The Physics of Wall Street covers the history of math from Bernoulli through Mandelbrot and how we arrived at modern algorithmic trading.http://www.amazon.com/The-Physics-Wall-Street-Unpredictable/...It doesn't cover the actual math. Not sure if there's a good book for that.

 > The Physics of Wall Street covers the history of math from Bernoulli through Mandelbrot and how we arrived at modern algorithmic trading.The way you put it makes it sound like algorithmic trading is the pinnacle achievement of probability theory. Surely not?
 It's great when you can use math, and remove humans from the process once the algorithms are written, to do something people thought was impossible. Was never trying to imply that's the pinnacle. You've got Gordon Gecko and then you have James Simons.http://youtu.be/QNznD9hMEh0Maybe the U.S. wouldn't rank so low in math and science if kids saw a few more career possibilities.
 The algo part of High frequency trading is generally overstated. The real value is the number of transactions as even really simple algorithms can make money if there are no transaction fees.EX: Hold one stock for every cent in the current price. 1.53\$ = 153 shares. Buy one stock per cent if the stock price moves down one cent and sell one stock if it moves up once cent. You now make a half cent every time the price moves up or down one cent. Note, this only works well for stocks with low share prices, lots of movement, and no transaction fees, and if the price doubles you have sold all your shares.
 You have this exactly backwards: you are guaranteed to lose 0.5 cents/transaction. Suppose the price is \$1.00 so you hold 100 shares. The price move up to 1.01 so you buy a share (at 1.01). Now the price moves back to 1.00 so you sell a share (at 1.00). You just lost 0.01.There is no algorithm that guarantees you will make money in a fair market. The only way to guarantee making money is arbitrage or trading on inside information.
 Ops, flipped it. If the price moves up you sell, if the price moves down you buy.The reason this is not 'optimal' is if the price moves up to far you have sold every stock and run out, also you need a reserve to handle the price moving close to zero and if the stock goes bust you now own a lot of worthless stock.PS: So, you will make money on a bounded random walk, but potentially far less money than holding the stock. In the end all this does is trade unbound potential gains for a finite income stream. Which is what all algo provide there effectively choosing which game to play in Vegas or shifting risk around etc.
 > Ops, flipped it. If the price moves up you sell, if the price moves down you buy.But then you aren't maintaining the invariant that you hold a number of shares equal to the current price.> So, you will make money on a bounded random walkYes. If you know ahead of time what the price is going to do (even probabilistically) then you can make money. If you don't, you can't.
 aren't maintaining the invariant Yes, it’s just the starting point for a really simple example.If you know ahead of time what the price is going to do (even probabilistically) then you can make money. Exactly, you pick a model and make money ‘if’ reality fits that model. However, fast dumb models can make lots of money and complexity adds risks.Anyway, many people get stuck with the idea you need the smartest people in the room while ignoring how much it costs to have the ‘smartest people’ in the room. Arguably, many companies are simply doing this to attract investors not because it maximizes returns as managing money is a great way to make money.
 > fast dumb models can make lots of moneySure, and they can also lose a lot of money.> complexity adds risksNot necessarily. Modern portfolio theory is a lot more complex than (say) buying and holding a single stock. But it's a lot less risky.> you need the smartest people in the roomMaybe you don't need the smartest people in the room, but it helps not to have too many stupid ones. The real problem is that it can be really hard to tell which is which.

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