok, I tricked you into that last one), but often the precision is way overstated as well.

You won't find the term 'mathematical blindness' in Wikipedia and you may be asking, what exactly is mathematical blindness? Blind math is the usage of math without the understanding of the assumptions undertaken to make this math work. To some degree, this blind math is the applied math you've always done! It's not your fault! You followed examples to do what your teachers taught you to do.

Does it then follow that the person who knows the most math is the least blind?
Perhaps not. Quant funds, run by people with incredible mathematical understanding, have a reputation for getting killed in the marketplace. However, many quant funds also have a reputation for making a killing in the marketplace. Math makes them confident (and tells them to lever up). What is more dangerous than a person willing to stake his life on something? But then again, who is more bound for success than the confident?

Sometimes blindness will be mathematical and other times will be caused by assumptions in another field (eg. financial assumptions). The blindness is in misunderstanding anything! It is the magnitude of assumptions made from all sources.

Blind financial math (sometimes called model risk) comes about from (but not limited to):

• applying mathematical tools to the wrong types of problems
• Running the wrong regression
• Mistaking covariance for correlation or vice-versa
  
• applying unrealistic (but sometimes necessary) financial assumptions to the data
• The future will look like the past
• There is a distribution to this random element
  
• using the wrong data
  
• Should you keep the outliers?
• Should you take out certain days?
• Are the more recent days more important?
• Does the time of the year matter?
  
• not keeping track of assumptions / not challenging assumptions often enough
  
• You diversify like you are told [The assumption is that your skill in selection isn't better than the value of further diversification]
  
• You take over a trading spot and inherit plenty of assumptions you didn't even know were made

A smart mathematician (name your highest math professor) is not a financial genius and a financial guru (take Buffet for example) is not a mathematical genius. First of all, note that Buffet is not a trader, but an investor. Also examine the assumptions because the future does not have to act like the past! These two, once separate fields, are converging into one understanding -- called financial mathematics. Trading uses financial mathematics. Assumptions in what was okay to consider two different fields can kill you in this one.

Focus

The rules of the blog have changed:

1) I am now more deeply involved in more company specific topics that require sensitivity and filtering. To maintain clarity, I will filter my topics instead of presenting only half-thoughts. Thank you to all that have read my material pertaining to Optiver; there will be no more.

2) My traffic, though nice, has few dedicated members (sorry to those I am offending!). Q: Why post something online? Ans: For others to see it. The point of this blog was to be as realistic as possible. In keeping with my dislike of being delusional, I am refocusing on only the core readership. My other thoughts can go into a word file as easily as they can go online.

Lastly, thank you to those that send me an email every so often asking for specific material. You make my work that much easier.

Three Weeks of Trading Finished (Volatility 2/3)

Quick comment: Since I haven’t posted for two weeks, I'll post twice as much this week. Thank you to everyone who complained and made me finally write something.

Call Vol = Put Vol for Strike K

Put call parity (for European options!) is an equation. Equations are amazing because they can have all sorts of mathematical operations done to both sides to deduce properties. Keep in mind that only the theoretical, strict definitions of arbitrage have equations. That makes put-call parity a great starting point for talking about implied volatility (Hull).

The following (slightly different) is found in Hull 5E, Chapter 15:

Buying a call and selling a put is the same as owning the stock and subtracting the carrying cost. Put-call parity holds in both the Black-Scholes world and the real world since it is based on a no arbitrage argument. Please take a look to the top of the post.

What is happening up there?! The first line is a formulation of put call parity for Black-Scholes option prices. The second line is the formulation for the market prices. The third line is a manipulation of the top two lines, specifically the first line less the second line. The forth line is just the third line rearranged. The outcome shows that the dollar pricing error in Black-Scholes is the same for calls and puts!

What's with the word implied?

The volatility that, given a particular pricing model, yields a theoretical value for the option equal to the current market price is implied because it must be inferred from the model. More simply, to imply the volatility means you have to guess and check to find it. There are a couple of ways to guess and check more efficiently than random.

Bisection Method: Guess in the middle and see if you are too low / too high. Then guess in the middle of the half that is better.

Newton-Raphson: Guess in the middle, guess again and see how much closer you are. If you increased the volatility by 1 point and got 2 dollars closer and you were 6 dollars off, you should try increasing the volatility by another 2 points.

Brent's Method: (If you would like to explain this clearly, please email me and I'll post it.)

Different Vols for Different Strikes

The first picture at the top of the post is an empirical result showing the volatility smile in DAX options at a certain time. Why does this smile exist? The following is a list of interrelated reasons:

1. Distribution adjustment - Since the underlying does not have to act lognormally (please see below posts for more on this), some parts of the distribution become more probable than others. This type of correction is very prominent in currency options because of the discrete nature interest rate moves.
2. Leverage adjustment - "As a company's equity declines in value, its leverage increases. This makes even lower stock prices more likely. As the company's equity increases in value, its volatility decreases in value, making higher stock prices less likely."
   
3. Reflexivity - Bad news triggers more bad news. The financial markets are reflexive. An example of positive reflexivity is giving a company a good credit rating. That good credit rating lowers their cost of borrowing. Since their cost of borrowing is less, the company's value increases. Since the company's value increases, it's credit rating increases. Understandably, the effect gets smaller and smaller with each iteration, but the positive feedback is clear. This is a possible explanation for fat tails commonly noticed in financial data.
   
4. Volatility clustering - "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes" - Mandelbrot. This is different from above two in that the direction does not matter.
   
5. Supply and Demand - Before the crash of 1987, the smile was not prominent in option prices. It is possible that investors are crashophobic and willing to pay a certain premium to have a back up. Since it is unclear how to arbitrage the smile (more on this in the next post), premium just builds up in a certain strike. Supply and demand also constantly updates the market's perception of the probability of the options ending up in different strikes.
   

Back by popular demand (Volatility 1/3)

Over the next three posts, I will try to address these questions:

1. What are the basic properties of volatility and implied volatility?
   
2. How do quants measure volatility? Why should there be different implied volatilities for options of different strikes? Why should the put volatility be the same as the call volatility on the same strike?
3. What is dynamic replication? Can I arbitrage volatility?

“Implied volatilities are the focus of interest both in volatility trading and in risk management. As common practice traders directly trade the so called "vega", i.e. the sensitivity of their portfolios with respect to volatility changes. In order to establish vega trades market professionals use delta-gamma neutral hedging strategies which are insensitive to changes in the underlying and to time decay, Taleb (1997). To accomplish this, traders depend on reliable estimates of implied volatilities and, most importantly, their dynamics.” -- Applied Quantitative Finance

Volatility and implied volatility are different

Volatility is the standard measure for how 'active' a stock is. Volatility:

• is linear
• measures the stock's spread of distribution
• is numerically debatable because the past isn’t supposed to fully reflect the future
• has an associated time frame. Vol = V per year. Assuming the volatility of the stock stays constant over more years, Vol per t years : V * √(t)
• has market components. Even though the company itself has not announced any news, the volatility can grow or shrink due to the market becoming more or less volatile. This is why Principal Components Analysis sounds like a good idea (covered in next post)
  

Since it seems unsettling to draw conclusions from the past to expect market behavior, the focus shifted to implied volatilities, Dumas, Fleming and Whaley (1998). Implied volatility is inferred from option prices given some sort of formula for option prices like Black-Scholes. Implied Volatility shares the above properties but has a couple of its own properties:

• is different for different strikes
• is the same for puts and calls of the same strike
• sometimes doesn’t make sense for exotic options
• incorporates a distribution adjustment due to stocks not being totally lognormal. This is because Black-Scholes formula assumes lognormality
• incorporates volatility premiums due to supply and demand. This means that the stock isn’t expected to have this volatility but the price has been bid up and the implied volatility is higher to match the price. This is somewhat controversial and mostly original with me, so don’t think that it is right. This is due to the unavailability of pure arbitrage and further difficulty of balancing in the short term. (What I’m saying is the dynamic recreation doesn’t work completely. The problem with the last idea of course is knowing when the 4^th reason is driving premiums and when the 5^th reason is driving up premiums.)