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.