## Wednesday, October 24, 2012

X%- a concept to unify different betting strategies
Following last post, a similar trading cousin for the Martingale gambling strategy is the sit and wait strategy (buy and hold with no stop losses and exit when you reach a predefined take profit level). This trading strategy is similar to the gambling strategy in the sense that both depend on the market/game to eventually do something (if I wait for long enough/try enough times it will eventually happen and I will be profitable overall). Sit and wait is like a more conservative version of Martingale. Lets say for the Martingale, instead of doubling down each time you lose, you increase your bet by x%. For example, x could be 200% (triple down) or 0% (sit and wait for casino bets). The stock market equivalent is re-balancing and increasing your original exposure by x%. (ie. let's say you had invested \$100 in stock, it goes to \$75. For Martingale strategy you invest another \$125 to make it 2x original exposure = \$200) This rebalancing assumes you are still under the same return distribution situation.

As we increase x, we are more likely to win/get back to breakeven when we are losing. For example, let's say we just lost k in a row. For doubling down on a 50/50 bet, we are 50% likely to get back to winning \$1 overall by the next turn, but for sit and wait, we are only (1/2)^k likely to get back to breakeven.

At the same time, our losses increase at an exponential rate (1+x%) if we keep losing. This creates an embarrassing problem- how do you avoid bankruptcy if your losses increase exponentially? This goes back to the "chunking" mentioned in the first post. In trading this would be how much extra margin/loss buffer should you budget for this trade before it starts to win. And if you are going to double down/add, at what point should you do this? ie. if you go bankrupt/can't take the pain/extreme exposure after the third time you are wrong, then you need to split the potential worst case scenario into 3 segments and add as you cross between segments. In general you would split these segments by P&L, but it could also be possible to split it by underlying price movements, or even time or a function of all of the above. In the case of the stock mkt, it is also meaningful to vary x as a function of the underlying stock price movement, p&l change, time and other parameters.

Let's derive the formula for how to chunk. For gambling, this is relatively easy. You lose b, b*(1+x), b*(1+x)^2,...,b*(1+x)^(n-1) So our total loss is a geometric series and equal to b*[(1+x)^n-1]/x. So given n (how many times you lose before you bankrupt) and x (the double down amt), you can calculate how big your bankroll has to be relative to bet size b:  bankroll is [(1+x)^n-1]/x times that of bet size b. For example, if x = 100% (martingale; doubling down each time), n = 10 (will go bankrupt if wrong 10 times in a row), you need 2^10 - 1 = 1023x initial bet.

It is however more complicated for trading, since you don't lose everything that you bet. Let's define a set {p_0, p_1...p_n} to stand for the stock price at which we would dial up our exposure another 1+x. We could have the set be in arithmetic progression (eg: keep adding per every \$1 price drop), or it could be a function of P&L (eg: keep adding per every \$100 loss), or time (eg: keep adding every 1min- here we would not be able to predict what p_i is before time t).

Let's start with the common “Martingale-like” case where p_i is an arithmetic sequence, we have a defined stop loss pt p_sl determined by the stock price terms (not P&L), and we rebalance by doubling the # of shares that you own each time. Then p_i = p_0 + (p_sl - p_0) * i / n. And the shares you buy each time b_i = b_0 * 2**i. (note ** means exponential).
Your worst loss is sum[b_i * (p_sl - p_i)] where i is the set of integers from 0 to n-1. Courtesy ofWolfram Alpha, we see:

Since m = n-1, this simplifies to b_0 * (p_sl - p_0) * [2**(n+1) / n - 1 - 2/n]. So let's say you had 1 mil budgeted to buy this stock, and you wanted to bet 4x before you go bankrupt, and you expect p_sl - p_0 = 100-70 = \$30. Then we can look at how much (# of shares) to bet initially if we want to be able to rebuy n times.

We can also create a table showing how the intial bet size (both in terms of shares- b_0 and also as a % of total risk budgeted/worst case loss) increases with n.

 Initial Bet N Shares % of Budget 1 33,333 333% 2 16,667 167% 3 9,091 91% 4 5,128 51% 5 2,924 29% 6 1,667 17% 7 945 9% 8 531 5.3% 9 296 3.0% 10 164 1.6%

Note that, for example, for n = 1 (just bet once and don't redouble ever), then you would be buying stocks with a notional amt that is 3x your risk budget. Notice that this does not necessarily mean that you are "taking on leverage"- for example, if you had 10mil, and budgeted 1mil for this trade, and so bought ~3mil, then you are not going over your total account equity. The traditional concept of "leverage" is insufficient to describe your trades/risk levels when it comes to more complicated trading strategies.

A more useful version for trading is perhaps this:

 Initial Risk Loss N (as % of budget) Multiplier 1 100% 1x 2 50% 2x 3 27% 4x 4 15% 7x 5 8.8% 11x 6 5.0% 20x 7 2.8% 35x 8 1.6% 63x 9 0.9% 113x 10 0.5% 204x

And then to get how much notional to invest, you simply divide this percentage by the worst case scenario return. eg: N = 2, worst case is -25%, then you should buy notional amt of 0.50 / 0.25 = 2x that of risk allocation initially. The loss multiplier is simply the inverse of the risk %. You can also use this to figure out what your max loss will be if you bought some amt and you are not sure how many times you will redouble:

eg: you bought 100 shares, and prices went from 120 to 110. You have been redoubling every \$5 (so this will your third buy). If you are wrong again, then by 105, your total loss will be 1 / 0.27 * (100 * 15) = 5.5k. Let's check this:

 Price Shares Loss 120 100 -1500 115 200 -2000 110 400 -2000 Current Price 105 Total Loss -5500

So this checks out.

Food for Thought- Thinking about Risk
This x% concept can be thought of as a trading style/bias. People who trade by x% (ie. doubling down on losses and taking profits vs cutting losses and adding to winners) are essentially trying to convert a payoff distribution that is linear to stock prices (ie. if stock goes up \$1, you make b, if stock goes down \$1, you lose b, where b = # of shares), to a distribution that is non-linear and path dependent. It is more complicated to understand and incorporate the risks involved with this sort of dynamic trading strategy when you are evaluating your portfolio's risk characteristics. You might realize that what I described is very similar to a description of options. There are interesting ways to integrating traditional methods of looking at options risk and concepts presented here (eg: chunking and taking into account your trading style):

(1) Bringing options concepts to x%. Your trading style (especially in the extreme case where you preset limit/stoploss orders religiously) can essentially be converted into discretized chunks of gamma. Using this, you can do standard risk analytics taking into account your intended future trades (eg: the increased tail risk? the expected theta?). Another possibility is to use x% as a way to normalize for some portfolio measures. eg- if you use a high +x% for this bucket of trades, then you would expect high success rates and probably equal-ish avg profit vs avg loss but also significant long tail losses. In fact, you could also back out an implied x% from your success rates or your avg profit vs avg loss.

(2) Bringing risk budgeting to portfolio analytics on derivative positions. One problem with current scenario analysis and stress testing techniques is that they is usually purely based on past data, and that they are used as a control- ie. someone/some software will give us this number and let's just keep our losses under this threshold. One thing to do is to be more logical when it comes to estimating the risk budget/max loss. For example, we developed a formula for max loss (given n) above. I believe that a model based on fundamental characteristics (eg- for stocks this could be using a P/E range to derive a max loss) could be an interesting supplement to standard risk models nowadays. Unfortunately, traditionally, the risk manager is not involved in the investment research process and so lacks the in-depth fundamental knowledge required to create such a model.