probability of five measurements all being above the median is 55
= 0.03125. Because there is the same chance of five measurements
being below the median, the total probability of being outside the
range is 0.065. We have a 93.5% chance of the median being
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within the range. This idea is generalizable to different confidence
levels and by considering only certain subranges.
As with all heuristics, the Rule of Five is only approximate, but if it
increases your knowledge by any amount, it is worth using.
Rule of Three
This is a method to quickly estimate the probability of something
that has never happened before. Obviously, in some situations you
will have prior knowledge and won't need to rely completely on a
purely mathematical bound, but this is a useful method to at least
establish a base rate. For more discussion refer to Hanley and
Lippman-Hand (1983) and Louis (1981).
A 95% upper bound of the occurrence rate is given by
(A2.3)
where n is the number of observations.
So if we haven't seen an event in 30 observations, a 95% bound on
the chance of the event happening in the next period is 3/30, or
10%. This might seem high but remember this is an upper bound
and we are using no specific information about the particular
situation.
To derive the result, take the chance of the event to be p, which is
what we want to estimate. In each separate time period the chance
of the event not happening is 1− p. So, after n periods the chance of there being no events is
(A2.4)
We want to find p such that this probability is less than 5%. This
gives the bound on the event not happening.
So we solve the equation
(A2.5)
for p.
Taking logs
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(A2.6)
The logarithm of 0.5 is about −3. If p is small, which it has to be if
the event hasn't been observed, ln(1− p) is roughly - p (the first term of a Taylor series). So we get
(A2.7)
which gives the critical value of p.
Clearly, this idea can also be used with other confidence intervals
and, less clearly, can be generalized to the case when one
observation of the event has occurred.
We have made two very important assumptions in our derivation:
p needs to be constant and the successive observations need to be
independent. So this heuristic works well for a question such as,
“Given that I've run through the fireworks factory with a lighted
candle 20 times with no explosion, what is the chance I can do it
again?” But it couldn't be applied to the question, “Given that I've
been alive for 70 years, what is the chance of me surviving another
year?” Here the normal process of aging means the probability of
death increases with age, so p isn't constant, and any health issues
mean the observations aren't independent.
In questions a trader might be interested in, things probably won't
be this clear. For example, think about the case of a company
going bankrupt next year. Is the probability of bankruptcy
independent of what has happened in the past? In some cases, this
is a decent guess. It could be valid in the case of a company that is
very dependent on one product and is most vulnerable to a
disruptive technology. But in the case of a company like Uber,
where the current business model clearly needs to be changed if it
is going to survive, the assumption probably isn't valid. Similarly,
a company with an established business model might have a
relatively constant probability of bankruptcy, whereas a startup
won't.
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APPENDIX 3
Execution
After deciding what trade to do, we still need to do it. Trade
execution is a complex subject. Many open-outcry traders made a
lot of money purely because of their ability to execute well.
Execution was such a valuable skill that being good at it could
mask a lot of other trading weaknesses. It was possible to be a
good floor trader if your only skill was execution. Similarly, on the
electronic exchanges many firms have done very well with
algorithmic trading, order-type arbitrage, and latency advantages.
Even if execution ability isn't going to be a source of alpha (which
it absolutely can be), a large enough trader may need to seriously
consider optimizing execution. But most traders won't be in this
situation. They will obviously want to minimize trading costs but
won't trade enough to need to invest in an algorithmic system. For
these people, using the built-in execution algorithms in the widely
available (and free) brokerage-provided trading systems should be
fine. Twenty years ago, it was reasonably easy to beat most VWAP
systems (volume-weighted average price). Now it isn't.
However, it is still important to understand how to think about
transaction costs generally. All financial decision-making is about
balancing risk and reward. In the case of trade execution, the issue
is how much we should pay to do a trade. If we are too aggressive,
we will kill our returns by paying too much, but if we are too
passive, we won't ever make any trades at all. This is true no
matter what size we are trading or how liquid the product is. The
situation may differ by degree, but the principles will be the same.
The mathematics of balancing expected return and trading cost
can become complex but there are also some broadly applicable
rules of thumb that we can use.
The decision to make a trade is contingent on the instrument
being at some particular price, the “decision price.” This is often
the price in the middle of the bid-ask spread but it could be any
price at all. A transaction cost is the price premium paid above the
decision price for buyers and the discount below the decision price
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for sellers. The total cost has several components. Here they are in
rough order of how evident they are:
Commissions and fees. Commissions are paid to brokers
and fees are charged by exchanges and regulatory bodies, but
they are both fixed and visible. Who they go to doesn't matter
to us as traders, so we will consider them to be the same.
Bid-ask spreads. The bid-ask spread is the difference
between the highest bid price and the lowest ask price. It exists
to compensate the market-makers for providing liquidity.
Because market-makers have their own set of problems, the
bid-ask spread is highly variable, both during the day and in
different market regimes. Further, the displayed bid-ask
spread will often not be the true spread. The visibly quoted
spread will be for a given size. Orders smaller than this can
often be filled inside the spread and larger orders will usually
pay a wider spread.
Price change. Price change is the change in value of the
instrument between our decision to trade and the execution.
This can be positive or negative depending on whether we are
buying or selling and whether the
market is rallying or
dropping. Generally, when we are entering a trade the cost will
be negative because we will be selling something we expect to
drop or buying something we expect to increase in value. But
the cost will be random when we exit, because if we had a price
view, we wouldn't be exiting. This is visible and variable.
Market impact. Market impact is the price change due to
our order. Because the price will also be changing of its own
accord, market impact is invisible. It is also variable and
depends on what other traders are doing at the same time.
Some of the market impact will be temporary and lasts only
until the market has absorbed the new trade, but some is
permanent as the market processes the new information from
the more aggressive price takers.
Opportunity cost. Opportunity cost is the lost profit when
we do not enter a trade. This is generally due to insufficient
liquidity. This cost is invisible and variable.
The trader's execution dilemma is to maximize return given these
costs. If he is too aggressive, the realized costs will be high. If he is
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too passive, there will be significant timing risk and opportunity cost because many trades will be missed.
Commissions, fees, and taxes are inescapable by-products of our
trading strategy. A given strategy can't just be made to trade less
often or to take a different holding period. Then it is a different
strategy. These costs should be considered when the strategy is
being planned and tested but once we are trading, they are what
they are. Changing these would require changing the strategy.
The bid-ask spread is also in some ways a fact of nature. If you
demand to be filled, you will pay the spread. Many traders have a
hard time even accepting this, thinking that they can buy on the
bid and sell on the offer. Of course, it is possible to try to do this,
but if you want a guaranteed fill, you will need to pay the spread.
But the effective spread will probably not be the difference
between the low offer and the high bid. To see this, look at the
order book in Table A3.1.
TABLE A3.1 The Order Book of All Bids and Offers for UVXY
(ProShares Ultra VIX Short-Term Futures ETF) on the Morning of
August 10, 2016
Bid
Pric
Ask
Size
e
Size
20.71 1,200
20.70 1,200
20.69 1,100
20.68 1,800
20.67 600
20.66 1,400
20.65 800
200
20.64
1,000
20.63
900
20.62
500
20.61
700
20.60
1,300
20.59
The best bid is 20.64 and the best offer is 20.65, but the 20.65
offer contains only 800 shares for sale. So, if we wanted to buy up
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to 800 shares, we could say that the spread was $.01 (the
difference between 20.64 and 20.65). But if we wanted to buy
5,000 shares, we would need to pay up to 20.69, with an average
purchase price of 20.6692. Similarly, to sell 5,000 shares would
give an average fill price of 20.608. So, for an order of 5,000
shares the effective bid-ask spread is $0612. The bid-ask spread is
contingent on the size of the order. In some cases, such as where
we have a populated order book, this is self-evident. But it is
always the case.
The example of the UVXY order book that we just gave showed the
case in which the actual, effective spread is wider than the
difference between the best bid and best offer. But there are also
cases in which the actual spread is narrower than the currently
posted spread. This can often happen in option markets. The
market-makers don't want to run the risk of showing tight prices
for large size, but they will probably trade tighter for smaller size.
For example, the indicated spread might be a bid of 9.0 for 100
options and an offer of 9.5 for 100 options but the market-maker
might be prepared to pay 9.2 for 5 and sell 5 at 9.3.
Sometimes you can see the actual spread and sometimes you can't.
Sometimes you have to “fish” by placing a small order and seeing
where you get filled. But remember that the spread is a fact of life.
If the market is showing 9.0 bid for 100 and 100 offered at 9.5, it
is highly unlikely you are going to get filled if you try to sell 1,000
at 9.2.
Generally speaking, the following are true:
The fill-price for an order of infinitesimal size will be the mid-
point of what you see.
You need to pay a spread if you demand to be filled.
The bigger your order the further away from the “zero size”
price you will need to go.
Somewhat related to the bid-ask spread is the idea of market
impact. This is the amount that a given order changes the market
price. It is useful to further split market impact into temporary
and permanent impacts.
First, let's think about temporary market impact. This is just the
fact that by doing a trade we will take out some of the orders in the
book. Most of this is temporary because we would expect the book
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to fill back in again. But it won't completely, and this difference is permanent impact. There are a couple of ways to see that
permanent impact has to exist. The first is that all trades convey
information. If someone is buying, they must think the price is
going up. The price of a security is just the aggregation of all of
this information and every new order will cause some adjustment.
The second reason is to prevent arbitrage. If all of the market
impact was temporary, we could split our order into smaller pieces
and always be guaranteed to pay a smaller spread than by doing a
single large trade. We would just wait for the market to
repopulate. In the UVXY example, the impact when buying 500
shares is only $.01. Why not just buy this many, wait for the book
to fill back in, then do it 10 more times? Even if we ignore fixed
fees and price appreciation, the effect of permanent market impact
means doing this is not automatically advantageous.
If we are to make a sensible decision about whether to execute in
slices or all at once, we will need to have a model of the order book
dynamics. At any random time, what is the spread as a function of
order size?
If we had such a model we could decide on the optimal “slicing”
procedure, dividing the total order into suborders that minimize
impact. But although there are many such models, most are far
too complex to be of any use to a non-quantitative trader (and it
also isn't clear to me that they add enough value to make all the
work worthwhile anyway). If you are trading large enough size for
such a model to be useful (a decent guess for “large enough” would
be an order about 1% of the volume in a given time period), you
should probably use one of the execution algorithms provided by
&n
bsp; most professional-level brokerage firms. If you aren't trading this
big, it probably isn't too important exactly how to split an order.
The class of algorithms that is designed to minimize market
impact is VWAP trading, which aims to match the volume
weighted average price. The VWAP itself is a measure of the
average transaction price for all market activity in a given amount
of time. It is often considered a sort of fair execution price and
many traders have their own execution benchmarked to the
VWAP price.
Achieving VWAP is theoretically easy, but obviously is harder in
real life. By definition, to perfectly match VWAP a trader would
need to participate in every single trade. For example, if a trader's
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order was 10% of the volume in a given time, he would need to
participate by having 10% of every trade. This is impossible for
several reasons:
Most exchange filling algorithms won't just let you get on any
trade you want. A lot operate on a time-priority basis.
You can't know what volume will trade until after the event.
It is impossible to enter enough individual orders to
participate on every trade.
However, the volume profile is stable enough that we can get a
good idea of volume per unit time by looking at historical
numbers. The numbers are so stable that many brokers offer a
guaranteed VWAP execution where they promise your fill price
will be the actual realized VWAP. They will never actually execute
at this price, but their tracking errors are small and bias free.
VWAP strategies are good at lowering market impact because they
are trading proportionally to the amount of volume in the market.
They don't try to push large volume into thin markets, which
would move the price, the definition of market impact. But market
impact is just one trading cost and VWAP strategies are not the
best at managing the most important cost for the active trader:
timing cost.
The timing cost is the amount the market moves in the time
between the trading decision and the end of the execution. This
can either be positive or negative. As an example, consider the VIX
Fed trade. Here my thesis is that the VIX futures rally in the 15
minutes before the Fed announcement and crash immediately
Positional Option Trading (Wiley Trading) Page 24