Forwards and forward prices
In a forward contract, one party agrees to deliver a specified amount of a specified
commodity – the underlying asset – to the other at a specified date in the future (the
maturity date of the contract) at a specified price (the forward price). The commodity
may be a commodity in the narrow sense, e.g. gold or wheat, or a financial asset, e.g.
foreign exchange or shares. The price of the underlying asset for immediate (rather
than future) delivery is called the cash or spot price.
The party obliged to deliver the commodity is said to have a short position and
the party obliged to take delivery of the commodity and pay the forward price for it
is said to have a long position.
A party with no obligation offsetting the forward contract is said to have an open
position. A party with an open position is sometimes called a speculator. A party
with an obligation offsetting the forward contract is said to have a covered position.
A party with a closed position is sometimes called a hedger.
The market sets forward prices so there are no cash flows – no money changes
hands – until maturity. The payoff at maturity is the difference between forward
price, which is set contractually in the market at initiation, and the future cash
price, which is learned at maturity. Thus the long position gets STñFt,T and the short
gets Ft,TñST, where Tñt is the maturity, in years, of the forward contract (for
example, Tó1/12 for a one-month forward), ST is the price of the underlying asset
on the maturity date, and Ft,T is the forward price agreed at time t for delivery at time
T. Figure 1.4 illustrates with a dollar forward against sterling, initiated at a forward
outright rate (see below) of USD1.60. Note that the payoff is linearly related to the
terminal value ST of the underlying exchange rate, that is, it is a constant multiple,
in this case unity, of ST.
1.45 1.50 1.55 1.60
ST
0.05
0.10
0.05
0.10
payoff
forward rate
Figure 1.4 Payoff on a long forward.
No-arbitrage conditions for forward prices
One condition for markets to be termed efficient is the absence of arbitrage. The
term ‘arbitrage’ has been used in two very different senses which it is important to
distinguish:
Ω To carry out arbitrage in the first sense, one would simultaneously execute a set
of transactions which have zero net cash flow now, but have a non-zero probability
of a positive payoff without risk, i.e. with a zero probability of a negative payoff in
the future.
Ω Arbitrage in the second sense is related to a model of how asset prices behave. To
perform arbitrage in this sense, one carries out a set of transactions with a zero
net cash flow now and a positive expected value at some date in the future.
Derivative assets, e.g. forwards, can often be constructed from combinations of
underlying assets. Such constructed assets are called synthetic assets.
Covered parity or cost-of-carry relations are relations are between the prices of
forward and underlying assets. These relations are enforced by arbitrage and tell us
how to determine arbitrage-based forward asset prices.
Throughout this discussion, we will assume that there are no transactions costs
or taxes, that markets are in session around the clock, that nominal interest rates
are positive, and that unlimited short sales are possible. These assumptions are
fairly innocuous: in the international financial markets, transactions costs typically
are quite low for most standard financial instruments, and most of the instruments
discussed here are not taxed, since they are conducted in the Euromarkets or on
organized exchanges.
Cost-of-carry with no dividends
The mechanics of covered parity are somewhat different in different markets,
depending on what instruments are most actively traded. The simplest case is that
of a fictitious commodity which has no convenience value, no storage and insurance
cost, and pays out no interest, dividends, or other cash flows. The only cost of holding
the commodity is then the opportunity cost of funding the position.
Imagine creating a long forward payoff synthetically. It might be needed by a dealer
hedging a short forward position:
Ω Buy the commodity with borrowed funds, paying St for one unit of the commodity
borrowed at rt,T, the Tñt-year annually compounded spot interest rate at time t.
Like a forward, this set of transactions has a net cash flow of zero.
Ω At time T, repay the loan and sell the commodity. The net cash flow is
STñ[1òrt,T(Tñt)]St .
This strategy is called a synthetic long forward.
Similarly, in a synthetic short forward, you borrow the commodity and sell it,
lending the funds at rate rt,T,: the net cash flow now is zero. At time T, buy the
commodity at price ST and return it: the net cash flow is [1òrt,T(Tñt)]StñST.
The payoff on this synthetic long or short forward must equal that of a forward
contract: STñ[1òrt,T (Tñt)]STóSTñFt,T. If it were greater (smaller), one could make
a riskless profit by taking a short (long) forward position and creating a synthetic
long (short) forward. This implies that the forward price is equal to the future value
of the current spot price, i.e. the long must commit to paying the financing cost of
the position: Ft,Tó[1òrt,T(Tñt)]St .
Two things are noteworthy about this cost-of-carry formula. First, the unknown
future commodity price is irrelevant to the determination of the forward price and
has dropped out. Second, the forward price must be higher than the spot price, since
the interest rate rt,T is positive.
Short positions can be readily taken in most financial asset markets. However, in
some commodity markets, short positions cannot be taken and thus synthetic short
forwards cannot be constructed in sufficient volume to eliminate arbitrage entirely.
Even, in that case, arbitrage is only possible in one direction, and the no-arbitrage
condition becomes an inequality: Ft,TO[1òrt,T(Tñt)]St .
10 Şubat 2011 Perşembe
Autocorrelation of returns
The distribution of many asset returns is not only kurtotic and skewed. The return
distribution may also change over time and successive returns may not be independent
of one another. These phenomena will be reflected in the serial correlation or
autocorrelation of returns. Table 1.1 displays evidence that asset returns are not
typically independently and identically distributed. The rightmost column displays a
statistic which measures the likelihood that there is serial correlation between
returns on a given day and returns on the same asset during the prior five trading
days. High values of this statistic indicate a high likelihood that returns are
autocorrelated.
Table 1.1 Statistical properties of selected daily asset returns
Standard
Asset deviation Skewness Kurtosis Autocorrelation
Dollar–Swiss franc 0.0069 0.3472.485 6.0
Dollar–yen 0.0078 0.660 6.181 8.0
Dollar–Mexican peso 0.0132 ñ3.015 65.94756.7
Dollar–Thai baht 0.0080 ñ0.461 25.879 87.4
Crude oil 0.0204 0.249 4.681 41.1
Gold 0.0065 ñ0.165 4.983 21.3
Nikkei 225 average 0.0138 0.213 3.131 26.3
S&P 500 average 0.0087 ñ0.578 8.391 25.6
distribution may also change over time and successive returns may not be independent
of one another. These phenomena will be reflected in the serial correlation or
autocorrelation of returns. Table 1.1 displays evidence that asset returns are not
typically independently and identically distributed. The rightmost column displays a
statistic which measures the likelihood that there is serial correlation between
returns on a given day and returns on the same asset during the prior five trading
days. High values of this statistic indicate a high likelihood that returns are
autocorrelated.
Table 1.1 Statistical properties of selected daily asset returns
Standard
Asset deviation Skewness Kurtosis Autocorrelation
Dollar–Swiss franc 0.0069 0.3472.485 6.0
Dollar–yen 0.0078 0.660 6.181 8.0
Dollar–Mexican peso 0.0132 ñ3.015 65.94756.7
Dollar–Thai baht 0.0080 ñ0.461 25.879 87.4
Crude oil 0.0204 0.249 4.681 41.1
Gold 0.0065 ñ0.165 4.983 21.3
Nikkei 225 average 0.0138 0.213 3.131 26.3
S&P 500 average 0.0087 ñ0.578 8.391 25.6
Skewness
The skewness of a distribution is a measure of the frequency with which large returns
in a particular direction occur. An asset which displays large negative returns more
frequently than large positive returns is said to have a return distribution skewed to
the left or to have a ‘fat left tail’. An asset which displays large positive returns more
frequently than large negative returns is said to have a return distribution skewed
to the right or to have a ‘fat right tail’. The normal distribution is symmetrical, that
is, its coefficient of skewness is exactly zero. Thus a significantly positive or negative
skewness coefficient is inconsistent with the assumption that returns are normal.
Figure 1.3 compares a skewed, but non-kurtotic, distribution with a normal
distribution with the same variance. Table 1.1 presents estimates of the kurtosis and
skewness of some widely traded assets. All the assets displayed have significant
positive or negative skewness, and most also have a coefficient of kurtosis significantly
greater than 3.0.
The exchange rates of the Mexican peso and Thai baht vis-a` -vis the dollar have
the largest coefficients of kurtosis. They are examples of intermittently fixed exchange
rates, which are kept within very narrow fluctuation limits by the monetary authorities.
Typically, fixed exchange rates are a temporary phenomenon, lasting decades in
rare cases, but only a few years in most. When a fixed exchange rate can no longer
be sustained, the rate is either adjusted to new fixed level (for example, the European
Monetary System in the 1980s and 1990s and the Bretton Woods system until 1971)
or permitted to ‘float’, that is, find a free-market price (for example, most emerging
market currencies). In either case, the return pattern of the currency is one of
extremely low returns during the fixed-rate period and extremely large positive or
negative returns when the fixed rate is abandoned, leading to extremely high kurtosis.
The return patterns of intermittently pegged exchange rates also diminishes the
forecasting power of forward exchange rates for these currencies, a phenomenon
known as regime-switching or the peso problem. The term ‘peso problem’ has its
origin in experience with spot and forward rates on the Mexican peso in the 1970s.
Observers were puzzled by the fact that forward rates for years ‘predicted’ a significant
short-term depreciation of the peso vis-a` -vis the US dollar, although the peso–dollar
exchange rate was fixed. One proposed solution was that the exchange rate peg was
not perfectly credible, so market participants expected a switch to a new, lower value
of the peso with a positive probability. In the event, the peso has in fact been
periodically permitted to float, invariably depreciating sharply.
in a particular direction occur. An asset which displays large negative returns more
frequently than large positive returns is said to have a return distribution skewed to
the left or to have a ‘fat left tail’. An asset which displays large positive returns more
frequently than large negative returns is said to have a return distribution skewed
to the right or to have a ‘fat right tail’. The normal distribution is symmetrical, that
is, its coefficient of skewness is exactly zero. Thus a significantly positive or negative
skewness coefficient is inconsistent with the assumption that returns are normal.
Figure 1.3 compares a skewed, but non-kurtotic, distribution with a normal
distribution with the same variance. Table 1.1 presents estimates of the kurtosis and
skewness of some widely traded assets. All the assets displayed have significant
positive or negative skewness, and most also have a coefficient of kurtosis significantly
greater than 3.0.
The exchange rates of the Mexican peso and Thai baht vis-a` -vis the dollar have
the largest coefficients of kurtosis. They are examples of intermittently fixed exchange
rates, which are kept within very narrow fluctuation limits by the monetary authorities.
Typically, fixed exchange rates are a temporary phenomenon, lasting decades in
rare cases, but only a few years in most. When a fixed exchange rate can no longer
be sustained, the rate is either adjusted to new fixed level (for example, the European
Monetary System in the 1980s and 1990s and the Bretton Woods system until 1971)
or permitted to ‘float’, that is, find a free-market price (for example, most emerging
market currencies). In either case, the return pattern of the currency is one of
extremely low returns during the fixed-rate period and extremely large positive or
negative returns when the fixed rate is abandoned, leading to extremely high kurtosis.
The return patterns of intermittently pegged exchange rates also diminishes the
forecasting power of forward exchange rates for these currencies, a phenomenon
known as regime-switching or the peso problem. The term ‘peso problem’ has its
origin in experience with spot and forward rates on the Mexican peso in the 1970s.
Observers were puzzled by the fact that forward rates for years ‘predicted’ a significant
short-term depreciation of the peso vis-a` -vis the US dollar, although the peso–dollar
exchange rate was fixed. One proposed solution was that the exchange rate peg was
not perfectly credible, so market participants expected a switch to a new, lower value
of the peso with a positive probability. In the event, the peso has in fact been
periodically permitted to float, invariably depreciating sharply.
Kurtosis
The kurtosis or leptokurtosis (literally, ‘fat tails’) of a distribution is a measure of
the frequency of large positive or negative asset returns. Specifically, it measures the
frequency of large squared deviations from the mean. The distribution of asset
returns will show high kurtosis if asset returns which are far above or below the
mean occur relatively often, regardless of whether they are mostly above, mostly
below, or both above and below the mean return.
Kurtosis is measured in comparison with the normal distribution, which has a
coefficient of kurtosis of exactly 3. If the kurtosis of an asset return distribution is
significantly higher than 3, it indicates that large-magnitude returns occur more
frequently than in a normal distribution. In other words, a coefficient of kurtosis well
over 3 is inconsistent with the assumption that returns are normal. Figure 1.2
compares a kurtotic distribution with a normal distribution with the same variance.
the frequency of large positive or negative asset returns. Specifically, it measures the
frequency of large squared deviations from the mean. The distribution of asset
returns will show high kurtosis if asset returns which are far above or below the
mean occur relatively often, regardless of whether they are mostly above, mostly
below, or both above and below the mean return.
Kurtosis is measured in comparison with the normal distribution, which has a
coefficient of kurtosis of exactly 3. If the kurtosis of an asset return distribution is
significantly higher than 3, it indicates that large-magnitude returns occur more
frequently than in a normal distribution. In other words, a coefficient of kurtosis well
over 3 is inconsistent with the assumption that returns are normal. Figure 1.2
compares a kurtotic distribution with a normal distribution with the same variance.
Behavior of asset prices
Efficient markets hypothesis
The efficient market approach to explaining asset prices views them as the present
values of the income streams they generate. Efficient market theory implies that all
available information regarding future asset prices is impounded in current asset
prices. It provides a useful starting point for analyzing derivatives.
One implication of market efficiency is that asset returns follow a random walk.
The motion of the asset price has two parts, a drift rate, that is, a deterministic rate
at which the asset price is expected to change over time, and a variance rate, that
is, a random change in the asset price, also proportional to the time elapsed, and
also unobservable. The variance rate has a mean of zero and a per-period variance
equal to a parameter p, called the volatility. This assumption implies that the
percent changes in the asset price are normally distributed with a mean equal to the
drift rate and a variance equal to p2.
The random walk hypothesis is widely used in financial modeling and has several
implications:
Ω The percent change in the asset price over the next time interval is independent
of both the percent change over the last time interval and the level of the asset
price. The random walk is sometimes described as ‘memoryless’ for this reason.
There is no tendency for an up move to be followed by another up move, or by a
down move. That means that the asset price can only have a non-stochastic trend
equal to the drift rate, and does not revert to the historical mean or other ‘correct’
level. If the assumption were true, technical analysis would be irrelevant.
Ω Precisely because of this lack of memory, the asset price tends over time to wander
further and further from any starting point. The proportional distance the asset
price can be expected to wander randomly over a discrete time interval q is the
volatility times the square root of the time interval, p q.
Ω Asset prices are continuous; they move in small steps, but do not jump. Over a
given time interval, they may wander quite a distance from where they started,
but they do it by moving a little each day.
Ω Asset returns are normally distributed with a mean equal to the drift rate and a
standard deviation equal to the volatility. The return distribution is the same each
period.
The Black–Scholes model assumes that volatility can be different for different asset
prices, but is a constant for a particular asset. That implies that asset prices are
homoskedastic, showing no tendency towards ‘volatility bunching’. A wild day in
the markets is as likely to be followed by a quiet day as by another wild day.
An asset price following geometric Brownian motion can be thought of as having
an urge to wander away from any starting point, but not in any particular direction.
The volatility parameter can be thought of as a scaling factor for that urge to wander.
Figure 1.1 illustrates its properties with six possible time paths over a year of an
asset price, the sterling–dollar exchange rate, with a starting value of USD1.60, an
annual volatility of 12%, and an expected rate of return of zero.
Empirical research on asset price behavior
While the random walk is a perfectly serviceable first approximation to the behavior
of asset prices, in reality, it is only an approximation. Even though most widely
traded cash asset returns are close to normal, they display small but important ‘nonnormalities’.
In particular, the frequency and direction of large moves in asset prices,
which are very important in risk management, can be quite different in real-life
markets than the random walk model predicts. Moreover, a few cash assets behave
very differently from a random walk.
The random walk hypothesis on which the Black–Scholes model is based is a good
first approximation to the behavior of most asset prices most of the time. However,
even nominal asset returns that are quite close to normally distributed display small
but important deviations from normality. The option price patterns discussed below
reveal how market participants perceive the distribution of future asset prices.
Empirical studies of the stochastic properties of nominal returns focus on the
behavior of realized asset prices. The two approaches largely agree.
The efficient market approach to explaining asset prices views them as the present
values of the income streams they generate. Efficient market theory implies that all
available information regarding future asset prices is impounded in current asset
prices. It provides a useful starting point for analyzing derivatives.
One implication of market efficiency is that asset returns follow a random walk.
The motion of the asset price has two parts, a drift rate, that is, a deterministic rate
at which the asset price is expected to change over time, and a variance rate, that
is, a random change in the asset price, also proportional to the time elapsed, and
also unobservable. The variance rate has a mean of zero and a per-period variance
equal to a parameter p, called the volatility. This assumption implies that the
percent changes in the asset price are normally distributed with a mean equal to the
drift rate and a variance equal to p2.
The random walk hypothesis is widely used in financial modeling and has several
implications:
Ω The percent change in the asset price over the next time interval is independent
of both the percent change over the last time interval and the level of the asset
price. The random walk is sometimes described as ‘memoryless’ for this reason.
There is no tendency for an up move to be followed by another up move, or by a
down move. That means that the asset price can only have a non-stochastic trend
equal to the drift rate, and does not revert to the historical mean or other ‘correct’
level. If the assumption were true, technical analysis would be irrelevant.
Ω Precisely because of this lack of memory, the asset price tends over time to wander
further and further from any starting point. The proportional distance the asset
price can be expected to wander randomly over a discrete time interval q is the
volatility times the square root of the time interval, p q.
Ω Asset prices are continuous; they move in small steps, but do not jump. Over a
given time interval, they may wander quite a distance from where they started,
but they do it by moving a little each day.
Ω Asset returns are normally distributed with a mean equal to the drift rate and a
standard deviation equal to the volatility. The return distribution is the same each
period.
The Black–Scholes model assumes that volatility can be different for different asset
prices, but is a constant for a particular asset. That implies that asset prices are
homoskedastic, showing no tendency towards ‘volatility bunching’. A wild day in
the markets is as likely to be followed by a quiet day as by another wild day.
An asset price following geometric Brownian motion can be thought of as having
an urge to wander away from any starting point, but not in any particular direction.
The volatility parameter can be thought of as a scaling factor for that urge to wander.
Figure 1.1 illustrates its properties with six possible time paths over a year of an
asset price, the sterling–dollar exchange rate, with a starting value of USD1.60, an
annual volatility of 12%, and an expected rate of return of zero.
Empirical research on asset price behavior
While the random walk is a perfectly serviceable first approximation to the behavior
of asset prices, in reality, it is only an approximation. Even though most widely
traded cash asset returns are close to normal, they display small but important ‘nonnormalities’.
In particular, the frequency and direction of large moves in asset prices,
which are very important in risk management, can be quite different in real-life
markets than the random walk model predicts. Moreover, a few cash assets behave
very differently from a random walk.
The random walk hypothesis on which the Black–Scholes model is based is a good
first approximation to the behavior of most asset prices most of the time. However,
even nominal asset returns that are quite close to normally distributed display small
but important deviations from normality. The option price patterns discussed below
reveal how market participants perceive the distribution of future asset prices.
Empirical studies of the stochastic properties of nominal returns focus on the
behavior of realized asset prices. The two approaches largely agree.
19 Haziran 2009 Cuma
Your Web Site
Even if your business has annihilation to do with the Internet, you cannot
pass up the adventitious to actualize an online image. Indeed, a Web armpit has become
a business necessity. Not alone is it an bargain way to abutment your image
and acquaint bodies who you are, but it is additionally an befalling to advertise more, get
more customers, accomplish added money, and affect strangers.
And you charge not be Amazon.com to be successful. In fact, you probably
don’t appetite to be. Your business Web armpit should, in all likelihood, be a clean, simple,
elegant abode that does a few things actual well. Your home folio should explain
what your business is and what the Web armpit is about. It should be simple
and accessible to load. Inside, your business abode and acquaintance advice should
be accessible to find. Appearance and allowances of alive with you should be prominent.
Beyond that, what you do with your armpit is up to you. You may appetite to
consider accepting some appearance that accumulate bodies advancing back, because the
more they appear aback to your site, the added acceptable it is they will buy from you.
You can action such things as:
• Interactivity. E-commerce interactivity agency accouterment interactive
tools that accredit abeyant barter to apprentice added about your products.
It could additionally beggarly alms babble rooms, bulletin boards, or newsletters.
Streaming video is a possibility.
• Associates alone areas. Some businesses action associates alone domains
on their Web sites, area they action admission to exceptional information,
tools, and services. Think about AOL for a moment. It is annihilation but
a huge associates alone Web site; not a bad model.
• Content. On the Internet, agreeable is king. A armpit after good, arresting,
useful, appropriate agreeable is a armpit that is apparently activity nowhere.
Think about the sites you like. What is it that draws you there? In all
likelihood, acceptable agreeable is abreast the top of your list.
Even sites that are artefact aggressive can use acceptable content. Dan Harrison is
the buyer of
difference. When he started his Web armpit aback in 1994, visitors could purchase
thousands of spa and basin articles and, at the aforementioned time, learn
what to do if their basin angry algid or their hot tub got moldy.
Articles are accounting by Harrison and his agents and accommodate “Ask the Pool
Guy” and “Ask the Spa Guy” features. Babble apartment and bulletin boards are
also available. Harrison charge be accomplishing article right; he saw revenues
double to added than $2 actor in the aboriginal bisected of 2001.
Where do you get your content? You can address it yourself or appoint someone
to actualize agreeable for you. Another advantage is to buy amalgamated content.
Syndicated columns, news, horoscopes, weather, sports, and comics can be
an economical way to go. Consider the afterward options:
•
•
•
How do you get a good, apple-pie able armpit after spending a fortune?
You can appoint a Web designer, you can do it yourself, or you can go to a
one-stop shop. Web designers are big-ticket for best businesses ($2,500 and
up). If you can allow one, great, because they can accord you a abundant look. Web
designers can be begin in the Yellow Pages or online by attractive for one on a
search engine.
Unless you are accustomed with a Web architecture program, designing your site
yourself is not easy. Not alone is it time-consuming, but it can be frustrating
and ultimately end with a poor result.
The best cost-effective band-aid may be to acquisition a one-stop boutique that
hosts your armpit and designs it too. Many e-commerce solutions providers make
it actual accessible for you to acquisition a one-stop band-aid for accomplishing business on the Internet.
These new partnerships generally amalgamate armpit hosting, abundance setup, and
credit agenda processing into a distinct amalgamation accurately advised for small
businesses. Many bounded Internet account providers action these services, such as:
•
• Yahoo! Store
•
•
•
•
However you accept to go, it is capital that your business acquisition its way
onto the Web.
T H E B O T T O M L I N E
Creating an angel that bodies bethink is a amount of consistently
applying a contemporary architecture to your front-line business materials.
Everything from your jotter to your business cards and
your Web armpit needs to reinforce the angel you appetite to create.
146
16 Haziran 2009 Salı
Your Brochure
Not every business will charge or use a brochure. Even if a advertisement is not
traditionally allotment of businesses like yours, it still ability be a abundant way to create
a able angel and accompany in business. The affair to be alert of is
spending money on a advertisement if it absolutely does annihilation to add to your business.
A advertisement can be an big-ticket account and appropriately not account the money if
you absolutely don’t charge it.
When creating a brochure, abstain the following:
• Making it too busy. Creating a advertisement that is so accommodation with information
that it is faltering to the eye and difficult to apprehend is a sure
way to decay money. It is abundant bigger to accumulate paragraphs short, use
white space, use bullets, and accumulate it simple.
• Making the awning boring. Too abounding businesses anticipate that headlining
their advertisement with their business name is a abiding way to attract people
to apprehend more. If you appetite bodies to apprehend your brochure, you must
catch their absorption (usually with some account they could get by
reading more) and draw them in.
Ask yourself: What is the purpose of this brochure? Is it an introduction
to your business, a affairs tool, both, or more? Whatever your answer, your
brochure needs to reflect the aforementioned values, tone, and affair that will be found
in your added image-creating materials. Use your logo. Use your colors. Reinforce
your adapted angel with argument and cartoon that reflect your business
image.
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