Commodity (Gold) Trading
Report 





The purpose of writing this report
is to give reader depth knowledge of the derivative markets like
Futures, Forwards and Options and their prices derivation and future
price analysis. The choice of the Futures, Forwards and Option markets
together was made because they are fundamentally well related to each
other and they are now a day most important instrument for risk
management. In this decade, these instruments have grown rapidly and
understanding and knowledge of pricing of these instruments would be the
advantage.
I will focus on understanding of price derivation for each market. I
will undergo basic concepts of Future, Forward and Option markets as
they are vital for understanding market function and their theories on
determining and forecasting price.
At the starting of all market introduction there are understandings of
concepts used in the market, than there are brief discussions on the
pricing theories; basically, there are two pricing theories on Options
market and there are three theories on Futures market. At the end of an
essay, after giving very good idea of all theories and concept, you will
find an analysis about which theory is appropriate to predict the price
or which are very helpful for trader to reduce his risk for particular
contract or for particular market. With the help of Reuters 3000 Xtra we
will go throw with practical example for pricing the contract. 




Introduction 





Trading of basic commodities is one of the oldest commercial
activities found in every civilization. For all basic and omissible
needs of the humankind, nearly every civilization developed some sort of
central marketplace where people could meet and trade. This is when
physical or cash market born. Over time, more participants entered this
physical market directly or indirectly, thus price (value) of the
commodity was directly depend on prolong market chain (e.g. processed,
transported, stored, refined, manufactured, etc.). Each participant in
the chain added value to the commodity and this affected the price to
move up or down according to the supply and demand of the commodities in
the chain. A sudden change in the supply/demand equation can create move
in the price. If anyone in the chain needing to acquire a commodity then
it causes sudden jumps in value and if he required holding a commodity
decrement in value occurs.
The modern civilization introduced another level of risk. Merchants
began writing individual contracts for expected deliveries, “forward
contracts”, containing specific terms of delivery (quantity, quality,
anticipated date of delivery) all for a fixed price. By this
merchandisers start taking risk of weather, production, quality and
natural disasters. With this increasing cash market risk, merchants for
several basic commodities joined together in the late nineteenth century
to establish a more organized and specialized marketplace at the
TransAtlantic route (e.g. London and New York) for each commodity where
they could meet and negotiate these forward contracts. The forward
contract, however, was still a cash market instrument whereby a price
was negotiated for an actual physical commodity transaction at a future
date of delivery. The contract applied to the unique terms of one
transaction, but the existence of this piece of paper added another
possible level of trade – the idea of buying and selling the contract
itself.
As a binding instrument that committed the holders to a transfer of the
physical commodity, the contract itself could change hands many times as
long as its terms remained outstanding (until the actual delivery took
place). As the buying and selling of an existing contract became an
accepted practice, the standardization of that contract became the next
logical step. The standardization of the forward contract led to the
creation of the futures contract that added a whole new dimension to the
trade. The cash market continued its daytoday business of selling and
buying an asset at that day’s price. The organization of merchants
buying and selling the physical asset evolved into an organization that
standardized the contracts and the trading practices, and became the
futures exchange. The acceptance of the standard contract allowed
organized trading of the value of the asset for delivery at some future
date through a futures market for that asset. The futures contract had
specific terms, amount, type, price, timeframe, etc., but unlike the
forward contract, it did not apply to any specific transaction. A
“standard” contract with standard terms that applied across the board
was developed. The underlying asset could change hands under the terms
of the contract, but that was not its purpose – its purpose was to
establish a price for the underlying asset for a defined period of time
(the term of the contract). This price became a benchmark for
determining the daytoday cash market price. It also established the
futures contract as an instrument that had its own value.
For traders who do not want to take any risk in the market, a new tool
of risk management was introduced. Such a contract which is similar to
futures contract when prices are in favour but when the price of the
underlying asset was not in the favour, traders were given option of not
bearing loss more then they choose to. An Option contract was introduced
to limit the loss in futures contract. As a writer of the option
contract was bearing the risk of unfavourable price changes the premium
was paid to writer of the contract, this led this contract to have its
own value. In the prior period when contract was introduced it was
traded OTC (Over the Counter), these contracts were not standardized and
both party involved in the contract had to bear the credit risk. 




Contracts 




Forward Contracts 





A Forward contract is a transaction made now to purchase an asset, a
specified amount of cash asset at a specific price with the exchange of
funds or other contract as an underline asset at an agreedupon time in
the future. Each forward contract can be made on different terms.
For¬ward transactions are completed every day for agricultural
commodities, Treasury se¬curities, foreign currencies, and interest rate
agreements made all over the world. For example, a farmer often makes a
forward contract with an intermediary (called a "grain eleva¬tor")
whereby the farmer agrees to sell grain to the elevator after
harvesting. This con¬tract specifies the number of bushels of grain, the
price per bushel, and the delivery date of the grain. This forward
contract allows the farmer to plan for the future with the certainty of
a profit, barring a natural weather disaster. 




Futures
Contracts 





Futures contracts standardize the agreement between a buyer and a
seller, specifying a trade in an underlying cash asset for a given
quantity at a specific time. Two impor¬tant advantages of a futures
contract are its tradability and its liquidity (i.e., one can trade
large positions without affecting prices). In addition, one can profit
with a futures contract without having to buy the cash asset. Futures
exist because they provide risk and return characteristics that are not
available solely by trading cash instruments such as stocks and bonds.
Speculators can obtain very high rates of return with futures due. 




Options Contracts 





An option is a contractual agreement that gives the holder the right
to buy or sell a fixed quantity of a security or commodity (for example,
a commodity or commodity futures contract), at a fixed price, within a
specified period of time. May either be standardized, exchangetraded
and government regulated or overthecounter, customized and
nonregulated. 




Differences between Forward, Futures and
options 






Forward 
Futures 
Options 
Contract 
Future agreement that
obliges the buyer and seller 
Future agreement that
obliges the buyer and seller 
Future agreement
where the seller is obliged, but the buyer has an "option" but
not an obligation 
Contract Size 
Depending on the
transaction and the requirements of the contracting parties. 
Standardised 
Standardised 
Expiry Date

Depending on the
transaction 
Standardised 
Standardised.
American style options can be exercised at any time. European
style options can only be exercised at expiry. 
Transaction method 
Negotiated directly
by the buyer and seller 
Quoted and traded on
the Exchange 
Quoted and traded on
the Exchange 
Guarantees 
None. It is very
difficult to undo the operation; profits and losses are cash
settled at expiry. 
Both parties must
deposit an initial guarantee (margin). The value of the
operation is marked to market rates with daily settlement of
profits and losses. 
The buyer pays a
premium to the seller. The seller deposits an initial guarantee
(margin) with subsequent deposits made depending on the market.
The underlying asset can be used as guarantee. 
Secondary Market 
None. It is difficult
to quit the operation; profit or loss at expiry. 
Futures Exchange.
Operation can be quit prior to expiry. Profit or loss can be
realised at any time. 
Options Exchange.
Operation can be quit prior to expiry. Profit or loss can be
realised at any time. 
Institutional
Guarantee 
The contracting
parties 
Clearing House 
Clearing House 
Settlement

Cash settled. 
Contracts are usually
closed prior to expiry by taking a compensating position. At
expiry contracts can be cash settled or settled by delivery of
the underlying. 
When a long position
is exercised it may be settled by delivery or cash settled. A
long position which is outofthemoney is usually cancelled
prior to expiry. 





Futures/Forwards Market 




Concepts 




Hedging 





Hedging is a risk reduction strategy whereby investors and traders
take offsetting positions in an instrument to reduce their risk profile.
Hedging is the process of managing the risk of metal change by
offsetting that risk in the futures market.
Hedger is an individual or a firm who undertakes in hedging process.
Usually a big firm who has big stock in hand or a big gold mine that has
to protect itself for price risk.
A buy position or "long" in the underlying asset is covered by a sell
position or "short" position in futures. Conversely, a "short" position
in the underlying asset is covered by a buy position or "long" in
futures. The greater the correlation between the changes in prices of
the underlying asset and the futures contract the more effective the
hedge. As such, the loss in one market is partially or totally
compensated by the profit in the other market, given that the traded
positions are equal and opposite. 




Speculation 





Speculation involves the buying, holding, and selling of stocks,
commodities, futures, currencies, collectibles, real estate, or any
valuable thing to profit from fluctuations in its price as opposed to
buying it for use or for income ( via dividends, rent etc). Speculation
represents one of three market roles in western financial markets,
distinct from hedging and arbitrage. 





Type of Speculation 


Passive Speculation: When a spot position is held or expected to be
held without any type of hedge, it can also be classed as passive
speculative.
Active or Dynamic speculation: A speculative operation aims to profit
from expected differences in quotations, based on taking positions on
the basis of with expected trends. 





Speculators 


The speculator tries to maximise profits in the shortest period of
time, thus reducing the investment of personal funds.
The high degree of financial leverage that is obtained with futures
contracts makes them particularly appealing to speculators, as the
multiplier effect on profits in active speculative trades are seen as
especially attractive when the trend in quotations is correctly
predicted.
Most nonprofessional traders lose money on speculation, while those
that do make money tend to become professional. 





The
Economic Role of Speculation 


The roles of speculators in a market economy are to absorb risk and
to add liquidity to the marketplace by risking their own capital for the
chance of monetary reward. A speculator will exploit the difference in
the spread and, in competition with other speculators, reduce the spread
thus creating a more efficient market. It is positive for the overall
operation of the market as it brings greater liquidity and stability, as
well as greater range, flexibility and depth in contract quotations.





Arbitrage 





An arbitrage is the practice of taking advantage of a state of
imbalance between two (or possibly more) markets: combinations of
matching deals are struck that exploit the imbalance, the profit being
the difference between the market prices. A person who engages in
arbitrage is called an arbitrageur.
An arbitrage is an opportunistic operation that usual exists for very
short time periods. Arbitrage trading includes a wide range of crossed
operations, of which the most frequent and representative are arbitrages
on futuresspot, futuresoptions, futuresoptions with different
expiries and matching or similar futures and options quoted in different
exchanges. 





The Economic
Role of Arbitrage 


In Economics, The activity of the arbitrageur ensures that prices
tend to efficiency. Thus, we should consider the role of the arbitrageur
as both positive and necessary for the overall operation of the market.





Pricing Futures Market 




Capital Asset Pricing Model 





The Capital Asset Pricing Model (CAPM) has been widely applied to
all kinds of financial instruments. It states that the return on a
security is a function of the market (systematic) risk. 





Assumptions of
CAPM 


[1.] Capital markets are perfect, there is only a single borrowing
and lend¬ing rate no transaction costs, all capital assets are perfectly
divisible (one can buy fractions of a security) and there are no taxes,
investors can sell short and information is freely available to all
market participants.
[2.] Investors attempt to maximize their utility, which consists of
maximiz¬ing returns for a given level of risk. Investors are riskaverse
and measure risk in terms of standard deviations of returns.
[3.] Investors use a common oneperiodahead time horizon for investment
decisions. All investment decisions are made at the beginning of the
period and no changes are made during the investment horizon.
[4.] Investors have identical expectations about the risk and return.
[5.] There exists a single riskfree asset at which borrowing and
leading can take place. 





The Model 


E(Rj) = Rf +ßj[E(Rm)  Rf]
Where,
E(Rj) = the expected return on security j
Rf = the riskfree rate
E(Rm) = the expected return on the market portfolio
ßj = cov(Rj, Rm)/σ2 = The Beta of the instrument 





The Capital Asset Pricing Model is denned by in above equation,
where the measure of systematic risk (ß) is the covariance of the asset
return (futures price change) with the return on the market portfolio
(index), divided by the variance of the market return (index return);
It is an important theory for pricing stocks and portfolios of stocks.
For futures pricing it states that futures price is directly related in
a proportional to the “market”, the proportional is systematic risk and
it is measured by ß. The beta is usually estimated from a regression
equation where ß measure the systematic risk and standard deviation of
the error tells us about the unsystematic of the market: 





Rj = α + ßjRm + ej
Where,
ßj = the systematic risk of security j
σ(ej) = the unsystematic risk of security j 





According to the CAPM, only unavoidable risk should be compensated
in the marketplace, and traders can avoid much risk through
diversification. Even after diversification, risk remains because the
returns are correlated with the market as a whole. This remaining risk
is systematic. So if ß = 1 then asset has the same degree of systematic
risk as the market portfolio. So the asset should earn same return as
the market portfolio. If ß = 0 then asset should only earn risk free
rate of interest.
In contexts to futures market if we put this logic then if the ß is more
then zero then expected futures price is positive which means
expectation of futures pries should be more then spot price. If the ß is
zero then there shouldn’t be any change in future price as there is no
investment on it. However, there is a margin to pay but it is not so
called investment as it is in a form of the deposit against the price
fluctuation and price risk. 




The Hedging Pressure Theory 





Assume for the moment that speculators are rational and risk
adverse. The hedging pressure theory of future pricing comes from the
view point of speculators. If the futures price reaches the expected
price of the commodity when the futures contract matures, then there is
no reason for speculators to speculate in futures. Speculators will only
come to contract if they have been compensated for the risk they have
been taking. When net long hedger exist net short speculators, this
require speculators to take more short position or when net short hedger
exist net short speculators this require speculators to take more long
position, for to bear this an extra risk the speculator will want
aboveaverage returns to enter in contract. These aboveaverage returns
will be compensated by moving futures price to the favour to speculator.
This theory work simply works like demand and supply theory demand goes
up price go up.
The hedging pressure theory of future pricing is if net short hedging
exceeds net long speculation, then long speculators require
aboveaverage returns compensation for purchasing additional futures
contracts in order to equate supply and demand the relationship is known
as normal backwardation; that is, futures prices must be underpriced
relative to their true value to encourage speculators to buy futures.
Similarly contango means that futures must be overpriced for short
speculators to earn abnormal return when net long hedging is greater
then net short speculation, by this supplyand demand factor could cause
the futures to be consistent under –or overpriced relative to its true
value.
Although the number of long positions must equal the number of short
positions for trading to exist, the hedging pressure theory states that
when a net short or long hedging position exist, the futures contract
becomes a biased estimate of the spot price. These biased prices
encourage additional speculators to enter the market and create the
needed activity to offset the hedger’s activity in the market. Now let’s
take the real example of the backwardation theory in real market.
.(Robert T. Daigler,1994) 





Heating Oil
Futures Pricing: 


Heating Oil futures price is the best example the theory of the
normal backwardation Even though the costs of storage and transport are
high, the heating oil future pricing is regional, inventory is low
relative to consumption; demand is seasonal; and the risks of a supply
interruption (OPEC supply cutbacks, a refinery explosion) are real, many
key market participants can’t afford the risk of selling inventory and
going long on heating oil futures if a spotfutures pricing arbitrage
opportunity presents itself. Certain participants have to hold
inventory. For holding the inventory the participant is giving
convenience yield to the speculator. And because of the convenience
yield it is very hard execute the arbitrage and maintain price
efficiency. It is all these factors, but especially the last one, that
weaken the force of arbitrage to maintain price efficiency with
consumption commodities. The result is the following inequality for
pricing consumption commodity futures contracts: 





F_{0T}<S_{0}e^{(r+w)T} 





The convenience yield is the amount paid to the speculators by the
hedgers to enter in the contract. It is the difference between
arbitragefree efficient futures price and the spot price. The
convenience yield reflects the market’s expectations concerning the
future availability of the commodity. The greater the possibility that
shortages will occur during the life of the futures contract, the higher
the convenience yield. If we take out the convenience yield (y) from the
spot price then we can get the arbitrage free, efficient futures price. 





F_{0T}
= S_{0}e^{(r+wy)T} 





Summery of Studies of Normal Backwardation: 





Study 
Key Results 
Houthakker(1957) 
Found positive returns for cotton, wheat, and corn.

Rockwell (1967) 
Found little
support for normal backwardation. 
Dusak(1973) 
Found returns near
zero for wheat, corn, and soybeans. 
Bodie and Rosansky (1980) 
Found positive returns for many commodities.

Carter, Rausser, and Schmitz (1983) 
Found positive
returns in wheat, corn, and soybeans. 
Raynauld and Tessier (1984) 
Results for corn, wheat, and oats did not support normal
backwardation. 
Chang (1985) 
Examined wheat, corn, and soybeans, but results were
inconclusive for normal backwardation. 
Baxter, Conine, and Tamarkin (1985) 
Found no positive returns for wheat, corn, or soybeans. 
Park (1985) 
No
normal backwardation in currencies and plywood, but normal
backwardation in metals. 
Hazuka (1984) 
Evidence from applying the consumptionbased CAPM to 14
commodities, supported normal backwardation. 
Fama and French (1987) 
Found positive returns that weakly support normal backwardation. 
Ehrhardt, Jordan, and Walkling (1987) 
Found no support for normal backwardation in wheat, corn, or
soybeans. 
Hartzmark (1987) 
Found that hedgers made money and speculators lost money, which
is inconsistent with normal backwardation. 
Yoo and Maddala (1991) 
Large hedgers pay some risk premium, and profits of large
speculators are due mostly to risk bearing. 
Kolb(1992) 
Of 29 commodities tested, only four conform well to the
backwardation hypothesis. Backwardation is not a general feature
of futures markets. 
Deaves and Krinsky (1995) 
Provided a more detailed examination of the commodities that
Kolb (1992) found to be candidates for backwardation. Found
that futures prices are good predictors of subsequent spot
prices. 





Costofcarry Model 





The Model



The Costofcarry is the total cost to carry a good forward in time.
For example gold on hand in June can be carried forward up to, stored
until, December. So the price of December contract will be June contract
plus cost of carrying gold to December. It is an arbitragefree pricing
model. Its central theme is that futures contract is so priced as to
prevent arbitrage profit. In other word, investors are not paying less
or more in either spot or future market for executing their buying and
selling contract or underline asset. It is because future prices are
effectively priced by considering the interest gain or loss via holding
or selling contract and other expenses paid for holding the underline
assets. It is true that expectations do influence the price, but they
influence the spot price and, through it, the futures price. They do not
directly influence the futures price. According to the costofcarry
model, the futures price is given by 





Futures price = Spot Price + Carry Cost  Carry Return 





F = S + C – R 


Where,
F = Future price
S = Spot Price
C = Carry Cost
R = Carry Return 





Carry cost has basic four categories: storage costs,
insurance costs, transportation costs and financing cost. Storage cost
involves the cost of warehousing of the commodity in the appropriate
facility. Storage cost plays key role in physical goods and a goods
which could be stored for long period of the time like gold, silver,
wheat, lumber etc. For most of the goods in storage insurance is also
necessary. For example insurance on lumber for fire, on gold or silver
for theft, on wheat for water damages.
Financing cost is the cost of risk free interest that investors have to
pay or have to receive according to their position. Most of the time in
futures market instead of the taking physical deliveries of an underline
asset investor go for the cash settlement, investor simply take the
opposite position and get out of the contract. Because of it insurance,
transportation and storage cost are irrelevant but even if it is cash
settlement investor has to pay financing cost directly or indirectly.
That’s why financing cost is most important cost for the costofcarry
model.
Carry return is the income (e.g., dividend, interest gain) derived from
underlying asset during holding period. Thus, the futures price should
be equal to spot price plus carry cost minus carry return. If it is
otherwise, there will be arbitrage opportunities as follows. 





Examine the Model 


To prove that futures price has to be equal spot price plus carry
cost minus carry return lets see what happen when it is not so:
Lets take when F > (S+C –R) then we can sell the overpriced futures
contract and buy the underline asset in the spot market and hold until
the maturity. On maturity we give the delivery of the underline asset
and we earn the difference. This is called "cashandcarry" arbitrage.
Now as per assumption of perfect market everybody knows this and
everybody tries to take advantage of the "cashandcarry" arbitrage. Now
at time 0 everybody tries to sell the futures contract in futures market
and tries to buy the underline commodity on spot market because of this
the supply for the futures contract in futures market will be increased
and demand for the underline asset will increase in spot market. Now as
per demand and supply law the access supply will lead to drop the price
of future contract in futures market and access demand will lead to
increase in the price in the spot market. The increment in price of spot
market and decline in the futures market will continue until the futures
price should be equal to spot price plus carry cost minus carry return.
So we can say that this can’t hold in long run.
Lets take when F > (S+C –R) then we can buy the underpriced futures
contract and shortsell the underlying asset in spot market and invest
the proceeds of shortsale until the maturity of futures contract. On
maturity we take the delivery of the underline asset and sell the
futures contract. This is called "reverse cashandcarry" arbitrage.
Now as per assumption of perfect market everybody knows this arbitrage
and everybody tries to take advantage of the "reverse cashandcarry"
arbitrage. Now at time 0 everybody tries to buy the futures contract in
futures market and tries to shortsell the underline asset on spot
market because of this the demand for the futures contract in futures
market will be increased and the supply of the money in money market
will increase. Again, as per demand and supply law the access demand
will lead the increment of price of future contract in futures market
and access supply will lead the decline in the interest rate in money
market. The increment in price in futures market and decline in interest
rate will continue until the futures price should be equal to spot price
plus carry cost minus carry return. So we can say that this can’t hold
in long run.
Thus, in long run futures price will be same as spot price plus carry
cost minus carry return which means it makes no difference whether we
buy or sell the underlying asset in spot or futures market. If we buy it
in spot market, we require cash but also receive cash distributions
(e.g., dividend) from the asset. If we buy it in futures market, the
delivery is postponed to a later day and we can deposit the cash in an
interestbearing account but will also forego the cash distributions
from the asset. However, the difference in spot and futures price is
just equal to the interest cost and the cash distributions. 





Cost of Carry
with Different Markets 





Basic CostOfCarry



Costofcarry model can also determine the price relationships that
can exist between futures contracts on the same good that differ in
maturity. According to Costofcarry model future price of distance
contract must be equal to future price of nearby contract plus carry
cost. 





F_{d}
= F_{n} + C  R 


Where,
Fd = Distance Future contract
Fn = nearby Future Contract
C = Carry cost 





As we have proved above, we can also prove that Futures price of
distance contract can’t be either less than or more than the future
price of the nearby contract plus carrying cost.
The pricing of futures contract in every market is also different form
each other because the fundament for underline asset in each market is
different. These Fundaments create the equation of the carry cost. Most
of the time carry cost is the interest we pay or give in every market.






Storable Non Income
Generating Commodities 


There are a number of interest rates in the economy, even for the
same period. There are deposit rate, lending rate, repo rate, Treasury
bill yield, etc. Since the Clearing Corporation guarantees the futures
contract, it is a riskfree asset, like treasury bills. Accordingly, the
interest rate factored in futures price should be the interest rate on
riskfree assets like treasury bills. This is called the "riskfree
rate." There is no subjectivity or uncertainty about carry return. Most
commodities fall into this category and the fair value (or theoretical
value) of the futures contract is determined by the following equation: 





F = S(1 + r)t/365 





Where,
F = Fair value of a Futures contract expiring in t days using
Compounding Interest Rate
P = Current spot price of underlying assets
r = Carry cost (largely interest charges)
t = Days to settlement 





Stock Market Futures 


In stock market as well interest rate has the largest proportion of
carry cost. The equation for futures price gives the current price
whereas the cash dividend is payable sometime during contract life. To
bring all terms on a common footing, we will have to use the
presentvalue of cash dividend rather than the dividend amount itself.
Let us now examine the carry cost, which is essentially the interest
rate. We can now translate the futures pricing equation in computable
terms as follows. 





F = S + (S r T)  (D  D
r t) 





Where,
F = futures price
S = spot price
r = riskfree interest rate (pa)
D = cash dividend from underlying stock
t = period (in years) after which cash dividend will be paid
T = maturity of futures contract (in years) 





It is customary to apply the compounding principle in financial
calculations. With compounding, the above equation will change to 





F = S(1+r)^{T }
D(1+r)^{t} 





Alternately, using the continuous compounding or discounting, 





F = S^{erT}  D^{ert
} 





There are two good reasons why continuous compounding is preferable
to discrete compounding in stock market. First, it is computationally
easier in a spreadsheet. Second, it is internally consistent. For
example, interest rate is always quoted on an annual basis but the
compounding frequency may be different in different markets. 





Equity Stock Index 


In Equity Stock Index, we consider the portfolio of shares held in
the index. Then we discount it by shortterm rate and the dividend yield
of the market. 





F = P(1 x r/100 
d/100) t/365 





Where,
P = AllShare Index times the contract size
r = shortterm rate
d = dividend yield
t = number of days between futures trading date and expiry date 





Stock index futures closely follow the price movement of their
respective indexes, typically referred to as the “underlying” or “cash”
indexes. Intraday, monthly, and yearly correlations between cash indexes
and futures are very close. On some occasions, the futures may diverge
from the cash index for short periods of time, but market forces (such
as arbitrage) usually work to bring these brief variances back into
line. 





Currency Futures 


The fair value of a currency futures contract is largely determined
by the interest rate differential between deposits in different
currencies (interest rate parity).
The following formula is used to establish the fair value of a currency
futures contract (which is also the forward exchange rate): 





F_{x }
= [1 + (r_{f} x t/365)]/[1 + (r_{d} x t/365)] x F_{s} 





Where,
Fx = forward exchange rate expressed in foreign currency units
Fs= current spot exchange rate expressed in foreign currency units
rf = current foreign interest rate
rd = current domestic interest rate
t = number of days between futures trading date and futures 





Currency trade nowadays done on forward exchange rather then futures
exchange so futures price tend to be same as forward price. 





Calculating the Currency Futures price in Real world 


Now By using Cost and carry model for currency futures, let us
calculate the future price of the Swiss franc in the real world it
always tends to be the interest rate differential between deposits in
different currencies (interest rate parity).
For example, assume we are in March, so to get Swiss Franc 3 month
forward rate means June’s forward rate. Now by using the Reuters Xtra
3000 pro, we can get the spot rate of the Swiss franc (CHFF=) the Spot
price for sell Swiss franc is 1.6771 and for the bid price of the 3
months interest in Switzerland check (CHF3M=) which 0.00129 rounded
0.0013.
Forward rates’ upper range is calculated as
1.6771  0.0013 = 1.6758
For lower range buy price in spot market (CHFF=) is 1.6781 and ask price
of the 3 months interest is 0.0009.
Forward rates’ lower range is calculated as
1.6781  0.0009 = 1.6772. 





As Swiss franc forward price are quoted in the Swiss franc we have
inversed our result. The result we will get is 0.5967 and 0.5962. Which
means forward rate must be in range of 0.5967 to 0.5962. If we see the
June Forward’s last price (JUN2: 0#SF) then find that It is 0.5964 which
is in the rage. 










Options 




Introductions 





An option is a contract between two
parties, where buyer of an option contract acquires right but not an
obligation of buying or selling the underline an asset on a determined
quantity of at certain price at any point of time in future from writers
of an option contract. 





There are two type of Option contract
traded in markets:
• Option contract to buy (call).
• Option contract to sell (put). 





Likewise in futures there are two basic
strategies, namely to buy and to sell contracts, in options there are
four basic strategies as follows: 





• Buy option contract to buy (long
call).
• Sell option contract to buy (short call / Write call).
• Buy option contract to sell (long put).
• Sell option contract to sell (short put / Write Put). 





As in futures contract the buyer of a
contract has a right as well as the obligation of buying or selling the
contract at expiry date. Whereas, this obligation is broken in options
where the buyers of the contract has the right but not obligation to buy
(call) or sell (put), where writer (seller) of the option only has the
obligation to sell (call) or to buy (put) the underline asset. As the
seller of the writer the price risk, he requires a premium which has
been paid by the buyer of the contract up front. There always been a
credit risk while dealing with any contract related to any future
transaction. To make an option popular there has to be done something to
eliminate the credit risk. Here the solution came from the old concept
“the exchange place”, which is a common place where all the buyers and
seller come for trading and exchange barriers the credit risk for the
both parties.
An option has been made up with five fundamental characteristics like:1)
type of option (buy call or sell  put), 2) the underlying asset or
reference, 3) the amount of the underlying that the option gives right
to buy or sell, 4) the expiry date and 5)the exercise price of the
option.
An Option which can be exercised at any moment up to expiry is called an
American options or and which can be only exercised at expiry is a
European options. 




Jargons of Option Market 





In every market there are my words used
as some specific term or concept. In option market there are most
popular jargons are in the money, at the money, or out of the money. At
any define time t; an option may be in the money, at the money, or out
of the money.
At the money: When the Strike Price of call is same as the current stock
price then that call is said to be at the money. Call in most liquid at
this position.
In the money: When the strike price of call is more then current stock
price, which means the call can be excised on write. That call is said
to be in the money call. If the difference between strike price and
stock price is too large then that call is said to be deepinthemoney.
Out of the money: When the strike price of call is less then strike
price current stock price, which means the call can not be excised on
write in near future. That call is said to be out of the money call. If
the difference between stock price and strike price is huge, then that
call is said to be deepoutof themoney.
For put, these terms are reversed means out of the money call is in the
money for put and so on so for. 




Call Premium Value 





We can brake an option's premium into
two parts: intrinsic value (some¬times called parity value), and time
value (sometimes called premium over parity). 





Table: Out, In and At the money 






Calls 
Puts 
In the money
Out of the money 
S>K
S<K S~K 
S<K
S>K
S~K 






Intrinsic Value: Difference between
strike price and the stock price is called intrinsic value, if the value
of the difference is more then zero then and then intrinsic value is
taken into the account and that’s why intrinsic value comes into account
in premium calculation when option is Inthemoney. The intrinsic value
of the premium will be zero if the call is at the money or out of the
money. 





S – K if S >
K
Intrinsic Value =
0
if S ≤ K
Which means the intrinsic
value of a call is the greater of 0 or St — K. 





Time Value: Time value is the difference
between the option premium and the intrinsic value. The structure of the
time value can also be broken in two parts first is present value of the
amount of stock price and second is the compensation amount that buyer
of the call pays to the writer of the call for taking the price risk.
Usually, the maximum time value exists when the call (or put, for that
matter) is at the money because writer of the call is having more price
risk then to Out of the money or In the money call. As buyer of the in
the money call takes price risk so there is no time value in the
Inthemoney call C = S – K and other way round in out of the money call
writer is barring the price risk the time value is higher then the
intrinsic value that’s why C > S – K, the longer the time period the
greater time value all else equal. 





Time value of a call = C,  {max[0, St 
K]} 





Similar is with put: 





K  S
if K > S
Intrinsic Value =
0
if K ≤ S 





It means puts intrinsic Value will be
Max (0, K – S) and about time value of the put all put which is In the
money or Out of the money has it but put at the money may or may not
have the time value. (David A Dubofsky, 1992) 




Pricing Options 





Pricing options is a foundation of the
finance. There are popular two methods used for the pricing Options.
First in 1970s The Black Schole model coincided with initiation of
exchangetraded options on the Chicago board of exchange in 1973. Then
the innovation of Black Schole model came which called Binomial Model is
now more popular and more used. 




The Binomial Option Pricing
Model (BOPM) 





Introduction 


The binomial option pricing model has
proved over time to be the most flexible, intuitive and popular approach
to option pricing. It is based on the simplification that over a single
period (of possibly very short duration), the underlying asset can only
move from its current price to two possible levels. Among other virtues,
the model embodies the assumptions of no riskless arbitrage
opportunities and perfect markets. Neither does it rely on investor risk
aversion or rationality, nor does its use require estimation of the
underlying asset expected return. It also embodies the riskneutral
valuation principle which can be used to shortcut the valuation of
European options. In addition, we show later, that the BlackScholes
formula is a special case applying to European options resulting from
specifying an infinite number of binomial periods during the
timetoexpiration. 





Assumptions 


Perfect Market Assumptions
• Equilibrium
• Perfectly competitive
• Existence of risk free asset
• Equal access to the capital market
• Infinitely divisible securities
• Perfect Shortselling allowed
• No transaction costs or taxes
Other Assumptions:
• Only one Interest rate for landing and borrowing
• Periodic interest rate and size of up tick and down tick know in every
future period.
• Stock moves according to “Geometric Random Walk”
• Investor prefers more wealth to less. 





One Period Pricing Model 


The basic assumption in the any Binomial
pricing model is the stock price follows a multiplicative binomial
process over discrete periods. The rate of return on the stock over each
period can have two possible values: 1+ u with probability p, or 1+d
with probability 1 – p. Thus, if the current stock price is ST1, the
stock price at the end of the period will be either (1+u)S or (1+d)S.
This movement can be represented with the following diagram: 





(1+u)St_{ T1 }=S_{T,u} with
probability p
S_{T1}
(1+u)S_{T1}
=S_{T,d} with probability 1  p 





If “ r” represent one plus the riskfree
interest rate on the stock price S, over one period of time then to make
no arbitrage it has to hold u > r > d, if these inequalities did not
hold, there would be profitable riskless arbitrage opportunities
involving only the stock and riskless borrowing and lending.
To find out the price of the call on this stock whose expiration date is
just one period away lets take CT1 be the current value of the call,
C(1+u) = CT,u = max ( 0, ST,u – K) be its value at the end of the period
if the stock price goes to ST,u and C(1+d) = CT,d = max ( 0, ST,d – K)
be its value at the end of the period if the stock price goes to ST,d.
where max ( 0, ST,d – K) comes from the concept of call premium
calculation on page 28. Therefore, 





C_{(1+u) }= C_{T,u }= max ( 0, S_{T,u}
– K) with probability p
C_{T1}






Now suppose a portfolio containing
shares of stock and B dollar amount in riskless bonds is formed. So the
portfolio will cost ST1 + B. At the end of the period, the value of
this portfolio will be: 





D(1+u)S_{T1}
+ (1+r)B =
DS_{T,u
}+ (1+r)B
DS_{T1}
+
B
D(1+d)S_{T1}
+ (1+r)B =
DS_{T,d
}+ (1+r)B






Where r is the riskfree interest rate u
is the uptick and d is downtick. The value of the and B in shows the
risk adversity of the investors. As assumed that the investors are very
risk adverse and wish not take any risk, so the endofperiod values of
the portfolio and the call for each possible outcome must be same to
attract the investor. So Value of and ß will be: 





D(1+u)S_{T1}
+ (1+r)B
= C_{T,u}
D(1+d)S_{T1}
+ (1+r)B
= C_{T,d} 





Solving these equations, 





(1) 





The value of refers to how many shares
of stock to buy in order to replicate a call, where 0≤ ≤ 1.The value
of B, where B≤ 0, specifies how much to borrow to finance the investment
in the stock. If the call and the debtequity portfolio both offer
exactly the same payoffs at time T, then the price of the call at time T
 1 must equal the investment in the equivalent portfolio at time T –
1.If there are to be no riskless arbitrage opportunities, the current
value of the call, C, cannot be less than the current value of the
hedging portfolio, S + B. If it were, we could make a riskless profit
with no net investment by buying the call and selling the portfolio. But
this overlooks the fact that the person who bought the call, has the
right to exercise it immediately.
Suppose that St1 + B < S – K. If a person try to make an arbitrage
profit by selling calls for more than St1 + B, but less than S – K,
then he will soon find that he is the source of arbitrage profits rather
than the recipient. Anyone could make an arbitrage profit by buying our
calls and exercising them immediately.
The conclusion is that if there are to be no riskless arbitrage
opportunities, it must be true that
If St1 + B is greater than S – K then, 





(2)
Equation (2) can be
simplified by defining
And
So new equation is:
C
= [pC_{u }+ (1 – p)C_{d}]/(1+r
) (3)
, and if not, C =
S – K. 





The TwoPeriod Pricing Model 


The next possible situation in binomial
model can be interpreted by two period pricing mode: pricing a call with
two periods remaining before its expiration date. In keeping with the
binomial process, the stock can take on three possible values after two
periods. 





(1+u)^{2}S_{ T2} = S_{T,uu
}
(1+u)S_{ T2} =S_{ T2}
S_{T2}
(1+d)(1+u)S_{ T2} = S_{T,ud}
(1+d)S_{
T2} =S_{ T1}
(1+d)^{2}S_{ T2} = = S_{T,dd}






In above diagram (a) S T2 is current
stock price at the end of one period, T1, the stock price can take two
places, depending if it is an up tick or downtick. At the end of second
period, T, the stock price can take three prices. Tick upup which will
be St,uu, tick downdown will be St,dd and note here tick can go updown
or downup which will give the same answer but can be written different
in ways which are St,du or St,ud. Here you will find it as a S. 





Pricing Process for a call 


C_{T 1,u}
C_{T2}
C_{T,ud }= max[0,
(1+d)(1+u)S_{ T2} – K]
C_{T 1,d }
C_{T,dd
}= max[0, (1+d)^{2}S_{ T2} – K] 





In Diagram (b) CT,uu stands, for the
value of a call two periods from the current time if the stock price
moves upward each period; CT,dd for the stock price moves downward in
both periods and CT,ud for one period going up and another period going
down from current time here as well CT,du and CT,ud are giving same
answers.
Now to find out the value of a call at current time CT2 we start our
calculation from left to right. So to get the values of CT,uu, CT,ud and
CT,dd we will just get the difference between ST2 and the strike price.
Now the price of the CT1,u and CT1,d can be determined by using the
one period model. In other word pretend current time is T1 time and
stock goes uptick or downtick. As said before the answer for both CTdu
and CTud is same so modified formula (3) for this situation will be: 





C_{T1,u }
= [pC_{T,uu }+ (1 – p)C_{T,ud}]/(1+r)
And
(4)
C_{T1,d }
= [pC_{T,du }+ (1 – p)C_{T,dd}]/(1+r)






Again, a portfolio with ST2 in stock
and B in bonds whose endofperiod value will be CT,uu if the stock
price goes to ST,uu and CT,dd if the stock price goes to ST,dd. The
functional form of and B remains unchanged so to get the new values of
and B, simply use same steps as on to get equation (1) and change the
CT,u and CT,d with the new values of CT1,u and CT1,d respectfully.
So now portfolios offer the exact same payoffs at time T1. It means
they are selling at same price at T1. If they are selling at same price
T1 then they must sell for the same price at time T2. So now we have






C _{T2}_{
= }
DS_{T2}
+
B 





If value of and B is replaced in above
equation: 





C_{T2}
= [p^{2}C_{uu }
+ 2p(1 – p)C_{ud }+ (1 – p)^{2}C_{dd}]/(1+r)^{2
}(5) 





Where 





And 





A little algebra shows that this is
always greater than S – K if, as assumed, r is always greater than one,
so this expression gives the exact value of the call. 





Multi Period BOPM 


If the about formula (3) and formula (5)
are observed then it can be seen that there is only one thing change in
the formula(3) which led to formula (5) and that is number of period n.
Now if the formula is statistically changed for any n period of time, by
starting at the expiration date and working backwards, the general
valuation formula can be determined for any n: 





(6) 





Where j is how many ways can an
underline asset’s price reach a terminal value in a binomial process. In
other word we say in n periods, how many ways can the stock realize j as
an uptick. 





Now if let a = minimum number of upticks
needed to result in the call finishing in the money then if j ≤ a then
call is worthless and value of the call became 





(6) 





This gives us the complete formula, but
with a little additional effort we can express it in a more convenient
way. 




BlackScholes’ Option Pricing
Model 





Introduction 


Now let take new step to words
continuoustime version of binomial model by making the time periods
smaller and smaller which tend n to be larger this is the basic insight
of the BlackScholes’ model. In this continuoustime version 





• The individual interest payments
become continuous compounding;
• The random walk becomes geometric Brownian motion;
• The binomial distribution of the number of price rises becomes a
normal distribution. 





In binomial pricing model price movement
was specified by the side of the price factors u and d. In the
continuoustime black scholes’ model price movement is specified as
drift parameter µ and the volatility parameter price σ. Under the
standard assumption of no arbitrage 





µ
= r −σ
2 





(Thomas, 2003,STX 2020) 





Assumptions 


[1.] Perfect Market
a. Capital Market is perfect. There is no transaction cost and
taxes. There is no restriction on short selling. All assets are
infinitely divisible.
b. All investors car borrow and land money at same riskless interest
rate, which is constant over the life of the option.
c. Market is always open, full liquidity on buying and selling the
contract and it is continuous.
[2.] Volatility
The volatility of the stock is accurately known and it is constant
for the life time of the option contract.
[3.] Continuous time
One of Black and Scholes's main ideas was to work within a
continuous timeframe. As a result, they obtained differential equations,
which are easier to work with than discrete equations.
[4.] Brownian motion
The second key assumption was that relative price movements should
be represented by Brownian motion. In fact, this was a default option
because Brownian motion is the most "natural" process within a
continuous time framework.
[5.] Arbitragefree condition
The arbitragefree condition (or absence or arbitrage) states that
it is not possible to gain for sure without an initial outlay. Although
this precept may seem trite, it is in fact the cornerstone of the whole
argument. 





The Model 





C=
SN(d_{1}) – Ke^{rT}N(d_{2}) 





Where S = The price of the underlying
asset
K = The strike price of the call Option
r = risk free interest rate
T = time to expiration
N(d) = the cumulative standard normal distribution function
= the standard deviation of the underlying asset’s return
In(S/K) = the natural logarithm of S/K
erT = the exponential function of –rT. erT is the present value of the
factor when r is continuously compounded for T period of time KerT is
the present value of K. 











The model assumes that there value of σ
and r remains constant. T is the number of months to expire. 





Interpretation
of N(d1) and N(d2) 


The N(d1) and N(d2) terms are the
cumulative probability functions which take into account the risk of the
option being exercised. N(d1) is the cumulative probability relating to
the current value of the underline assets it indicates by increasing one
unit to the price how much the risk premium increases. The value of
N(d1) lies between 0 and 1. If option is deeply outofthemoney, then
any unit rise in the price of underline asset will have little effect on
the value of the call since it remains unlikely that the option will be
exercised. If the option is currently atthemoney, then values of N(d1)
will be 0.5 as there is a 50 per cent chance it will end up inthemoney
and a 50 per cent chance it will end up outofthemoney which means
underline asset’s price increase by one unit price increase by 0.5
units. If the option is already deepinthemoney, then unit rise in the
price will have same unit rise in option price, and the values N(d1)
will get closer to 1. The higher the stock price in relation to the
price, the higher the value of N(d1). The value of the N(d1) is closely
to value of delta of the stock.
N(d2) says the cumulative probability relating to the exercise price, it
is the probability of call option will be actually exercised if N(d2) is
0.70 then there is a 70 percent chance that the option will be
exercised. Normally the value of N(d1) is greater then N(d2) but when it
is certain that option will be exercised then the values of both N(d1)
and N(d2) is 1. 





The Volatility 


The most controversial thing in the
model is to measure volatility. Ideally, to get the an efficient pricing
of option from this model one has to measure volatility that is likely
to reflect the volatility that will occur in the future. There are three
major way of calculating volatility: 





• Implied volatility
• Expected volatility
• Historical volatility 





Expected Volatility: The Problem with
Expected volatility is; it differs from one market participant to
another, and therefore the view of the appropriate market price of an
option will vary between market participants. So there won’t be standard
price of the option.
Implied volatility: Implied volatility is the volatility implicit in the
current option price, this is found by taking the current price of the
option. When this volatility plugged into the option pricing then
formula gives the current market price of the option. So it is useless.
Historical volatility: Historical volatility may be a useful measure for
this purpose but it could prove to be defective as the past is not
necessarily a good guide to the future. In addition, there are different
ways to get the historical volatility like on the last month, the last
three months, the last 6 months or last year, which one should be taken?
The answer for this was the standardisation. The method used to
calculate this is the annualized standard deviation of daily, weekly or
even monthly changes in prices. The annualized price volatility is
obtained by multiplying the calculated sample standard deviation by the
number of periods. For daily data (based on 252 trading days per annum): 





σ =
x
daily standard deviation
for weekly data,
σ =
x
weekly standard deviation
for monthly data,
σ =
x
weekly standard deviation 




Conclusions 




Futures Pricing Analysis 





While futures market has a reputation
for high risk and wild price swings, we cannot deny that prices vary
suddenly and sharply. It is very difficult to measure the real price of
the futures contract but it is quite possible to measure range of the
nonarbitrage price with the theories that have been covered. Both the
CostofCarry and the Hedging Pressure Theory provide rational
procedures for thinking about the behaviour of futures price. It must
also be admitted that futures prices, on the whole, do not only conform
to these theories there many other fundament, technical and
psychological factors do have impact on the price with great extend.
After Analysising and researching on the theories, I found that some
market follow more over to one particular theory. Below Table gives
examples of which contract follow which market. 





Capital Asset Pricing Model (CAPM) Theory 
The
Hedging Pressure Theory 
The
Cost of carry Model Theory 
Example :
INDEX
·
NASDAQ 100 Futures
·
Nikkei 225 Futures
·
S&P 500 Futures 
Example:
FOOD/FIBER
, GRAIN/OILSEED
·
Flaxseed WCE
·
Oats CBOT
·
Rice Rough
·
Soybeans ecbot
·
Coffee
·
Coffee Mini
·
Orange Juice
·
Sugar #11

Example:
METAL, INTEREST RATE
·
Gold
·
Silver
·
Eurodollar CME
·
Eurodollar GBX
·
TNote 5
·
Yr
CBOT 






Capital Asset Pricing Model (CAPM) is
more suitable for the stock futures as the concept of capital asset
pricing model is to measure and managing the systematic risk of the
single stock/ a portfolio of stock in comparison with market portfolio.
It is also handy with as Equity futures index as Equity futures index is
a portfolio of many no. of shares.
The Hedging Pressure Theory is mostly used for market where futures
price must consider convenience yield. The commodities contract on
wheat, orange juice, Crude oil, soybean, corn, coco, coffee, meat which
comes under the basic need of the human being can be valued very well
with this theory. It is because this is the only theory that takes
consideration of speculator’s point of view for coming in the market and
taking the unwilling risk.
The Cost of carry Model is the widely used model all over the world for
pricing financial instrument like futures contract, Interest rate
contract, currency forward contracts stock futures contracts. It is most
accurate on nonincome generating storable assets like Gold, silver,
platinum etc. 




Option Pricing Analysis 





Option pricing is a relatively complex
area. There are some crucial assumptions that need to be made for a
valid application of the BlackScholes pricing formula and Binomial
pricing model. But I found that Binomial is more practical and has
better predicting capability then BlackScholes Model. 





The Assumptions like stated below makes
the BlackScholes model weaker in terms of predicting the option price: 





• Volatility: In the real market the
market participant measures the volatility by strike price of the
option, not by the statistical estimates so it varies from participant
to participant.
• Continuous time: The model assumes continuous readjustments, but in
practice traders can readjust only at discrete intervals.
• American Option: The formula we have looked at is only applicable to
European options,
• LogNormal Distribution: The formula assumes that the log of the share
price follows a lognormal distribution. In the real world,
distributions tend to have fatter tails than a normal distribution,
meaning that there are better chances of an option being exercised that
suggested by the BlackScholes formula. Hence, in real world option
prices are exceeding the BlackScholes formula price. 





Binomial pricing model presents much
greater transparency to the users of the Options prices. The Binomial
Option has been able to leave some of these assumptions to get the
realistic price of the Options. They are: 





• Volatility: The Upward/downward stock
movements are governed by underlying asset’s volatility, and the model
can use a term structure of volatility, a different volatility during
each distinct measurement period (i.e. implied volatility during first 3
months and longterm volatility thereafter)
• Continuous time: The BlackScholes model assumes continuous
readjustments, but in practice traders can readjust only at discrete
intervals. The Binomial model calculates option values during each
distinct measurement period the way calculated in real world.
• RiskFree rate: The model can use different riskfree rate of return
during each distinct measurement period. 





Because of these benefits the binomial
model indeed give better result then BlackScholes’ model, in fact the
binomial model can also describe the insight of the BlackScholes model.
If we take n to infinity then the binomial model becomes more over same
like blackScholes’ model and results of that has same problem which we
found in Blackscholes model. 




Next Step 





With the limitation of researching time
and limitation of words, I was able to cover only the basics of the
derivative markets. 




Options Pricing 


The next step towards analysis of the
option pricing would be the consideration of the dividend in the stock
market for stock option, Analysising the put call parity, Blackscholes
model for American call, Uses of Greek letters and analysis of Option
strategies for risk management and mathematic example for all the
theories. 




Futures Pricing 


The next step toward analysis of the
Futures/Forward contract pricing would be having more understanding of
interest rate futures contacts, analysising changes in formulas of
costofcarry model whilst interest rate changes frequently and
mathematic example for all the theories. 




Reuter 3000 Xtra 


Because of unable to get the software
called Reuter 3000 Xtra for real market quotes and news, it was
difficult to explain the real market pricing. With the help of this
software in depth real market price analysis and real market examples
would have been possible. 




References 





John C. Hull (2004). Options Futures and
Other Derivatives. (5th Ed.) PrenticeHall of India.
Anthony Saunders and Marcia Millon Cornet. Financial Institutions
Management. (4th Ed.) McGrawHill Irwin
Robert T Daigler, (1993). Financial futures markets: concepts, evidence,
and applications HarperCollins College Publishers.
Clewlow, Les, Strickland, Chris, Enron Corp (2000). Energy derivatives:
pricing and risk management Lacima
Robert W. Kolb (1997). Understanding futures markets. Blackwell.
Stein, Leon Jerome (1986) The economics of futures markets Blackwell.
Keith Pilbeam (1998), Finance & Financial Markets Macmillan
Robert T Daigler, (1994). Financial futures markets: concepts, evidence,
and applications HarperCollins College Publishers.
David A. Dubofsky (1992) Options and Financial Futures Valuation and
Uses, McGramHill, Inc.
John C. Cox, Stephen A. Ross, Mark Rubinstein (1979) , Option Pricing :
A Simplified Approch Journal of Financial Economics
Thomas Bending 2002, Lectures notes STX 2210, Middlesex University
Websites
http://www.nybot.com/educationalResources/ brochures/files/underst.pdf
http://sify.com/finance/equity/fullstory.php?id=13211183
http://www.abnamroprivatebanking.com/
http://www.meff.com/
http://www.wikipedia.org/
Glossary (http://biz.yahoo.com/f/g/aa.html) 
