Originally Posted by
Data
I am having hard time realizing this. I am really trying to find a flaw in my reasoning but you were not very convincing above. I think you need to define "profit" better because I do not immediately believe in profits gained due to "arbitrary defined" prices. On a side note, I was talking about buy prices, not buy/sell prices.
I was actually hoping that you'd be able to convince yourself by examining the above equations.
"Profit", you see, just corresponds to accounting profit which is merely the negative of the difference between the buy price and the "current" price, however you choose to define it. As (I suspect you're aware, this is how P&L reporting exists within the financial trading industry.) If the mark-to price also happened to be the "fair" price then the accounting profit will also equal the "expected" profit. If the mark-to price also happened to be the market transaction price (as in the example I gave above) then the benefit that the Kelly result can be obtained using only the standard single-bet Kelly formula without any further calculus. From an accounting perspective neither one of these options are necessary.
I think you're also getting hung up on the concept of a changing bankroll. This again is simply an accounting construct used to satisfy the Law of Conservation of Unrealized Bet Capital (a phrase coined by yours truly not 15 seconds ago). To wit:
Y = x + Π
or in plain English:
bet stake after price change =
bet stake prior to price change +
unrealized profit after price change
Anyway, let's go back to the original example. The player acquires 4.545% of bankroll worth of a binary bet at +110 that is expected to win with probability 50%. The market price then moves to +120.
You choose, however, to mark-to-the-first-number-uttered by cousin-Steve. This number is -1000.
I'm now going to demonstrate that using the mark-to-the-first-number-uttered by cousin-Steve method, yields the same additional bet (3.977% of original bankroll) as both mark-to-market and a full optimization using the line change spreadsheet.
So here's what we have so far:
o = 2.1
n =
x = 4.545%
which then gives us:
Π = x/n * (o-n) = 4.132%
Y = x/n * o = 8.678%
So in other words, if we mark to a price of -1000 we have unrealized accounting profit of 4.132% of initial bankroll and a current position at -1000 of 8.678% initial bankroll.
We're presented with the opportunity to increase our bet size at +120. However, since, we're marking to -1000 each unit at +120 yields:
o2 = 2.2
n =
Π2 = x*(2.2-1.1)/1.1 = 100%*x profit
Y2 = 2.2/1.1 = 200%*x exposure to the bet at -1000
So in other words, after marking to -1000, each dollar wagered at +120 yields an immediate $1 of accounting profit and an immediate $2 of exposure to the bet at -1000.
So if δ represents our additional stake at +120 then our full-Kelly utility function looks like this:
U(δ) = 50%*log(1+(4.132%+δ)+(1.1-1)*(8.678%+2*δ))
U(δ) + 50%*log(1+(4.132%+δ) - (8.678%+2*δ))
which simplifies to:
U(δ) = 0.5*log(0.95454 - δ) + 0.5*log(1.05 + 1.2*δ)
Differentiating wrt δ and setting to zero then yields:
0.5/(0.95454 - δ) = 0.6/(1.05 + 1.2*δ)
Solving for δ we find, lo and behold ...
δ = 3.977%
Imagine that!