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Modeling Luck & High Frequency Trading

Recently while working on a stochastic model, a friend asked what we are achieving by introducing randomness in our model and wouldn’t it be simpler to  just assume the expected value. At that moment I responded that expected values do not help us in modeling situations with a finite time horizon, but I don’t think I got my point across. That lead to this post to explain why stochasticity and randomness is important in our models. To remove jargon I renamed stochasticity or randomness as Luck (Good or Bad). To give a perspective I would like to illustrate how we can evaluate high frequency trading using this model.

To illustrate let us model a simple game of luck. We toss a fair coin and head will mean earning a dollar and tails would mean losing a dollar. The expected value is of course zero, but what if I play that for a finite number of times say 30 times. So I simulated 30 rolls of the die as a game and checked what my total pay off was at the end of 30 rolls.

The first time I played this simulation I got a gain of $20. (Good Luck I would say). I played it a few more 30 roll cycles and found that I lost anywhere between -4 and -20.  I repeated this a 1000 cycles of 30 rolls of die each time and found that I ended up with neutral or $0 only 14% of the time and the rest of the time I had around 41% of good luck and 45% of bad luck.

So you may say Ashwin, that is obvious what does this tell us. Well running it once doesn’t tell us but by using the model I came to understand that changing the pay off matrix had a great impact even if I didn’t have any influence on the randomness of the die roll.

So in other words if I could keep my losses at -0.5 and my gains at 1.0 then I could consistently experience periods of good luck. In fact my good and bad luck is shown below.

Randomness is definitely important but it is how we design the pay off matrix that shapes luck.

Life may not be fair, it may all be up to chance, but turning chance to Luck is probably within our control by altering the payoff matrices.

In my next post we will look at how failure associated with learning may improve the payoff matrix with time.

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