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.)

First Week of Trading

Please note: I will be talking about strictly screen trading in this article. I know nothing about floor trading or the crazy symbols they use.

Trading is a major rush for me. Given how much fun I had, it is hard to believe that traders get paid (besides compensating for stress). Every trainee loves trading, but not all of us loved how low our P/Ls were on the first week. I lost a good deal of money, down about 700 (this is pretty consistent among the trainees).

I know many of you read this blog so I would finally get to the part where I talk about: what is trading like? I will split the below in as many perspectives as I have time to write.

Trading as Math
You do second grade math all the time. You need add / subtract quickly and consistently. The more difficult math (some of which may be found here) is for thinking at home and developing strategy. You are unlikely to use it throughout the day.

Often, you will want to recalculate the theoretical value of what you want to trade. You will need to instinctively be able to find the middle of the spread. Sometimes, you want to be in the middle of one and lifting the offer on another. That means the high end of the spread is worth entering the transaction for you, so you transact with the person and buy high. (Otherwise, you would join the bid, which basically means that you are willing to transact if someone wants to come sell to you)

Trading as a Game
Trading is just like playing a computer game, except your mom won't get upset if you play. Many of the concepts I apply get good at a real time strategy game, I am trying to apply to trading.

Knowledge

1. Learn all the statistics - what is the correlation between futures price falling and vol going up; how much deviation is likely in each greek; what news affects the prices; how does it affect the price; etc.
   
2. Understand your role in the game - what is your advantage (if you think you have one); how can you exploit your core competency fully?
3. Weakness - once you understand your strength, you should understand how your strategy can be gamed; are you doing mental math fast enough, are you misreading what the market is saying
4. Analogies - what is trading like? What do thinking or non thinking beings do to survive / thrive in these types of environments
5. Next Steps, Frameworks, and Discussion - people have theories, learning them is good
   

I used to play a game called Rise of Nations. At a job interview, my interviewer told me his children played the same game. Since I was an expert in the game, he asked which nation I liked to play. The answer is that I play every nation on random. In a real game, every strategy has a weakness and telling your opponent anything, is too much.

Preferences

1. Change screen content - the more you see the more you know, but also the more you slow down the computer
2. Visualizations - a friend of mine wanted to switch the cursor icon to all black so that he could see it better, I changed the font size so that I could see more on the screen
3. Window Placement - self explanatory
   

Speed and Consistency

1. Hot key usage - you should know most of the hotkeys after week; there are hundreds of key combinations that can do the same thing; some will be easier at first but don't have the speed potential of others
   
2. Mouse sensitivity - you should get used a mouse with a higher sensitivity (you can change this in the control panel)
   
3. Mechanical Weakness - you will sell futures when you need to buy futures or hedge correctly but too slow; you will accidentally increase vol when you should decrease vol; you will enable your computer to do trades when you want to stop (these get less and less likely as I continue to list them).
   
4. Hand Eye Coordination - Programs like EyeQ and THINKfast may help you improve your mechanical skills
   

Trading as a Psychological Experience
The market is risk-neutral. You will often face the market with less than perfect objectivity. You will probably lose upside potential due to fear. I will list some examples of each type of feeling. There will be times where you follow an example and you are not feeling any of these. There are always exceptions.

• Fear / Nervousness - Example 1: Lets say you got a good deal on a call option that you bought for 2,15 when you thought it was worth 2,18. You are sitting with some greeks that could potentially move against you. The only way to hedge is to sell your call option. Some time passes and the market is willing to buy the option for 2,20 from you. You decide you will sell and just take your profits. Is this a good move? If you took the 'fear' header seriously then you probably know that I'm going to say not to do this. It is likely that the market wants to buy the option for 2,20 because it is worth a bit more than 2,20. This fear is very dangerous because it can be rationally justified (although poorly) with arguments due to mean reversion. Or even more poorly with the gamblers fallacy.
  
• Greed - This is the exact opposite of fear. You are holding a position that you think is overvalued, but you think that the market might continue to pay you off due to some sort of momentum argument. Way back in your head you know that if you think something is overvalued, you should hedge it. When you are greedy though, you are likely to believe that you are smarter than everyone else and reject the signs that tell you to get out of a position.
  
• Wishful thinking - You just lost a lot of money because a position that you were about to hedge swiftly moved against you. You decide you should wait because the market must return to its original levels and hedging now, for a loss, would be stupid. More often than not, this is just wishful thinking.
  
• Frustration - You just lost a lot of money unexpectedly and someone is whistling a terrible rendition of Bach. Resist the urge to break things. It is unlikely that you will get violent , but you will have trouble making sound decisions.
• Everyone has there own 'pet peeves.' Mine are vibrations that I can feel (especially my chair being touched, humming, tapping on the desk or by feet). It is a fact that it is impossible to live without making vibrations, and yet -- I don't like feeling other's vibrations. You might have an aversion to gum smacking. These things need to be communicated clearly and without malice if you hope to overcome this weakness as easily as possible. Combined with regular frustration, pet peeves can seriously debilitate your ability to perform at the highest level.
  
• Happiness / Laziness - You just made a ton of money without even trying. You decide to sit back, relax, and stop trading.
  
• Sadness - I just lost a lot of money. I am not going to trade any more today.
  

I went to the streets to see what the traders had to say about trading:
"awesome," said one. "fantabulous," said another. The average trader doesn't seem to like to talk all that much. The average wholesale trader -- that's another story.

Who would be good?

• Strong self discipline
• Consistent, quick mental math
  
• Likes computer games
• Likes chance games (most people do not like chance games)
  
• Interested in finance
• Can accept the risk of failure
• Likes computer screens
• Interested in making money
  

Answers to previous Math Finance Questions:
1) You own a stock. It exhibits 0% vol for 3 months and then 30% vol for the next three months. What's your average vol for the 6 months? (hint: variance is additive) I was surprised that no one answered this question correctly. Volatility is the square root of variance, so you have to square the volatility and then average to get the average variance. Then you have to take the square root to get the average vol. As the picture at the top of the post shows, vol is just another name for the standard deviation.
There are two reasons why I am stressing this:

1. Everyone seems to be able to say "volatility grows at the square root of time," but can't prove it. Take year 1 vol = v, year 2 vol = v as well. Square both to get v^2 + v^2 = 2v^2 for the 2 year variance. Now just take the square root.
   
2. Natenberg's treatment of volatility is poor relative to the rest of the book. He just averages volatility. Clearly, vol does not average.
   

2 and 3) I am going to wait until some more people try to answer these.

4) According to Hami: covariance is the measure of how the deviations of two variables match each other. Correlation is an adjusted measure of covariance when the variables have been standardized, or made comparable in magnitude and dispersion.
- Exactly, measures like Beta are Covariance. Statistical correlations of say, vol and futures, do not speak of magnitude (be careful!).

Math of Finance Questions: (I don't know the answers)
1) What is a good predictor of market spikiness? (This can be mathematical or non mathematical)
2) On what days do you expect larger market movements? Why?

Note: Do you like writing or even hearing yourself talk? Do you like finance, math, or trading? I am considering opening up this blog to a couple who want to write good stuff. Please email me at igor.schmertzler@gmail.com

Ps: If I have time, I will write about adverse selection.

Financial Math

Money Management (sometimes incorporated within risk management)
"Ralph Vince did an experiment with forty Ph.D.s. He ruled out
doctorates with a background in statistics or trading. All others were
qualified. The forty doctorates were given a computer game to trade.
They started with $10,000 and were given a 100 trials in a game in
which they would win 60% of the time. When they won, they won the
amount of money they risked in that trial. When they lost, they lost the
amount of money they risked for that trial.

This is a much better game than you’ll ever find in Las Vegas. Yet
guess how many of the Ph.D’s had made money at the end of 100
trials? When the results were tabulated, only two of them made
money. The other 38 lost money. Imagine that! 95% of them lost
money playing a game in which the odds of winning were better
than any game in Las Vegas. Why? The reason they lost was their
adoption of the gambler’s fallacy and the resulting poor money
management." -Van Tharp (who also has a blog, which is not very good)

If you make 10% on your investment in the first year and lose 10% on your investment in the second year, how much money do you have? Ans: (1+ -.1) * (1+ .1) = .99 [y0ou lost 1% of your wealth]. Take aways:

1. Note that this doesn't matter whether you made the 10% first or if you lost the 10% first.
2. If you instead didn't reinvest your winnings, you would have broken even. [ex. you have $100, (100 * 1.10) = 110, save 10 dollars then (100 * .9) = 90. Then take the 10 out of savings and you have broken even.
3. What happens when variance increases? Lets say the stock loses 20% in one year and gains 20% in the other year. Like the example above, this is an arithmetic average of 0%. But the math, with reinvestment, looks like this : (1.2 * .8 = .96 ) You just lost 4% of your wealth. As variance increases, geometric returns fall. The geometric will always lag the arithmetic return with any variance / volatility. Your actual loss due to geometric growth lagging arithmetic growth is half the variance. (see kelly criterion formula at top, read fortune's formula, and check out wikipedia)
   
4. You can diversify out of this. Lets say you invest in both stocks in one year. That way, you do break even! Check it out.

Arithmetic of Geometric
If you reinvest your money, it gets multiplied. If a company reinvests it own money, why shouldn't its money get multiplied as well? This multiplication phenomenon is why stocks are said to grow geometrically. (The way that mathematics simplifies multiplication is with the log function.)

"One of the many hearts of this book is the broader concept of decision
making in environments characterized by geometric consequences.
An environment of geometric consequence is an environment
where a quantity that you have to work with today is a function of prior
outcomes. I think this covers most environments we live in! Optimal f is
the regulator of growth in such environments" - Ralph Vince, Mathematics of Money Management

Optimal f is related to the Kelly Criterion but is supposed to be more robust for different distributions. Hopefully, I will get a chance to write more about this interesting subject. Please see the above book for more.

Dimensionless Risk Measures [slash comparing apples to oranges]
When something is dimensionless, it is called a scalar. Suppose you sit down at a new desk and your risk measures show 200 vega and 2000 delta. As a trader, you do not have a stance on which way the underlying or vol is going to go, but unwinding your position may cost a lot of money. Between delta and vega, which is the bigger risk? Experienced traders will give you the same philosophical answer: it depends.

Lets work on a simpler problem. Can we even compare two different deltas?
If delta really acts log normally then the magnitude of its change will be scaled by the size of the underlying. That is, delta is supposed to be your change in price (or your change in wealth) with respect to a one point move in the underlying. But lognormal dynamics say that the chance of the stock changing by 1 percent not by 1 point. That means, that you have to scale delta by the underlying value to take out the percentage effect.

Let say you have delta on an index. In particular, you invest in the Dow and have a delta of 10. The index is at 14,000, so your delta * index = 140,000. Someone else has delta of 100 on Nasdaq, which happens (very conveniently) to be trading at 1,400. Her delta * index is 140,000 also. Who has the bigger risk here? (ans: Some stocks move with greater variation than others.) The scale of variation is called beta. This is the amount that a stock (or whatever) moves with respect to the market. If the Dow has a beta of one and the Nasdaq has a beta of two, then if the market moves by 1%, then nasdaq will move by twice as much as Dow. For that reason, you have to multiply beta into the equation. (From what I understand, multiplying by beta is not common in trading. Please send me an email or leave a comment if you have a good argument on why beta is not multiplied in.)

So how do we compare delta risk to vega risk. One thing is for sure, the variance of underlying (for delta) and variance of volatility (for vega) are of great importance. Each has, in theory, a distribution associated with them. Variance of delta is supposed to be log normal but the variance of volatility isn't. It would be nice to take the standard deviation of delta and the standard deviation of volatility to compare one to the other, but when the shape of distribution is so different, this leads to many problems. (can you think of any?)

With all of the theoretical problems associated with taking the standard deviation in the underlying and the standard deviation in volatility, a trader could move to new desks with a greater understanding of what is happening. The variance of volatility grows as the option get close to expiration so some help from projected risks might help a trader really get a good feel. I believe this would really help.

The arguments against this type of approach [as I see them. ie, there may be more :) ]:

1. What about the correlation between a change in underlying and the respective change in volatility? (This type of thinking leads instead to simulations and VaR)
   
2. The measure would give the trader the false feeling of confidence but would in reality be very unreliable
3. The computational intensity isn't worth the marginal return

Topics for Next Week

• How does it feel to trade
• Equation fitting: Taylor explained and little on Fourier (don't worry, I already wrote half)
  
• Functions of a trader that can and can't be automated (scary topic for many traders to consider)
  
• Adverse selection (those that transact with you are likely to know something you don't)
  

Math of Finance Questions

1. You own a stock. It exhibits 0% vol for 3 months and then 30% vol for the next three months. What's your average vol for the 6 months? (hint: variance is additive)
   
2. If the underlying makes 20% in one day. How should the implied volatility curve move? Keep in mind, this updates your information about the vol of the stock. There is no single right answer. (please email your thoughts)
3. How is question two altered when the underlying makes 20% but no news is released?!
4. What is the difference between correlation and covariance? (The untrained always say correlation when they mean covariance.)
   

Format
I have gotten some comments that the presentation is somewhat complex or unclear. I will do everything I can to improve clarity and remove mistakes, but I cannot simplify topics any further. I will post any links to the topics I discuss for those that don't understand but want to understand.

Trading on Monday

Lonya arrives
Lonya has been here for a couple of days now. The top picture is him in the train that goes from Schipol Airport (I picked him up because I am good guy). The picture below is meant to indicate how comfortable Amsterdam is for us. If he didn't have a life back home, he would stay here for quite a while longer. He likes the free feeling of Amsterdam as much as I do.

Art vs. Science
Many people, with all too much pride, explain that what they do is an art and not a science. This type of thinking assumes that science is something that is simple because it has rules and art is difficult because it requires the creativity. This implies that Newton engaged in art since his science was not yet science and implies that middle schoolers in art class who are taught to work with certain materials are actually performing science (because they are following the teachers rules). How boring science is!

I categorically disagree with this definition of science. Science is the way that people discover new things and check them so they can rely on their observations. Science is the way that people systematically add elements together to create something totally new. Science is the way that a choreographer reuses elements that elicit reactions from the audience or reuses elements to put dancers into a certain position from which they can express something better. There is no reason to assume that science covers only linear or expected outcomes. Combinations of elements in chemistry create compounds with entirely different properties.

Chemistry, art or science? Every pursuit is both art and science.

Finance: Art or Science?
I won't surprise you by saying that I believe finance, like pretty much everything else, is both art and science. First, you learn the rules (as could be established as scientifically as possible). Then you practice playing by the rules and learn what is not truly set in stone. Most things are not set in stone: volatility does not have to fall when the underlying price rises; stocks do not have to act log normally (more about this below); correlations are not a rule; a spread trade is a risky bet, not a sure thing.

And if you are a great artist, you develop something that people will use later. You will develop science. The better you think the more you'll be able to contribute to making what you do a "science." (this topic is covered quite exhaustively in Sparks of Genius)

How to discuss art. (slash why talking in outputs is terrible)
When discussing a financial topic with someone, keep in mind whether the particular topic you are talking about is a topic that is an established rule or something that just has arguments for and against. (A model does not have everything built in and does not work all the time). Unlike in engineering (or physics), you cannot just say some numeric or boolean answer and be satisfied. A true conversation about something that can act randomly requires balancing pros and cons, and re-balancing all the time. (Yes and no is very useful in operational mode, when you want someone to understand exactly what you want him or her to do.)

Since, the assumptions in finance are ever changing and the behavior of markets are under constant flux, you should strive to understand something as fully as possible. That means you should examine all the arguments that push something one way or the other. For example, if I ask what will happen to the underlying price when the volatility increases, and you answer the price drops, then you have missed the point. You must say why you think the price will drop.

In the same way, if you ask whether someone is democrat or republican, you do not yet know anything about them. You know whether they agree with what you currently are willing to believe but since you do not know why they made that choice you know too little to make any sort or judgment.

In that way, finance is a lot like marketing or psychology. Empirical studies are important but will not give you rules to follow. Instead, asking why is important.


Preschool Graduation
It has been 3 weeks since the program began. Most of the trainees are relatively fluent with the Option Volatility and Pricing book and many other topics like currency risk. We have spent a couple of days watching the previous two classes of trainees trade. The batch in front of ours is very talented and got promoted to the real trading floors. Though the graduated trainees (not my batch) are all pleased with which desk they will be sitting, they had no part in determining at which desk they will be working.

My batch gets to trade tomorrow, which means that it is now time to read all the books on how to control your emotions while trading. (Thank you poker for teaching me to deal with chance, especially bad beats.) Learning the theory well beforehand allows one the time to read about subjects that not all programs think are mandatory like 'how to get into the zone or emotion free trading.' Some of these subjects that interest me, but don't necessarily has established answers, are below.

Friends come in

Moving away from 'home', whether it be away from where your parents live, where you went to school, or where you have grown accustomed to, is always a hassle. The upside, of course, is that you get to make more places in the world your home. Friends and family are a phone call and a flight away.

Some choose to take that flight. Yesterday, a friend came in to see the US trader that I hang out with often. They went to a Casino and the Trader helped Phil Ivey sign in. If you know me, then you know how I dislike celebrity, but this is cool. Phil Ivey is one of the best Poker players in the world, regarded as highly loose and aggressive.

Today, we will experience my top 5 reasons why Amsterdam is the best place for friends to come visit.

1. The rules are different than any other place in the world. To visit other places is to experience other cultures firsthand. Where else can you visit the red light district and talk about the 'philosophical' implications at a coffeeshop.
2. Museums: Heineken museum, Van Gogh, Rembrandt, Anne Frank's house
   
3. All you need to get around is your own two feet. Take a tram anywhere you want to go in the city. Fly to Italy, Istanbul, or London for cheap. Take a train to Brussels, Germany, France, etc.
   
4. The culture is young and everyone speaks English. You will always find young tourists just like you.
   
5. Optiver's housing is incredible. While the rate at which things get fixed can be problematic, the actual intention of providing the best housing is clear.

Quick information on how to not ruin your laptop
If you come to Amsterdam, you will probably bring a laptop. If you are looking for a laptop, try xpbargains.com

• Computers overheat. And when they don't over heat they are still hot. Please turn off your computer every so often. If you do not have a cooling stand, you are likely to be overheating when you leave your computer on all day and all night. (My rec)
  
• Take out your battery when it is fully charged and your laptop is plugged in. (Tip due to Jerry, thanks Jerry.)
  
• If your computer is in "sleep mode" you can ruin your hard drive by carrying it around. Your hard drive is not turned off so it will spin and is vulnerable to shocks. Putting it instead into hibernate is preferable.
  
• Please look into downloading WinPatrol and AdAware

What happened to the Taylor expansion?
Friends, as soon as I figure out how to put pictures somewhere besides the top of the post, I will be much better armed to discuss the Taylor expansion.

Taylor Expansion

Think fast
Trading is fast. In trading interviews (with the exception of Citadel and Optiver), studying 'higher' mathematics is considered nearly useless. After all, what does a deep thinker have in common with a successful trader? Working on discipline or training your nerves is much more likely to lead to profits than developing a new strategy. What a mathematician may understand with a lot of work, the trader will feel.

The privilege of having a blog is the ability to say whatever I like. Math is incredibly helpful to gain intuition. Math may be more precise than feeling, but it's not the precision that I am emphasizing.

Estimating your wealth
Let's assume the adage that money makes money. In fact, a universal power has decided that your wealth will be given by f(x) = x*x or x squared, where x is your age in years and f(x) is your wealth in euros or dollars. You at 21 years old right now and you are making 42,000 a year. How rich will you be by the time that you are thirty? (please refer to the top picture)

In words
You could assume of course that you will be making 42,000 for the next couple of years. You would be saying, let's discard the higher order effects, like acceleration of salary. Your wealth is where your salary is coming from, where it derives from. Your salary is the first derivative of wealth because it is the first order effect that directly affects your wealth. With this approximation in mind, you can make the first calculation (see second picture). I will make forty two thousand every year for the next nine years so I will make 378,000. That means if I starting with 441,000, my wealth will be 819,000 by the time I am thirty.


In Math
f ' (x) = 42,000 every year (every 1 x that goes by)
f '' (x) = assumed to be 0
f (21) = 441,000
f(30) = f(21) + 9* f ' (21) = 819,000

Damn it
I know what many of you are thinking: 'igor, I want you to make more money since you have such a nice blog. After all, you don't even have any advertising on your site.' I have to go to work to make our next assumptions more probable.

In the next post, I will finish the example and introduce taylor's theorem to help us do the math in a second instead of belaboring the point.