Here is the lattice of the game’s possible outcomes:

If your first call is right, starting with $20 at S, you will win $5 and end up with $25 at A. In the second toss, if you win, you will have $30 (B); else your money will be reduced to $20 (E). After the third and last toss, depending on whether you start from B or E, you will end the game with $35, $25 or $15. In all cases, the condition of having at least $10 would be satisfied.

If your first call is wrong, you will end up with $15 (point D). If your second call is also wrong, you will be at X with only $10. That is the end of the line for you. Technically, the game is not over yet. One more toss is left. And if you guess right, you might end up with $15 at point G. But you cannot take that chance, because if you lose, you would violate the condition of leaving the game with at least $10. So there is no going forward from X, which is why it is no longer connected to the mesh “going forward”.

Taking the outcome of the coin toss as the proxy for change in the stock price, this game is the principle of portfolio insurance, which, as you can surmise from the name, aims to prevent the value of a portfolio from dropping below a prescribed amount. Virtually all portfolio managers employ a variation of this strategy to ensure that they would not suffer catastrophic losses.

There are a couple of differences between coin tossing and real life portfolio insurance, though.

One is that coin tossing is a discrete game; we “jump” from point to point – S to D to X – based on the outcome of coin toss.

Portfolio insurance is closer to continuous-time. A portfolio insurer arrives at his exit-from-market point only gradually and through piecemeal actions.

“Action” is the clue to the other difference between coin tossing and portfolio insurance.

In tossing coins, we were passive. There was nothing to do but wait for the outcome.

Portfolio insurers, by contrast, are active. They must constantly adjust the composition of their portfolio – sometimes daily, sometimes several times a day – in response to market conditions. As prices fall, they sell, in anticipation of markets moving towards point X. As prices rise, they buy, in anticipation of more profit.

This trading pattern is “pro-cyclical”. It creates a self-reinforcing, self-perpetuating process that exacerbates the market trend.

As for the mechanics of the portfolio adjustment – how much of which security to buy and sell – it is formulaic and algorithmic. A machine could be programmed to do it and, as a matter of fact, that is how it is done. Hence, the other name for portfolio insurance: program trading.

We now have the cause of speculative capital-driven market crash before us. It is a state when many portfolio managers simultaneously exit the market. When that happens, the bid side disappears. With few active buyers left, prices collapse. It is as if the synchronized withdrawal creates a resonance in the form of a market crash. From Vol. 2:

If the market falls [to point X), there is the chance that it could fall still further to [to point H.) That eventuality is unacceptable to the [portfolio] insurer. The only way to eliminate it is to leave the game, to stop playing. That is precisely what he does at that point. When the market falls [to X], he sells all his [shares] ... and stays on the sidelines ... We now have the meaning of the “riskless portfolio”. It is not a portfolio that earns a riskless rate of return but, rather, a portfolio that is kept out of the market. Risk [arises from] the presence in the market.To

*be*capital, in other words, is to have an inherent risk of diminution of size. The only way to eliminate the risk is

*not*to be capital. And the way to do

*that*is to leave the markets.

None of the luminaries of quantitative finance who devised portfolio insurance understood this point, but their methodologies nevertheless led to it, in the same way that the medieval irrigation methods of the European peasants conformed to the laws of fluid mechanics without the peasants being aware of the discipline or its laws.

Recall that we were able to take the coin toss game as a model for stock price movement because in short time intervals, stocks prices could also be assumed to be binary; they either go “up” or “down” by a “tick”.

Coin toss problems, as some of you may know from the example of a wheat grains on a chess board, begin simply but “branch” rapidly and multiply exponentially. As a result, the outcome of one particular sequence rapidly decreases.

To give you an indication of the magnitude, imagine the number of times you have to throw a coin to get 500 consecutive heads! That is a very crude approximation of the possible “up” and “down” scenarios a portfolio manager tracking S&P 500 index would have.

Now imagine the odd that a group of people doing the same would get the same result at about the same time!

The odd borders on impossible, which is why we do not have crashes every day. But when HFT is the dominant form of trading, the sheer number of “tosses” – trades – brings the near zero probability into life, so to speak. It turns a near zero probability not expected to be encountered in 100s of years – the so-called 100-year flood – into a not-so-insignificant probability with some chance of materialization every now and then, which is why we are beginning to have crashes every now and then.

This dynamic, if you know your probability and statistics, is the basis of simulation: a rapid sampling of random data to explore the size of the improbable areas. That’s how the so-called value-at-risk of trading portfolios is calculated. So, what is taking place every day in equities markets is the simulation of trading conditions that is meant to collapse hundreds of years into a single trading day.

As the purpose of simulation is to find the crash conditions (as well as unlikely sharp market surges), it eventually finds them, only that such findings are not virtual but very real. That’s how markets crash.

Crashes of this sort are purely technical. They are not driven by any underlying fundamental condition because stocks subject to such trading no longer reflect any fundamental underlying conditions; Recall the lament of Financial Times about the loss of the purpose of equities:

*What are the equities market for these days?*

Unsurprisingly, then, the damage from the crash does not reach the general economy. A short while later, the markets recover because “simulation” takes them to the price rise path. The recovery in Oct 87, for example, came the next day. On May 6, the last day, it took 20 minutes. That’s an indication of how much the speed of execution has “collapsed” the time: the old 24 hours is now a mere 20 minutes.

We now have all the pieces of the puzzle. I will return with concluding remarks.