 Let’s say the price of an asset behaves as follows:

• The asset price starts from 0
• At any time t, the price is equally likely to go up or down by 1 i.e. Price(t+1) can only be Price(t) + 1 or Price(t) – 1, and both of those are equally likely
• However, if the price goes to 10, it has to go back to 9 at the next instant. If the price goes to -10, it has to go back to -9 at the next instant. That is, if Price(t) = 10, Price(t+1) has to be 9. If Price(t) = -10, Price(t+1) has to be -9. ### Q1: What is the Fair Value of this asset?

It is fairly obvious that the long term expected value for this asset is 0. The probability distribution for prices looks like this The stock price can be anything between -9,9 with a probability of 5% and it can take values -10,10 with probability 2.5%

### Q2: Assuming there are no position limits and no fees to trade, what is the most ideal way to trade this asset so that you maximize profits?

Since the Fair Value is 0, Expected Profit from trading anywhere except 0 is +ve. Since trading is free and there are no limits, we should buy at all points < 0 and exit position at 0. Similarly, we should sell at all points > 0 and exit position at 0.

Note there is no value in holding a position at 0 (we make money if asset price moves favorably but lose the same amount if it moves unfavorably. Since both are equally likely, EV = 0), hence we should always exit position at 0.

### Assuming there are still no fees to trade, what is the most ideal way to trade this asset?

The position limit changes things. While long term EV of any individual trade is still zero, trades are not independent of each other. For example, if you sell at +1, while that trade has EV +1, it prevents you from selling at +3. So total expected profit in this case is lower than selling at +3, given the stock goes there. This suggests we can’t look at long term EV of trades independently. However, the stock is more likely to go to 1 v/s 3 from 0.

Another thing to note is that instantaneous EV of trading anywhere = 0, expect at the wall where stock price is guaranteed to revert. At the walls,  instantaneous EV of trading = 1.

So the only hard rule to follow is to make sure you have a long position at -10 and a short position at +10. In between, we can follow any trading strategy, since trading is free(instantaneous EV of trading anywhere = 0)! For example, if you choose to trade at 1,-1, you will trade more frequently, but make smaller profits. If you trade at 5,-5, you will hit those levels less frequently but make larger profits.

### Q4: What is the most ideal way to trade this asset?

Now, if we’re paying a fees to trade, the instantaneous EV of trading anywhere except the walls is now -0.2. At the wall where stock price is guaranteed to revert, instantaneous EV of trading = 0.8.

Now trading comes with a cost, which means if you trade at 1,-1, you trade more frequently, but you also make smaller profits and pay more to trade.If you trade at 5,-5, you will hit those levels less frequently but pay 1/5th of the fees. (It is important to note that you need to go from 0 to 1  and return back to 0 five times to make the same profit(not counting fees). What do you think is the ratio of probability of going to 5 from 0 and returning is and probability of going from 0 to 1  and retuning back to 0, 5 times). Now we need to balance the frequency of trading with fees to trade. We’ll find that higher values(8,9,10)  will result in higher overall profits. Obviously we still need to ensure we have a long position at -10 and short at 10. The plot of profits v/s trading bounds is as below: Now what should happen when we increase the fees, say to 0.3 per trade (trading fees are usually 1/3rd to 1/4th of profits)? Since it’s quite costly to trade, we should only try to trade where profits are guaranteed. In this case, we should only trade at -10,10 and exit position at 0. The plot of profits v/s trading bounds is as below: You’ll notice as fees gets higher and higher, it makes sense only to trade on the walls. And that’s the whole point of this exercise!

## Takeaways

• First, knowing the Fair Value/Expected Value of asset price is not enough in real world applications since we have position and fees constraints. We saw that while trading at -1,+1 were +ve EV trades in the long run, they’re not the best trades with constraints. Knowing probability distribution around Expected Value helps us determine these hard-like walls.
• Second, more important, when stock moves follow a random like walk, we sometimes get impatient and get stuck doing trades that seem like decently profitable (like the +/- 1,2,3,4,5 levels here). A lot of you suggested this strategy, saying we’ll trade more frequently. What we don’t realize is that we need to a make a lot of those trades and the probability of hitting those multiple times instead of breaking out of range is only about as high hitting the better levels. Infact, these marginal trades only stop us from making the really good trades (at +/- 10 here). They’re -ve EV trades!
• Obviously we don’t have as clean as distribution or as hard walls in real life as in the question here, but we should look for levels where we have a very high probability of stock reverting back before making trades.
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#### David

I'm the community lead at Auquan. Feel free to message me if you get into any problems. A fun fact about me is I actually studied medicine at university.