the growth rate is
(9.39)
This is maximized for f = 1 and drops as the investment fraction is
reduced from this (Figure 9.6).
Measures of risk and drawdown also decrease as the investment
fraction drops. The ratio of growth rate to drawdown is shown in
Figure 9.7 for the case where μ = 0.05 and σ = 0.3.
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FIGURE 9.6 The dependence of growth rate on the fractional Kelly ratio.
FIGURE 9.7 The growth rate to drawdown ratio as a function of the scaling factor.
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FIGURE 9.8 The probability of reaching 200% before being stopped out at 0% when trading at fractions of the Kelly ratio.
So, if drawdown is the primary risk factor, you should be very
cautious indeed.
Another way to quantify the effect of changing the investment
ratio is by looking at the chance of hitting a lower barrier (possibly
the stop level) before reaching a target. If the stop percentage level
is A, and the target is B, the probability is given by
(9.40)
(Interestingly, neither the return nor volatility of the return
stream is relevant. A “good” trade means the level will be hit
earlier but doesn't change the relative probability.) The
dependence of the probability on the scaling fraction, f, is shown
in Figure 9.8 for the case A = 50%, B = 200%.
A very conservative trading size will raise the probability of a good
outcome because the volatility associated with Kelly betting is
dampened to the point that hitting a stop is unlikely.
However, a very small scaling factor also means that the expected
time to hit the goal increases. The expected exit time is given by
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(9.41)
Expected exit time as a function of f, for the case A = 50% and B =
200%, and μ = 0.05 and σ = 0.3. is shown in Figure 9.9.
FIGURE 9.9 The expected time to reach 200% before being stopped out at 50% when trading at fractions of the Kelly ratio for
the case where μ = 0.05 and σ = 0.3.
(Note that the average exit time is monotonically decreasing in f,
even decreasing when expected growth rate is negative. At this
point, you will probably get stopped out, but you also might just
get lucky.)
In terms of risk mitigation, it is easy to make a case for trading at
only a fraction of Kelly. However, doing this will also affect the
good points of the method. Further, due to sampling issues when
we estimate the parameters, we may be betting in a negative
expectation game. We can mitigate these issues by combining the
Kelly concept with a stop.
The Effect of Stops
To many traders, the use of stops is seen as an essential part of
risk control and money management. And usually they take the
utility of stops to be self-evident. “How can you go broke if you
limit your losses?” “Cut your losses and let your profits run.”
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“Losers add to losers.” But the effect of stop use is quite complex.
In this section I will explain where stops help, where they don't,
their effect on profitability, and, if you are using them, where to
set them.
First, we examine what the use of stops does to the distribution of
our trading results. A hypothetical trade result example is shown
in Figure 9.10. The trade results are normally distributed with a mean return of 10% and a standard deviation of 15%.
FIGURE 9.10 The return distribution of our trading strategy.
However, we can also see that a significant number of trades were
losers (here nearly 11% of trades will lose more than 15%). A
natural thought would be to introduce a stop loss to somehow “cut
off” the left-hand side of the distribution.
It is tempting to think that use of a stop simply truncates the
downside of the distribution by capping losses at a certain level.
Redrawing the distribution to reflect this intuition situation gives
us Figure 9.11, where a stop has been introduced at the 15% down
level, so all loses are capped at 15%.
But a little thought will be enough to see that this distribution isn't
possible. The trades that get stopped out don't just disappear.
Their results still have to be accounted for (mathematically, the
integral of the probability density function must still be 1). So we
next surmise that these trades cluster around the level of the stop.
But this still misses an important point.
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FIGURE 9.11 The hoped-for distribution when a stop has been added.
Many trades that at the end of the period will be small winners
and will have been stopped out beforehand. Note that the large
winners will still remain because these largely consist of trades
that started as winners and never looked back, but the presence of
a stop will drastically reduce the number of small winners. This is
the hidden cost of using a stop.
To see the exact effect, I simulated 1,000 GBM paths that
represented checking the performance of the trade once a day for
a year. Again, the expected final return was 10%, the standard
deviation was 15%, and a stop was placed to cap losses at 15%.
Results are shown in Figure 9.12.
There are several things to note here. First, the average return is
negatively affected by using stops. The “unstopped” investment
has a mean return of 10% (as we planned for the simulation), but
when we add a stop this drops to 9.6%, and the median is now
8.6%. This is the mathematical inevitability of assuming that the
(unstopped) results follow the normal distribution. By adding a
stop, we eliminate the big losers (in this case of greater than 15%)
but we also eliminate trades that would have later recovered above
the stop level. And because the normal distribution has more
density around the mean than it does in the wings, there are more
of these marginal trades than there are big losers.
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This analysis has been for a fixed stop, set at a given distance from
our entry price. The other type of stop is the trailing stop, which is
set so it stays a certain distance from the highest amount the
investment has made. A trailing stop is a very comforting strategy.
It protects us from the painful experience of seeing our winners
turn into losers.
FIGURE 9.12 The true distribution when a stop has been added.
FIGURE 9.13 The return distribution of the simulated trade when using a trailing stop.
However, the trailing stop costs even more than the fixed stop.
This is because some positions that would be large winners don't
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get the chance to fully realize their potential. When using fixed stops, some investments benefit by getting away from the stop and
having the chance to develop. Trailing stops are always in play.
Adapting our previous simulation confirms this. The trailing stop
is always 15% below the high of the trade. The addition of a
trailing stop lowers the mean return of the trade to only 9.4%, and
the median is 8.0%. This is lower than the unstopped return and
also lower than when employing a fixed stop. Figure 9.13 shows
the dis
tribution when we employ a trailing stop. The distribution
in this case is now totally different from that of the unstopped
investment.
Stops don't just stop losses. They drastically change the shape of
the return distribution and can lower the average return. Adding
stops won't transform a losing strategy into a winning strategy.
The only reason that we would add a stop is that we prefer the
shape of the stopped distribution, that is, we prefer to trade lots of
small losses and fewer small wins for some large losses.
Although this is true, the real world is far more complex. Returns
are not normally distributed and the results will change over time.
Also, there is no reason a trader need only be interested in
maximizing returns. Safety and risk control are important for both
financial and psychological reasons.
Obviously, many traders like using stops. Indeed, some insist that
stops are absolutely essential and that their appropriate use is a
good predictor of overall, long-term success. Given that most tests
show that stops cost money, what is it that these traders are
thinking?
Let's take a quick look at some of the common arguments given for
using stops:
Stops limit losses. For any given trade this is trivially true.
Ignoring slippage and trading costs, we can't lose more than a
certain predefined amount on any investment when we use a
stop. But, as we have seen in our simple simulation, in
aggregate this is an illusion. Using stops will lower returns in
the long term so this reason at best needs more justification
and is probably just incorrect.
Stops are a form of discipline. It is true that if you always
use stops, then you have displayed discipline. But discipline
needs to be about applying a sensible methodology
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consistently, not just doing something consistently. If your idea has negative return, executing it diligently will result in
greater losses than doing it haphazardly.
Using stops means all trades risk the same amount.
Some traders have the idea that it is important to risk a
certain set amount on each investment (1% or 2% is often the
amount given). This is generally false. Different trades, even
within the same strategy, will have different projected risks
and returns. It is best to take this into account when trade
sizing. Not doing so will lead to lower total returns and
unnecessary risk. Further, trade sizing and risk control are a
different issue from setting a loss for each single position.
A stop is a predefined exit. The trade will be stopped out
when we don't want to be in the position anymore. It should
be obvious that exiting a position when we don't want to be in
it is an excellent idea. If we use stops to formalize this, then
they are perfectly sensible. But we need to think more about
what this means. If we are exiting a position purely because
the price has moved by a given amount, then we are assuming
the trade has positive autocorrelation: the move that has
already happened is predictive of a future move. Price-based
stops are a trend-following system. So they make a good deal
of sense if we are explicitly betting on momentum. Conversely,
they don't make a lot of sense if we are trading something that
we expect to revert. In this case we will be exiting trades at
points where we see potential for future profit. We should
never exit a position when a trader with the same strategy and
no position would want to enter.
Rephrased, the reason we should get out of a position that
has moved against us is if, and only if, we expect the move to
continue. The loss has already been incurred; we need to
think about our current risk, not the sunk cost of the incurred
loss. And if we are only basing the stop on the price action,
we are saying price direction alone determines future price
direction, that is, we are trend following.
A position should be exited when we are wrong. Sometimes
this will coincide with losing money. In this case a stop is
harmless. But sometimes losing money corresponds to
situations for which we have more edge. Here a stop is
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actively damaging and contrary to the idea behind the
strategy.
Stop Placement
Once we have decided we want to use a stop, we still need to
choose where to put it. There are two aspects to this: game theory
and statistics.
The role of game theory is often emphasized although it is the less
important consideration. The idea is to avoid placing stops in
“obvious” price levels such as around the highs or lows of the
previous session, key technical levels, or round numbers. Other
traders could take advantage of this predictability. For example,
imagine the market was quoted at 98 bid and 99 offered and a
trader knew there were likely to be buy stops set at 100. He could
get long at 99, print a trade at 100, and set off the stops. This
would drive the price higher, giving him an instant profit.
This sort of tactic was common on trading floors, but the higher
liquidity and different sociological structure of electronic markets
means the idea is less relevant than it once was. It probably
doesn't hurt to avoid placing orders at these levels, but it almost
certainly doesn't matter much.
The more important consideration is choosing a stop price that
correctly balances risk and costs. If a stop is too far away, it isn't
doing much to reduce risk and if it is too close, it will get hit too
often, increasing transaction costs and not giving trades enough of
a chance to become winners.
There have been many attempts to establish a theoretical basis for
stop placement. Unfortunately, the results are highly dependent
on both the assumed price process and the trader's utility
function. For example, a risk-neutral trader who is long an
instrument with a positive drift will never use a stop.
We can never be completely sure about either the price process or
parameters and our utility function is likely to be a function of
many things other than just wealth. So, these theories aren't
helpful.
There have also been a number of empirical tests of the efficacy of
stops. The problem is that these are tests of the interplay between
a stop and a particular strategy. They can possibly help as guides,
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but they will never apply particularly well to a different strategy or set of instruments. Some such studies are those by Lei and Li
(2009), Clare et al. (2013), and Han et al. (2016).
Instead we need to use a completely empirical method for each
strategy we are considering. Here, the only theoretical assumption
is the usual one that the past return distribution is predictive of
the future. It is never a great idea to base a trading decision only
on data analysis but at this point I don't think that there is a better
way to choose stop parameters than just testing various ideas.
Incorporating Stops
into the Kelly Criterion
Conceptually, we split the trading account into two parts: a risky
part that we trade according to the Kelly criterion and the riskless
part that we hold in cash. If we never transfer profits from the
risky part to the safe part, we are using a fixed stop. For example,
we split our $100 account into $80 cash and an active trading
subaccount of $20. We will never be able to lose more than $20
(in theory we can't even lose all of that because we will be trading
proportionally) but we can suffer larger peak-to-trough
drawdowns if we first make money and then lose those profits as
well as the original $20. This idea has been studied by Grossman
and Zhou (1993), Cvitanic and Karatzas (1995), and Browne (2000b).
Consider trading our $100 in two situations with identical growth
rates. Our trading strategy has a mean return of 5% and a volatility
of 30%. In one case we trade at one-quarter of the Kelly ratio,
which according to equation 9.39 gives a growth rate of 0.0061.
So, after one year the expected account value is $100.61, and the
volatility is 0.75, or $7,546. The complete distribution of results is
shown in Figure 9.14.
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FIGURE 9.14 The distribution of the final account after 10,000
simulations of a GBM where μ = 0.05 and σ = 0.3 when using a quarter of the Kelly ratio.
Full Kelly gives a growth rate of 0.139. To achieve the same
expected account value using a smaller subaccount, Wr, which
trades at full Kelly, we solve the equation
(9.42)
which gives Wr = $43.7. Initial volatility is 0.3 or $13. So, the
subaccount method gives the same growth rate but with a higher
dollar volatility. This doesn't seem promising. But things are
better when we consider the entire distribution. We have given
ourselves some downside protection while still retaining the
possibility of the extreme growth that Kelly can give. This is shown
in Figure 9.15.
Another method is to adjust the amount in our safe subaccount to
stay a constant percentage below the peak. We will be using a
trailing stop on the account value. This time we use a trailing stop
level of 43%. The distribution of results from using this method
Positional Option Trading (Wiley Trading) Page 20