Anti-Martingale - page 2

islandrock:
I have been mod-ing the EA "10points 3" but I have one problem.

I Would like for it to open the buy and sell trades independently.

if one sets the extern max trades to 4 and the the EA opens a buy trade. and the pair falls and it opens all 4 buy trade and the direction changes it now cannot open a sell trade to hedge the buys and counteract it... so I need to make the buys and sell independent of each other. can anyone help with that. or at least point me in the right direction?

drgoodvibe:
That's an interesting idea WNW I think it's worth looking into.

krelian99:
Anti-cyclical yes, anti-martingale no. It's dangerous. You can loose money ;)

mntiwana:
Nightingale,martingale and holygrail are cousin sisters :)

Mathematical analysis

One round of the idealized martingale without time or credit constraints can be formulated mathematically as follows. Let the coin tosses be represented by a sequence X0, X1, … of independent random variables, each of which is equal to H with probability p, and T with probability q = 1 – p. Let N be time of appearance of the first H; in other words, X0, X1, …, XN–1 = T, and XN = H. If the coin never shows H, we write N = ∞. N is itself a random variable because it depends on the random outcomes of the coin tosses.

In the first N – 1 coin tosses, the player following the martingale strategy loses 1, 2, …, 2^N–1 units, accumulating a total loss of 2^N − 1. On the N^th toss, there is a win of 2^N units, resulting in a net gain of 1 unit over the first N tosses. For example, suppose the first four coin tosses are T, T, T, H making N = 3. The bettor loses 1, 2, and 4 units on the first three tosses, for a total loss of 7 units, then wins 8 units on the fourth toss, for a net gain of 1 unit. As long as the coin eventually shows heads, the betting player realizes a gain.

What is the probability that N = ∞, i.e., that the coin never shows heads? Clearly it can be no greater than the probability that the first k tosses are all T; this probability is q^k. Unless q = 1, the only nonnegative number less than or equal to q^k for all values of k is zero. It follows that N is finite with probability 1; therefore with probability 1, the coin will eventually show heads and the bettor will realize a net gain of 1 unit.

This property of the idealized version of the martingale accounts for the attraction of the idea. In practice, the idealized version can only be approximated, for two reasons. Unlimited credit to finance possibly astronomical losses during long runs of tails is not available, and there is a limit to the number of coin tosses that can be performed in any finite period of time, precluding the possibility of playing long enough to observe very long runs of tails.

As an example, consider a bettor with an available fortune, or credit, of 2 43 {\displaystyle 2^{43}} (approximately 9 trillion) units, roughly half the size of the current US national debt in dollars. With this very large fortune, the player can afford to lose on the first 42 tosses, but a loss on the 43rd cannot be covered. The probability of losing on the first 42 tosses is q 42 {\displaystyle q^{42}} , which will be a very small number unless tails are nearly certain on each toss. In the fair case where q = 1 / 2 {\displaystyle q=1/2} , we could expect to wait something on the order of 2 42 {\displaystyle 2^{42}} tosses before seeing 42 consecutive tails; tossing coins at the rate of one toss per second, this would require approximately 279,000 years.

This version of the game is likely to be unattractive to both players. The player with the fortune can expect to see a head and gain one unit on average every two tosses, or two seconds, corresponding to an annual income of about 31.6 million units until disaster (42 tails) occurs. This is only a 0.0036 percent return on the fortune at risk. The other player can look forward to steady losses of 31.6 million units per year until hitting an incredibly large jackpot, probably in something like 279,000 years, a period far longer than any currency has yet existed. If q > 1 / 2 {\displaystyle q>1/2} 1/2">, this version of the game is also unfavorable to the first player in the sense that it would have negative expected winnings.

The impossibility of winning over the long run, given a limit of the size of bets or a limit in the size of one's bankroll or line of credit, is proven by the optional stopping theorem.^[1]

Anti-martingale

This is also known as the reverse martingale. In a classic martingale betting style, gamblers increase bets after each loss in hopes that an eventual win will recover all previous losses. The anti-martingale approach instead increases bets after wins, while reducing them after a loss. The perception is that the gambler will benefit from a winning streak or a "hot hand", while reducing losses while "cold" or otherwise having a losing streak. As the single bets are independent from each other (and from the gambler's expectations), the concept of winning "streaks" is merely an example of gambler's fallacy, and the anti-martingale strategy fails to make any money. If on the other hand, real-life stock returns are serially correlated (for instance due to economic cycles and delayed reaction to news of larger market participants), "streaks" of wins or losses do happen more often and are longer than those under a purely random process, the anti-martingale strategy could theoretically apply and can be used in trading systems (as trend-following or "doubling up"). (But see also dollar cost averaging.)

We show that, if we allow general admissible integrands as trading strategies, the three dimensional Bessel process, Bes 3 , admits arbitrage possibilities. This is in contrast with the fact that the inverse process is a local martingale and hence is arbitrage free. This leads to some economic interpretation for the analysis of the property of arbitrage in foreign exchange rates. This notion (relative to general admissible integrands) does depend on the fact, which of the two currencies under consideration is chosen as num eraire. The results rely on a general construction of strictly positive local martingales. The construction is related to the FF ollmer measure of a positive super-martingale.