點評  一個好的交易系統會有很明確的買賣信號！止損以自己所能承受的風險為限，引出資金管理的問題！不同的交易方法和投資理念對止損的設置各有不同，但終究一點，止損是貫穿交易的始終的必不可少的程序，當一個頭寸贏利了解時，止損就變成了止贏，其實質是一樣的。止損的目的是使虧損限於小額，讓利潤充分增長，所以止損也必須是不斷移動的，這樣才能保住既得得利潤。一個適合交易的點位首先必須是一個便於止損的點位！好的交易系統要和投資理念和交易技巧融合一體形成一種交易的信念，用信念交易，使各種紀律和交易守則成為一種交易的習慣！ 

我的止損理念

Pyramiding and Martingale

In the case of a random process, such as coin tosses, streaks of heads or tails do occur, since it would be quite improbable to have a regular alternation of heads and tails. There is, however, no way to exploit this phenomenon, which is, itself random. In non-random processes, such as secular trends in stock prices, pyramiding and other trend-trading techniques may be effective.

Pyramiding is a method for increasing a position, as it becomes profitable. While this technique might be useful as a way for a trader to pyramid up to his optimal position, pyramiding on top of an already-optimal position is to invite the disasters of over-trading. In general, such micro-tinkering with executions is far less important than sticking to the system. To the extent that tinkering allows a window for further interpreting trading signals, it can invite hunch trading and weaken the fabric that supports sticking to the system.

The Martingale system is a method for doubling-up on losing bets. In case the doubled bet loses, the method re-doubles and so on. This method is like trying to take nickels from in front of a steam roller. Eventually, one losing streak flattens the account.

Optimizing - Using Simulation

Once we select a betting system, say the fixed-fraction betting system, we can then optimize the system by finding the PARAMETERS that yield the best EXPECTED VALUE. In the coin toss case, our only parameter is the fixed-fraction.  Again, we can get our answers by simulation. See figures 3 and 4.

Note: The coin-toss example intends to illuminate some of the elements of risk, and their inter-relationships. It specifically applies to a coin that pays 2:1 with a 50% chance of either heads or tails, in which an equal number of heads and tails appears. It does not consider the case in which the numbers of heads and tails are unequal or in which the heads and tails bunch up to create winning and losing streaks. It does not suggest any particular risk parameters for trading the markets.

Figure 3:  Simulation of equity from a fixed-fraction betting system.

Figure 4:  Expected value (ending equity) from ten tosses, versus bet fraction,

for a constant bet fraction system,  for a 2:1 payoff game,

from the first and last columns of figure 3.

Optimizing - Using Calculus

Since our coin flip game is relatively simple, we can also find the optimal bet fraction using calculus. Since we know that the best system becomes apparent after only one head-tail cycle, we can simplify the problem to solving for just one of the head-tail pairs.

The stake after one pair of flips:

S = (1 + b*P) * (1 - b) * S₀

_{S - the stake after one pair of flips}

_{b - the bet fraction}

_{P - the payoff from winning - 2:1}

S₀ - the stake before the pair of flips

(1 + b*P) - the effect of the winning flip

(1 - b) - the effect of the losing flip

So the effective return, R, of one pair of flips is:

R = S / S₀

R = (1 + bP) * (1 - b)

R = 1 - b + bP - b²P 

R = 1 + b(P-1) - b²P

Note how for small values of b, R increases with b(P-1) and how for large values of b, R decreases with b²P. These are the mathematical formulations of the timid and bold trader rules. 

We can plot R versus b to get a graph that looks similar to the one we get by simulation, above, and just pick out the maximum point by inspection. We can also notice that at the maximum, the slope is zero, so we can also solve for the maximum by taking the slope and setting it equal to zero.

Slope = dR/db = (P-1) - 2bP = 0,  therefore:

b = (P-1)/2P , and, for P = 2:1,

b = (2 - 1)/(2 * 2) = .25

So the optimal bet, as before, is 25% of equity.

Optimizing - Using The Kelly Formula

J. L. Kelly's seminal paper, A New Interpretation of Information Rate, 1956, examines ways to send data over telephone lines. One part of his work, The Kelly Formula, also applies to trading, to optimize bet size.

Figure 5:  The Kelly Formula

Note that the values of W and R are long-term average values,

so as time goes by, K might change a little.

Figure 6:  Optimal bet fraction increases linearly with luck, asymptotically to payoff.

The Expected Value of the Process, at the Optimal Bet Fraction

Figure 7:  The optimal expected value increases with payoff and luck.

Finding the Optimal Bet Fraction from the Bet Size and Payoff

Figure 8:  For high payoff, optimal bet fraction approaches luck.

Non-Balanced Distributions and High Payoffs

So far, we view risk management from the assumption that, over the long run, heads and tails for a 50-50 coin will even out. Occasionally, however, a winning streak does occur. If the payoff is higher than 2:1 for a balanced coin, the expected value, allowing for winning streaks, reaches a maximum for a bet-it-all strategy.

For example, for a 3:1 payoff, each toss yields an expected value of payoff-times-probability or 3/2. Therefore, the expected value for ten tosses is $1,000 x (1.5)¹⁰ or about $57,665. This surpasses, by far, the expected value of about $4,200 from optimizing a 3:1 coin to about a 35% bet fraction, with the assumption of an equal distribution of heads and tails.

Almost Certain Death Strategies

Bet-it-all strategies are, by nature, almost-certain-death strategies. Since the chance of survival, for a 50-50 coin equals (.5)^N where N is the number of tosses, after ten tosses, the chance of survival is  (.5)¹⁰, or about one chance in one thousand. Since most traders do not wish to go broke, they are unwilling to adopt such a strategy. Still, the expected value of the process is very attractive, so we would expect to find the system in use in cases where death carries no particular penalty other than loss of assets.

For example, a general, managing dispensable soldiers, might seek to optimize his overall strategy by sending them all over the hill with instructions to charge forward fully, disregarding personal safety. While the general might expect to lose many of his soldiers by this tactic, the probabilities indicate that one or two of them might be able to reach the target and so maximize the overall expected value of the mission.

Likewise, a portfolio manager might divide his equity into various sub-accounts. He might then risk 100% of each sub account, thinking that while he might lose many of them, a few would win enough so the overall expected value would maximize. This, the principle of DIVERSIFICATION, works in cases where the individual payoffs are high.

Diversification

Diversification is a strategy to distribute investments among different securities in order to limit losses in the event of a fall in a particular security. The strategy relies on the average security having a profitable expected value, or luck-payoff product. Diversification also offers some psychological benefits to single-instrument trading since some of the short-term variation in one instrument may cancel out that from another instrument and result in an overall smoothing of short-term portfolio volatility.

The Uncle Point

From the standpoint of a diversified portfolio, the individual component instruments subsume into the overall performance. The performance of the fund, then becomes the focus of attention, for the risk manager and for the customers of the fund. The fund performance, then becomes subject to the same kinds of feelings, attitudes and management approaches that investors apply to individual stocks.

In particular, one of the most important, and perhaps under-acknowledged dimensions of fund management is the UNCLE POINT or the amount of draw down that provokes a loss of confidence in either the investors or the fund management. If either the investors or the managers become demoralized and withdraw from the enterprise, then the fund dies. Since the circumstances surrounding the Uncle Point are generally disheartening, it seems to receive, unfortunately, little attention in the literature.

In particular, at the initial point of sale of the fund, the Uncle Point typically receives little mention, aside from the requisite and rather obscure notice in associated regulatory documentation. This is unfortunate, since a mismatch in the understanding of the Uncle Point between the investors and the management can lead to one or the other giving up, just when the other most needs reassurance and reinforcement of commitment.

In times of stress, investors and managers do not access obscure legal agreements, they access their primal gut feelings. This is particularly important in high-performance, high-volatility trading where draw downs are a frequent aspect of the enterprise.

Without conscious agreement on an Uncle Point, risk managers typically must assume, by default to safety, that the Uncle Point is rather close and so they seek ways to keep the volatility low. As we have seen above, safe, low volatility systems rarely provide the highest returns. Still, the pressures and tensions from the default expectations of low-volatility performance create a demand for measurements to detect and penalize volatility.

Measuring Portfolio Volatility

Sharpe, VaR, Lake Ratio and Stress Testing

From the standpoint of the diversified portfolio, the individual components merge and become part of the overall performance. Portfolio managers rely on measurement systems to determine the performance of the aggregate fund, such as the Sharpe Ratio, VaR, Lake Ratio and Stress Testing.

William Sharpe, in 1966, creates his "reward-to-variability ratio." Over time it comes to be known as the "Sharpe Ratio." The Sharpe Ratio, S, provides a way to compare instruments with different performances and different volatilities, by adjusting the performances for volatilities.

S = mean(d)/standard_deviation(d)  ... the Sharpe Ratio, where

d = Rf - Rb   ... the differential return, and where

Rf - return from the fund

Rb - return from a benchmark

Various variations of the Sharpe Ratio appear over time. One variation leaves out the benchmark term, or sets it to zero. Another, basically the square of the Sharpe Ratio, includes the variance of the returns, rather than the standard deviation. One of the considerations about using the Sharpe ratio is that it does not distinguish between up-side and down-side volatility, so high-leverage / high-performance systems that seek high upside-volatility do not appear favorably.