1.How does the graph of a parabola differ from the graph of one branch of a hyperbola. Identify some real
world applications of parabolas and hyperbolas.
We know that orbits of planets are parabolas. Some applications in real life such as radio
antennas.
Applications of Hyperbolas
. Dulles Airport
Dulles Airport, designed by Eero Saarinen, is in the
shape of a hyperbolic paraboloid. The hyperbolic paraboloid is a three-dimensional
curve that is a hyperbola in one cross-section, and a parabola in another cross section.
2.Search the Cybrary or other Internet sources to find and discuss at least two practical uses of probability
today. Be sure to cite your sources.
I gave you two examples with applications of probability.
Example 1 http://www.financewise.com/public/edit/energy/weather00/wthr00-forecast.htm
Seasonal weather
forecasts and
derivative valuation
Knowing which analysis of forecast data to use can
have an important bearing on hedging statefy. By
Bob Dischel
Every buyer or seller of a weather derivative must
assess its value. In the weather risk market, the
instrument’s value is calculated from estimates of future
outcomes – the relevant data is the probability of future
meteorological events.
This article originally
appeared in the August 2000
Weather Risk supplement to
both Risk magazine and
Energy & Power Risk
management, published by
© Risk Publications
Back to Weather Risk
contents
Click here for a printer
friendly version of this article
To estimate these probabilities the weather is either projected forward based on
past decades of weather measurements – modelled, or it is predicted – forecast.
Weather derivative values calculated with these two alternative probability
estimates can differ because the information used in developing the estimates
only partly overlaps, and because the modeller’s and the forecaster’s disciplines
are different. As a result, hedging decisions viewed from these two perspectives
will differ.
Long-range or seasonal forecasts forecasts for the weather risk market are hard
to find, because most meteorologists are not yet aware of the weather market’s
need for them, and will have to learn new forecast skills if they are to add value in
this specialised forum. The meteorological community, until a few years ago,
considered forecasting difficult beyond a few days. Such forecasts never reached
the scientific standards of demonstrated skill.
Recently, however, the scientific community’s attitude to long-range forecasting
has moved from one of doubt to one of caution. Ever since the oceanic El Niño
events of the past two decades were successfully predicted, anticipation of better
long-range forecasts for some parts of the world seems justified. El Niño is a
periodic warming of the tropical Pacific ocean that affects weather around the
world.
Newly understood ocean–weather connections appear to be valuable for certain
kinds of forecasts, allowing skilled scientists who understand the uncertainty
inherent in such predictions to extract value for the public. Unfortunately, we too
often see these forecasts consumed whole by those outside the scientific
community, with the warning labels describing their experimental nature not
respected. The media hype surrounding El Niño and La Niña are examples of this.
The cautions are many and significant. Only during certain extreme oceanic
events can US seasonal weather begin to be confidently forecast – during
moderate to strong El Niño and La Niña events, for example – and only up to a
few months in advance. And for El Niño events, the forecast is different from, but
not opposite to, La Niña events – and forecast reliability varies across the country.
In general, during these extreme ocean events, statistical correlations show that
summer and winter forecasts for the US weather anomalies are more reliable than
either spring or autumn forecasts. Forecasting an ocean event and then
forecasting its effect on the weather in the US is a leap across a precarious divide
that is best left to the trained scientific long jumper. Long-range weather
forecasts are pushing at the outer limits of forecasting skill.
The picture for Europe may be even less clear than for the US because the
correlations between the El Niño-Southern Oscillation extremes, (Enso) – a
cyclical Pacific Ocean–atmospheric event – and European weather are even
weaker than they are for the US. Perhaps there are Enso effects in northern
Europe, but the current thinking is that the changes in atmospheric pressure
differences over the North Atlantic Ocean more strongly influence the seasons in
most of Europe. Confident and demonstrably reliable long-range forecasts for
Europe are yet to emerge.
The weather will always be unpredictable, so a useful forecast captures this
uncertainty in estimating the probabilities of multiple possible outcomes.
If a forecast states that it will rain, it will be either wholly right or wholly wrong.
By contrast, a forecast of an 80% chance of rain also admits a 20% chance of no
rain; the two forecast outcomes cannot both be correct, and yet the forecast can
be useful. This simple, two-way forecast begins to tell us about the uncertainty of
weather by quantifying weather probabilities.
The weather risk market is interested in forecasts that stretch from a few days to
a few months into the future. There are some critically important differences in
how forecasts are made, depending on how far into the future the forecast
meteorologist looks.
Short-range forecasts, such as those we routinely get from our broadcast and
print media, look out a few days at most, and depend heavily on the current
weather over the larger region that includes the site for which the forecast is
made. It describes the expected imminent weather events, detailing the range of
expected temperatures, rain and snowfall, sun and clouds, wind conditions and
more. Short-range forecasts have limited application in weather derivative
valuation, providing at most a few days of information whose certainty, in
general, declines with reach into the future.
By contrast, the long-range forecast incorporates more of a global view to look
out a month or more into the future. The long-range forecaster looks at factors
such as the current average state of the atmosphere and the ocean at distances
often thousands of miles away from the specific location of interest. Long-range
forecasts estimate only the average weather, not specific weather events. No-one
has the skill to forecast a specific weather event beyond a few days; therefore, in
seasonal forecasts, language such as “warmer than normal” and “wetter than
normal” is common.
The useful forecast for the weather risk market must go beyond the simple twoway forecast described above. It must quantify the probability of each possible
weather outcome throughout the full range of possibilities – from the highest to
lowest possible temperatures, from drought to flood, from calm to gale-force
wind. Additionally, we are looking for skilful descriptions of seasonal weather
probabilities focused on small geographical regions – sites that are as specific as
are the weather contracts. A forecast of the seasonal weather probabilities within
a very small region is called a ‘site-specific probability forecast’ (SPF).
The Climate Prediction Center (CPC) of the National Center for Environmental
Prediction of the US government provides experimental forecasts that meet many
of the requirements for an SPF. Monthly, the CPC issues graphics with forecasts of
the probabilities of weather deviations from the normal, for individual months and
for three-month periods. These outlooks, made in various formats, are issued
from two weeks to 13 months in advance. Currently, these experimental outlooks
can be found at the CPC website, at
http://www.cpc.ncep.noaa.gov/pacdir/NFORdir/HOME3.html. Figures 1 and 2 are
examples of one form of a probability distribution forecast.
In Figure 1, the solid black lines are the centres of the historical distributions of
temperature for recent July, August and September 2000 (JAS) periods. The
assumed distribution is the Gaussian distribution that best fits the historical data.
Shifts of the centre of the historical distribution to higher or lower-than-normal
temperatures is what is forecast, and the amount of the shift is colour-shaded and
keyed to the colour scale at the bottom of the figure. Notably, all shifts on this
particular temperature forecast map are positive – all shifts are to higher-thannormal temperatures. Where no confident forecast can be made, the area is left
unshaded, as it will be if the forecast is for a normal season. The correct
interpretation of an unshaded area is not possible with only this figure, and
requires a deeper look at the forecast process.
Figure 2 is the normal and forecast precipitation analogous to the temperature in
Figure 1. In Figure 2, there are unshaded areas, and both positively and
negatively shaded areas. The shaded areas of the east coast and the South-west
are where a wetter-than-normal season is predicted, and the shaded area in the
Midwest is where a drier-than-normal season is predicted.
Figures 3 and 4 are constructed from the same information used to construct
Figures 1 and 2. CPC calls figures 3 and 4 – ‘Probability of Exceedance Forecasts’
(PoE). The historical measurements for the site are shown by the stepped line to
which a normal curve has been fit. The three other lines are the forecast
distribution and its error envelope – the estimate of the amount of possible error
associated with the forecast curve. This experimental product gives the probability
that temperature (Figure 3), precipitation (Figure 4) and degree-days (not shown)
will be exceeded in a selected region for the selected season.
For example, in Figure 4, the probability estimated from historical measurements
for precipitation to exceed five inches in south-west Arizona in July, August and
September is about 15%, and to exceed six inches is about 5%. The
corresponding forecast probabilities for the same levels of precipitation are 20%
and 10%. A wetter-than-normal July, August and September is forecast.
CPC makes PoEs for 102 climate regions in the mainland US. The relatively large
geographical extent of these regions may limit their value to local weather
forecasters and to market players concerned with specific locations – site-specific
features may require site-specific resolutions. To address this concern, CPC
provides regression equations to bring the regional forecast to specific locations.
An alternative is to seek out privately made SPFs from consulting meteorologists.
Cautions are displayed at the CPC website about the experimental nature of these
forecasts and their inherent uncertainty. The CPC writes: “All of the forecasts...
have large uncertainties... “In some cases, uncertainties are gigantic, while in
other cases they are only moderately large… It is the responsibility of the user to
examine the product and its accuracy to their own satisfaction.”
Anyone using these experimental products is obliged to read the detailed
explanations at the website. Even those who are not meteorologists, reading
these cautions and recognising the experimental character of these forecasts,
would think that involving a meteorologist in interpreting these forecast products
would be prudent.
To show the application of weather forecast probabilities to weather risk
management, we revisit the faux Hot Air Gas Company, whose weather risk
management programme we described previously (see EPRM, March 1999). In
that article, using weather probabilities based on historical data alone, we
reviewed three of the company’s weather hedging choices – not hedging its
weather exposure, selling a heating degree-day (HDD) swap or buying an HDD
put. There, we cast the company’s weather-contingent revenue and the derivative
cashflows in probability terms, to compare better the impact of the alternative
hedges on weather exposure and enable reasoned decisions.
Here the situation is somewhat different – instead of reviewing alternative
hedges, we look at alternative estimates of weather probabilities.
In Figure 5, we show the weather probabilities for the coming winter season, both
as projections from historical weather and as a forecast. The chance of winter
being warmer than an HDD level, read on the horizontal axis, is indicated by the
cumulative frequency curves. For example, from the historical projections, we
estimate that a winter being warmer than average – about 5,175 HDDs – has a
53% probability of occurring. From the forecast curve, we estimate that a winter
being warmer than average has a 62% probability of occurring. This format is
called a ‘probability of warmer than normal’ distribution, or PoW.
Uncertainty
Figure 5 also shows the error envelope of the forecast, giving a measure of
forecast uncertainty. The forecast is for a shift to about 75 HDDs warmer than
history, and the historical distribution falls within the error envelope. Even though
the forecast view tilts to a warmer-than-normal season, the uncertainty is great
enough to admit that the normal winter distribution may also be a good forecast.
In Figure 6, we show the company’s weather contingent revenue and a measure
of its weather exposure. The contingent revenue is derived from the records of
how past winter intensities caused the company’s revenue to vary. Most winters
are near average and generate good revenue, but revenue declines if it is either
much warmer than normal or much colder than normal.
It is usual in value-at-risk (Var) analyses to assess financial exposures by
multiplying a contingent cashflow by the probabilities of the contingent outcomes.
The typical way of placing a value on a contingent cashflow is to calculate the
contingent payout, multiply it by the probabilities, and integrate the probabilityweighted cashflows across all possible outcomes. To estimate its weather
exposure, the Hot Air Gas Company multiplied the contingent revenue by the
weather probabilities. In the earlier article, the company then compared this
unhedged weather exposure with the weather exposure after hedging with each
alternative.
Again, our illustration has a different focus – alternative views of unhedged
weather exposures instead of alternative hedges.
In Figure 6, we also show these alternative views of weather exposure. To
produce these two views, we multiplied the contingent revenue by the two
estimates of weather probabilities.
This would be helpful in making a risk management decision, although we do not
take this analysis that far. Here, we are content to show how views of the future
affect the assessment of exposure and value. That is, the company’s weather
exposure – and the value of derivative cashflows – appears differently, depending
on the selection of the weather probability distribution.
If the company were to compare the impact of hedging with the swap or with the
put – as in the earlier article – with either view of the future, they would see little
difference in the hedging impact.
They would, however, come to a very different estimate of derivative price.
Rather than favouring one hedge over the other, the choice of view of the future
is more likely to affect the decision to hedge or not to hedge.
It is the company’s dilemma, given imperfect information, to choose how to weigh
these two uncertain views of the future in making risk management decisions.
Bob Dischel is a consulting meteorologist based in New York
Example 2. http://www.gold-eagle.com/editorials_03/pmtrader031303.html
Tomorrow's Price Probability:
A Serious Tool for the Trader
First let me make the usual disclaimers. Everyone must do his or her own due
diligence. Nothing in this note should be considered investment advice. Although
the stocks used in this article are for illustrative purposes only, in the interests of
full disclosure; I own long positions in GFI and AEM and periodically may take
long or short positions in INTC and the DOW. What I will present here is a bit of a
departure from some of my previous work. I have decided to make my
proprietary trading tool available to the general public. This essay is the first time
that this body of work has been formally presented.
First a bit of an introduction: The method used for the results discussed herein in
a very real sense embodies chart techniques varying from EW analysis to
candlesticks. It also contains many oscillator techniques. The formal description
of the technique is a trade secret owned by the author. It is up to you the reader
to determine whether this technique represents a tool that will be useful to you. I
might mention in passing that this author has an extensive scientific background.
The format of the rest of the article will be to present several charts and to
discuss several salient features of each chart. In order for this tool to be useful to
the trader, he must familiarize himself with the subtleties. The charts represent
the probability of a price being achieved tomorrow and are thus generated
before the trading day begins!
The first chart shows Agnico-Eagle (AEM). This chart was generated on
Thursday, March 6,2003 for the price probability of AEM for the next day March
7, 2003. After the trading day ended, the annotations shown above (green
lines, yellow and blue circles) were added. In particular, the green lines
represent the days high and low. The blue circle represents the open and the
yellow circle represents the close. In practice, the left axis represents the
probability of a price being obtained during the day. The related peaks and
valleys also indicate areas of support/resistance. Note that the price opened on
the high probability peak and then moved to the left (down in price). One
interpretation is that the gap to the right leading to a secondary price peak of
around 14.1 was larger than the gap to the left leading to a support level peak at
13.2.
One must develop an art to reading these charts. Although AEM's chart shows
that the high probability event happened, this is not always the case just as in
reality. i.e. Low probability events happen. To say otherwise would be to lose
credibility. Rather, what I present is a representation of the probability distribution
for tomorrow's price. This latter conclusion has been arrived at through extensive
back-testing and forward charting. If the high probability event always happened,
then one would have to almost certainly accept the deterministic philosophical
view.
The next chart shows Gold Fields (GFI) for the same time period as AEM given
above. It is not unusual for stocks in similar sectors to exhibit similar chart
performance, whether they be high probability hits or misses. Company specific
news is of course an exception. Notice in the GFI chart that although the low of
the day (left green line) moved into the lower probability area, the close (yellow
circle) moved back to support. In viewing these charts, generally the tick action of
the trading day follows the peaks and valleys of these charts. The slopes of the
peaks often equate to time spent traversing them. As you watch these charts
more and more closely, it really is amazing how precisely they often model the
intricacies of the intraday action.
Several more gold stocks performed similarly to the ones noted above. The next
chart shows the DOW for the same trading day. Note that the left green line
shows that the DOW bounced strongly off of the "deep valley".
Support/resistance at peaks and valleys are a recurring theme in these plots.
Once again, much of the high probability range was achieved during the day. It is
important to note that this is not always the case for the reasons mentioned in the
introduction (this is a probability distribution and not a crystal ball).
In order to highlight a low probability event and one other point, consider the
chart of Intel (INTC) for the same day. INTC on the Thursday evening referenced
earlier had some negative news. Thus, it was not surprising that a low probability
event occurred. This is evidenced by the location of the blue circle representing
the gap down open. What is encouraging is that once a point of reference in the
probability distribution was obtained, the chart gave good information for
resistance/support in the days trading range. That is, it was clear to the observer
that a low probability event was taking place and that due to the distance from
the high probability peaks and the various areas of resistance in between, it was
fairly clear that a 16.75 price was not to be obtained during the day. The
emphasis here is that the open gives one a needed point of reference on the
probability distribution. One may choose to simply ignore trading in a stock such
as INTC on a day like that shown where a low probability event is unfolding.
In closing, I might mention another point of interest. How often have we watched
the Stochastics or other indicator turn on a stock and wish that we had some way
to quantify the size of the move? Well these charts may provide just such an
edge.