Guide Study 2003/2004 – AI Modes in Options Pricing
Department of Computer Science
MSCS 2003/2004
Guide Study
AI Models in Options Pricing
Supervisor:
Dr. Andy Chun
Student: Hou Kwai Cheung, Brian
Student Number: 50355568
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Guide Study 2003/2004 – AI Modes in Options Pricing
Content
1. What is Options?
1.
2.
3.
4.
5.
6.
Options in Everyday Life
Why Options is Useful?
The Basics of Calls
The Basics of Puts
A Comparison of Calls and Puts
Options Price Levels
2. Vocabulary and Definitions
1.
2.
3.
4.
Expiration date
Exercise price (Strike price)
Bid-ask
Aspects of premium
i.
intrinsic value
ii.
time value
5. Moneyness
6. Duration and time decay
7. Long and short options position
8. Writing Options: Covered and Uncovered
9. European versus American style
10. Volatility and pricing models
i.
Historical volatility
ii.
Future Volatility
iii.
Implied Volatility
iv.
Forecasted Volatility
11. Geometric Brownian Motion (Random Walk)
12. Standard Wiener Process
13. Monte Carlo Methods
3. Current Modeling Method on Options Pricing
1.
2.
3.
4.
Black-Scholes Pricing Models
The Garch option Pricing Model
Implied Binomial Tree
Stochastic-Volatility Option Pricing
4. Intelligent System in Business: The Key Features
1. Learning
2. Adaptation
3. Flexibility
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4. Explanation
5. Discovery
5. Introduction to Intelligent Techniques (mainly on Neural Networks)
1. Neural Networks
2. Other Intelligent Techniques
• Genetic Algorithms
• Fuzzy Systems
• Expert Systems
3. Intelligent Hybrid Systems
6. Comparisons between Neural Nets and Other Time Series Methods
7. The Proposed AI Modeling on Options Pricing
8. Difficult in Training
1. Convergence
2. Optimal Stopping Point
3. Number of hidden nodes
9. Conclusion
10. Reference
11. Appendix
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Guide Study 2003/2004 – AI Modes in Options Pricing
What is Options?
The difference between a commodity, a futures contract, and an options contract is illustrated in
the following three paragraphs, which will take you 1.5 minutes to read.
Suppose you’re in the market for an oriental rug. You find the rug of your choice at a local shop,
you pay the shopkeeper $500, and he transfers the rug to you. You have just traded a commodity.
Suppose instead you wish to own the rug, but you prefer to purchase it in one week’s time. You
may be on your way to the airport, or maybe you need the short-term use of your money. You
and the shopkeeper agree, verbally or in writing, to exchange the same rug for $500 one week
from now. You have just traded a futures contract.
Alternatively, you may like the rug on offer, but you may want to shop around before making a
final decision. You ask the shopkeeper if he will hold the rug in reserve for you for one week. He
replies that your proposal will deny him the opportunity of selling the rug, and as compensation,
he asks that you pay him $10. You and the shopkeeper agree, verbally or in writing, that for a fee
$10 he will hold the rug for you for one week, and that at any time during the week you may
purchase the same rug for a cost of $500, excluding the $10 cost of your agreement. You, on the
other hand, are under on obligation to buy the rug. You have just traded an options contract.
A.
Options in Everyday Life
Puts
For example, most of us insure our home, our car and our health. We protect these, our assets, by
taking out policies from insurance companies who agree to bear the cost of loss or damage to
them. We periodically pay these companies a fee, or a premium, which is based in part on the
value of our assets and the duration of coverage. In essence, we establish contracts that transfer
our risk to the companies.
If by accident our assets suffer damage and a consequent loss in value, our contract gives us the
right to file a claim for compensation. Most often we exercise this right, but occasionally we may
not, for example, if the damage to our car is small, it has been incurred by our teenage son, and
filing a claim would produce an undesirable rise in our future premium level.
Upon receipt of our payment we might say that the cost of our accident has been ‘put to’ the
insurer by us. In effect, our insurance company had sold us a put option which we owned, and
which we have exercised.
In financial markets ‘puts’, as they are called, operate similarly. Pension funds, banks,
corporations and private investors have assets in the form of stocks and bonds that they
periodically protect against a decline in value. They do this by purchasing put options based on,
or derived from, their stocks and bonds. These options give them the right to put the amount of
an asset’s decline onto the seller of the options. They transfer risk.
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Calls
Suppose we need to purchase a washing machine. In our local newspaper we see an
advertisement for the machine that we want. It is ‘on sale’ at a 20 per cent discount from a local
retailer until the end of the week. We know this retailer to be reputable and that no tricks or
gimmicks are involved.
From our standpoint we have the right to buy this machine at the specified price for the specified
time period. We may not exercise this right if we find the machine cheaper elsewhere. The
retailer, however, has the obligation to sell the machine under the terms specified in the
advertisement. In effect, he has entered into a contract with the general public.
If we decide to exercise our right, we simply visit the retailer and purchase our washing machine.
We might say that we have ‘called away’ this machine from the retailer. He had given us a call
option which we had accepted and which we have exercised. In this case out option is
commonly known as ‘call’. It was given to us as part of the general public, free of charge. The
retailer bore the cost of the call because he had a supply of washing machines that wanted to sell.
Suppose we visit his store within the week and find that all washing machines have been sold.
The retailer underestimated the demand that the advertisement generated, and he is now short of
supply. He and his sales staff are anxious to meet the demand, and he has his good reputation to
uphold. Then, the retailer will try to rush delivery from a distributor, even at additional cost to
him. If no machines are available through the distribution network, he will give us a voucher for
the purchase of our machine when more arrive.
This voucher is, again, a call option. It contains the right to buy at the sale price, but its duration
has been extended. If in the meantime the factory or wholesale price of our machine rises, the
retailer will still be obligated to sell it to us at the sale price. His profit margin will be cut, and he
may even take a loss. The call option that he gave us may prove costly to him.
Suppose that we become enterprising with our voucher, or call option. Early the next week we
are talking to our neighbor who expresses disappointment at having missed the sale on washing
machines. The new supply has arrived, and the new price is above the old, pre-sale price. By
missing the sale, he will need to pay considerably more than he would have paid. We, after
refection, decide that we can live with out old machine. We offer to sell him a new machine for
an amount less than the new retail price but more than the old sale price. He accepts our offer.
We then return to the retailer, exercise the option, purchase the machine, and resell it to our
neighbor. He has a saving and we have a profit. We are now options traders.
Calls are a significant feature of commodity markets, where supply shortages often occur.
Adverse weather, strikes, or distribution problems can result in unforeseen rises in the costs of
basic goods. Petroleum manufacturers, transport companies, and grain distributors regularly
purchase calls in order to ensure that they have the commodities necessary to meet output
deadlines.
B.
Why Options is Useful?
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It is an unfortunate and costly reality that few investors know how to protect their investments
from downside risk. Their role investment strategy is to select a stock to buy, or a fund to buy
into. Over the long term, and if value is found at the time of purchase, this strategy makes sense.
It also makes sense for those who entrust the management of their portfolios to financial advisers.
But for those who take a more active role in their investments, options offer the two advantages
of flexibility and limited risk.
For those new to day-trading the markets, call and put purchases are excellent ways of
developing market awareness and building confidence. This is because with these strategies
traders can take either a bullish or bearish position while limiting their maximum loss at the
outset. Because the cost of options is paid for up front on most exchanges, the options buyer is
forced to be more disciplined than traders who must simply post margin.
Options are extremely popular among sophisticated investors who hold large stock portfolios.
Accordingly, institutional investors, such as mutual funds and pension funds, are prime users of
the options markets. By trading options in conjunction with their stock portfolios, investors can
carefully adjust the risk and return characteristics of their entire investment. A sophisticated
trader can use options to increase or decrease the risk of an existing stock portfolio. It is possible
to combine a risky stock and a risky option to form a riskless combined position that performs
like a risk-free bond. Moreover, investors prefer to trade options rather than stocks in order to
save transaction costs, to avoid tax exposure, and to avoid stock market restrictions.
Because options have lives of their own, they are indicators of market sentiment. Implied
volatility often anticipates changes in price activity in the underlying contracts. Simply knowing
about options can improve your market awareness.
The trade in options contracts continues to grow because more and more companies and
individuals need them to manage risk. Their needs are essentially very simple: the right to buy
with calls, and the right to sell with puts.
C.
The Basics of Calls
We saw that options are used in association with a variety of everyday items from which they
derive their worth. For instance, the value of our house determines, in part, the amount of our
insurance premium. In the options business, each of these items is known as an underlying asset,
or simply an ‘underlying.’ It may be a stock or share, a bond, or a commodity. Here, in order to
get started, we will discuss an underlying with which we are all familiar, namely stock, bond, or
commodity XYZ.
Owning a call
XYZ is currently trading at a price of 100. It may be 100 dollars, euros, or pounds sterling.
Suppose you are given, free of charge, the right to buy XYZ at the current price of 100 for the
next two months. If XYZ stays where it is or if it declines in price, you have no use for your right
to buy; you can simply ignore it. But if XYZ rises to 105, you can exercise your right: you can
buy XYZ for 100. As the new owner of XYZ, you can then sell it at 105 or hold it as an asset
worth 105. In either case, you make a profit of 5.
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What you do by exercising your right is to ‘call XYZ away’ from the previous owner. Your
original right to buy is known as a call option, or simply a ‘call.’
It is important to visualize profit and loss potential in graphic terms. Below is a profit/loss graph
of your call, or call position, before you exercise your right.
Owning a call for XYZ
12
10
Profit/Loss
8
6
4
2
0
95
100
105
110
XYZ Value
If you choose, you can wait for XYZ to rise further before exercising your call. Your profit is
potentially unlimited. If XYZ remains at 100 or declines in price, you have no loss because you
have no obligation to buy.
Offering a call
Now let’s consider the position of the investor who gave you the call. By giving you the right to
buy, this person has assumed the obligation to sell. Consequently, this investor’s profit/loss
position is exactly the opposite of yours.
The risk for this investor is that XYZ will rise in price and that it will be ‘called away’ from him.
He will relinquish all profit above 100. In this case, below represents the amount that is given up.
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Guide Study 2003/2004 – AI Modes in Options Pricing
Offering a call for XYZ
0
95
100
105
110
-2
Profit/Loss
-4
-6
-8
-10
-12
XYZ Value
On the other hand, this investor may not already own an XYZ to be called away. He may need to
purchase XYZ from a third party in order to meet the obligation of the call contract. Your
potential gain is his potential loss.
Buying calls
Obviously, the investor who offers a call also demands a fee, or premium. The buyer and the
seller must agree on a price for their call contract. Suppose in this case the price agreed upon is 4.
A correct profit/loss position for the buyer, when the call contract expires, would be graphed as
below: -
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Guide Study 2003/2004 – AI Modes in Options Pricing
Buying a call for XYZ
8
6
Profit/Loss
4
2
0
95
100
105
110
-2
-4
-6
XYZ Value
By paying 4 for the call option, the buyer defers his profit until XYZ reaches 104. At 104 the call
is paid for by the right to buy pay 100 for XYZ. Above 104 the profit from the call equals the
amount gained by XYZ. Between 100 and 104 a partial loss results, which equal to the difference
between 4 and any gains in XYZ. Below 100 a total loss of 4 is realized. A corresponding table
of this profit/loss position at expiration is tabulated as below: XYZ
95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
Cost of
-4 -4 -4 -4 -4
-4
-4
-4
-4
-4
-4
-4
-4
-4
-4
-4
call
Value of
call at
0 0 0 0 0
0
1
2
3
4
5
6
7
8
9 10
expiration
Profit/Loss -4 -4 -4 -4 -4
-4
-3
-2
-1
0
1
2
3
4
5
6
All options contracts, like their underlying contracts, have contract multipliers. Both contracts
usually have the same multiplier. If the multiplier for the above contracts is $100, then the actual
cost of the call would be $400. The value of XYZ at 100 would actually be $10,000. In the
options markets, prices quoted are without contract multipliers.
When trading options, it is important to know the risk/return potential at the outset. In this case,
the potential risk of the call buyer is the amount paid for the option, 4 or $400. The call buyer’s
potential return is the unlimited profit as XYZ rises above 104.
Calls can be traded at many different strike prices. For example, if XYZ were at 100, calls would
probably be purchased at 105, 110 and 115. They would cost progressively less as their distance
from the current price of XYZ increased. Many investors purchase these ‘out-of-the-money’
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calls, as they are known, because of their lower cost, and because they believe that there is
significant upside potential for the underlying.
Our 100 call, with XYZ were at 100, is said to be ‘at the money’.
In addition, if XYZ were at 100, calls could also be purchased at 95, 90, and 85. There ‘in-themoney’ calls, as they are known, cost progressively more as their distance from the underlying
increases. Where the underlying is a stock, many investors purchase these calls because they
approximate price movement of the stock, yet they are less expensive than a stock purchase. For
both stocks and futures, the limited loss feature of these calls also acts as a built-in stop-loss
order.
To summarize, a call is used primarily as a hedge for upside market movement. It is also used to
hedge downside exposure as an alternative to buying the underlying. The buyer and the seller of
a call contract have opposite views about the market’s potential to move higher. The call buyer
has the right to buy the underlying asset, while the call seller has the obligation to sell the
underlying asset. Because the call seller incurs the potential for unlimited loss, he must demand a
fee that justifies this risk. The call buyer can profit substantially from a sudden, unforeseen rise
in the underlying. When exercised, the buyer’s right becomes the seller’s obligation.
D.
The Basics of Puts
Put options operate in essentially the same manner as call options. The major difference is that
they are designed to hedge downside market movement. Some common characteristics of puts
and calls are as follows: —
—
—
—
—
—
The buyer purchases a right from the seller, who in turn incurs a potential obligation.
A fee or premium is exchanged
A price for the underlying is established.
The contract is for a limited time.
The buyer and the seller have opposite profit/loss positions.
The buyer and the seller have opposite risk-return potentials.
A put option hedges a decline in the value of an underlying asset by giving the put owner the
right to sell the underlying at a specified price for a specified time period. The put owner has the
right to ‘put the underlying to’ the opposing party. The other party, the put seller, consequently
incurs the potential obligation to purchase the underlying.
Buying puts
Suppose you own XYZ, and it is currently trading at a price of 100. You are concerned that XYZ
may decline in value, and you want to receive a selling price of 100. In other words, you want to
insure your XYZ for a value of 100. You do this by purchasing an XYZ 100 put for a cost of 4. If
XYZ declines in price, you now have the right to see it at 100.
Let’s consider the profit/loss position of the put itself. At expiration, this position would be
graphed as below: -
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Buying a put for XYZ
5
4
3
Profit/Loss
2
1
0
92
96
100
104
-1
-2
-3
-4
-5
XYZ Value
This graph should appear similar to the graph for a call purchase. In fact, it is the identical
profit/loss but with a reverse in market direction. Both graphs show the potential for a large
profit at the expense of a small loss. Here, profit is made as the market moves downward rather
than upward. In tabular form, this profit/loss position would be as below: XYZ
90 91 92 93 94
Cost of put -4 -4 -4 -4 -4
Value of
put at
10 9 8 7 6
expiration
Profit/Loss 6 5 4 3 2
95
-4
96
-4
97
-4
98
-4
99 100 101 102 103 104 105
-4
-4
-4
-4
-4
-4
-4
5
4
3
2
1
0
0
0
0
0
0
1
0
-1
-2
-3
-4
-4
-4
-4
-4
-4
The break-even level of this position is 96. There, the cost of the put equals the profit gained by
the right to sell XYZ at 100. Between 100 and 96 the cost of the put is partially offset by the
decline in XYZ. Above 100, the premium paid is taken as a loss. Below 96 the profit on the put
equals the decline of XYZ.
As the owner of XYZ, your loss is stopped at 96 by your put position. The cost of the put has
effectively lowered your selling price to 96. But if XYZ falls sharply, you have a substantial
saving because you are fully protected. In other words, you are insured. In the meantime, you
still have the advantage of potential profit if XYZ gains in price.
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The purchase of a put option can be profitable in itself. Suppose that you do not actually own
XYZ, but you follow it regularly, and you believe that it is due for a decline. Just as you may
have purchased a call to capture an upside move, you now may purchase a put to capture a
downside move. Your advantage, as an alternative to taking a short position in the underlying, is
that you are not exposed to unlimited loss if XYZ moves upward. The most you can lose is the
premium paid.
Again, note the risk/return potential. With a put purchase the potential risk is the premium paid,
4. The potential return is the full amount that XYZ may decline below 96.
Selling puts
Now let’s consider the profit/loss position of the investor who sells the XYZ put. After all, you
may decide that the put sale is the best strategy to pursue. Because the put buyer has the right to
sell the underlying, the put seller, as a consequence, has the potential obligation to buy the
underlying.
At expiration, the sale of the XYZ 100 put for 4 would be graphed as below: Selling a put for XYZ
5
4
3
Profit/Loss
2
1
0
92
96
100
104
-1
-2
-3
-4
-5
XYZ Value
This position should appear similar to that of the call sale. In fact, the profit/loss potential is
exactly the same, but the market direction is opposite, or downward.
In tabular form, this profit/loss position would be as shown below: XYZ
90
91
92
93
94
95
96
97
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98
99
100
101
102
103
Guide Study 2003/2004 – AI Modes in Options Pricing
Income
from put
Value of
put at
expiration
Profit/Loss
4
4
4
4
4
4
4
4
4
4
4
4
4
4
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
0
0
0
6
5
4
3
2
1
0
-1
-2
-3
-4
-4
-4
-4
The put seller’s potential return is a maximum of 4 if XYZ remains at or above 100 when the
contract expires. Between 100 and 96, a partial return is gained. 96 is the break-even level.
Below 96, the put seller incurs a loss equal to the amount that XYZ may decline.
Again, the risk/return potential for the put seller is exactly opposite to the put buyer. The
potential return of the put sale is the premium collected, 4. The potential risk is the full amount
that XYZ may decline below 96.
An investor may wish to purchase XYZ at a lower level than the current market price. As an
alternative to an outright purchase, he may sell a put and thereby incur the potential obligation
to purchase XYZ at the break-even level. The advantage is that he receives an income while
awaiting a decline. The disadvantage is that XYZ may increase in price, and he will miss a
buying opportunity, although he retains the income form the put sale. The other disadvantage is
the same for all buyers of an underlying: XYZ may decline significantly below the purchase price,
resulting in an effective loss.
For the investor who has a short position in XYZ, the sale of a put gives him the advantage of an
income while he maintains his short position. The disadvantage is that he may give up downside
profit if he must close his short position through an obligation to buy XYZ.
Practically specking, there are few investors who adopt the latter strategy, although many
market makers do, simply because they supply the demand for puts.
It is obvious that as with calls, the greater risk of trading puts lies with the seller. He may be
obligated to buy XYZ in a declining market. The put seller must therefore expect XYZ to remain
stable or slightly higher. He must demand a fee that justifies the downside risk.
To summarize, a put option is the right to sell the underlying asset at a specified price for a
specified time period. The put buyer has the right, but not the obligation, to sell the underlying.
The put seller has the obligation to buy the underlying at the put buyer’s discretion.
E.
A Comparison of Calls and Puts
The call buyer has the right to buy the underlying; consequently the call seller may have the
obligation to sell the underlying.
The put buyer has the right to sell the underlying; consequently the put seller may have the
obligation to buy the underlying.
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Options
Call
Put
Buyer, who long
Right to buy the underlying
Right to sell the underlying
Seller, who short
Oblige to sell the underlying
Oblige to buy the underlying
If the underlying is a futures contract, the above terms are modified.
The call buyer has the right to take a long position in the underlying; consequently the call
seller may have the obligation to take a short position in the underlying.
The put buyer has the right to take short position in the underlying; consequently the put seller
may have the obligation to take a long position in the underlying.
If these statements seem confusing, bear in mind that they are related to each other by simple
logic: if one is true, then the others must be true.
To summarize, markets can be bullish, bearish, or range-bound, and different options strategies
are suitable to each. Any particular strategy cannot be said to be better then any other. These
strategies, and those that follow, vary in terms of their risk/return potential. They accommodate
the degree of risk that each investor thinks is appropriate. It is the flexible and limiting approach
to risk that makes options trading appropriate to many different kinds of investors.
F.
Options Price Levels
Let’s begin with a straightforward options contract. Below is the Eurodollar futures contract: Strike Price
Call value
Put value
93.50
0.80½
-
93.75
0.56
0.01
94.00
0.32
0.02
94.25
0.12
0.06½
94.50
0.04
0.23
94.75
0.02
0.46
95.00
0.01
0.70
Suppose on this day the December futures contract settled as 94.30½, or an equivalent interest
rate of 5.695 per cent. As the interest rate falls, the futures contract increases; as the interest rate
rises, the price of the futures contract decreases. An investor wishing to hedge a rise in the
interest rate to 6 per cent could pay 0.02 for the 94.00. An investor wishing to hedge a fall in the
interest rate to 5.5 per cent could pay 0.04 for the 94.50 call. The contract multiplier is $25,
which means that the 94.50 call has a value of 4 x $25, or $100. There are 132 days until the
options contracts expire on December 14.
The number of different options contracts listed is designed to accommodate investors with
different levels of interest rate exposure. Each listed price level is known as a strike price, e.g.,
94.00, 94.25, 94.50, etc.
When an option is closest to the underlying, it is termed at-the-money (ATM). Here, both the
94.25 call and the 94.25 put are at-the-money. When a call is above the underlying, it is termed
out-of-the-money (OTM), e.g., at the calls at 94.50, 94.75 and 95.00. When a put is below the
underlying, it is also out-of-the-money, e.g. the puts at 93.75 and 94.00.
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When a call is below the underlying, it is termed in-the-money (ITM), e.g. the calls at 93.75 and
94.00. When a put is above the underlying, it is also in-the-money, e.g. all the puts at 94.50,
94.75 and 95.00.
Generally speaking, the options most traded are those at-the-money or out-of-the-money. If an
upside hedge is needed, then at-the-money, or out-of-the-money calls will work, and they are
less costly than in-the-money calls. For a downside hedge, the same reasoning applies to puts.
Vocabulary and Definitions
To a large extent, understanding options hinges on understanding their specialized vocabulary.
The following key terms cover the basics: 1. Expiration date
All options have a known, finite, life that ends on their expiration date.
2. Exercise price (Strike price)
Exercise price is the price at which the buyer of an option obtains the right to buy or sell the
underlying security.
3. Bid-ask
When posting option quotes, professional traders must give a two-sided market: a price at which
they are willing to purchase the option (the bid) and a price at which they are willing to sell the
option (the ask, or the offer). A buyer who purchases on option at the asked price is said to take
out (or lift) the offer. A seller who accepts the posted bid is said to hit the bid. The bid-ask
spread refers to the difference between these two prices, as in: “The spread on that option is too
wide.”
4. Aspects of premium
Premium is the price of an option, paid by the buyer, received by the writer. Quoted on a per
share basis, an option’s premium is composed of intrinsic and time value. Intrinsic value is the
amount by which an option is in-the-money. This is also known as the exercise value.
The premium of an option corresponds to its probability of expiring in the money. The 94.75call
and the 94.00 put are each worth only 0.02 because most likely the underlying will not reach
these levels before expiration. More specifically, the 0.02 value of each of these is termed the
time premium.
The premium of an in-the-money option consists of two components. The first of these is the
amount equal to the difference between the strike price and the price of the underlying, and it is
termed the intrinsic value. The second component is the time premium. The 94.00 call, with
the underlying at 94.30½, is worth 0.32; it has an intrinsic value of 0.30½ and contains a time
premium of 0.01½.
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When an option is deeply in the money, it will trade as a proxy for the underlying, and its
premium will consist of intrinsic value only. This kind of option is said to be at parity with the
underlying. The 93.50 call, with a value of 0.80½, is at parity with the underlying at 94.30½.
An at-the-money option will contain the most time premium because there two advantages to
owning an option are equal and greatest. A call that is exactly at-the-money, whose strike price
equals the price of the underlying, can profit fully from upside market movement, less the cost of
the call. As an alternative to purchasing the underlying, it can also save the call buyer the full
amount that the underlying may decline, less the cost of the call. With an at-the-money call, the
potential profit theoretically equals the potential savings. An at-the-money put has the same
profit/savings potential.
5. Moneyness
"Moneyness" is an option concept that refers to the potential profit or loss from the immediate
exercise of an option. An option may be in-the-money, out-of-the-money, or at-the-money.
A call option is in-the-money if the stock price exceeds the exercise price. For example, a call
option with an exercise price of $100 on a stock trading at $110 is $10 in-the-money. A call
option is out-of-the-money if the stock price is less than the exercise price. For example, if the
stock is at $110 and the exercise price on a call is $115, the call is $5 out-of-the-money. A call
option is at-the-money if the stock price equals (or is very near to) the exercise price.
6. Duration and time decay
Another aspect that determines the amount of an option’s premium is, quite reasonably, the time
till expiration. A long-term hedge will cost more than a short-term hedge. Time decay, however,
is not linear. An option loses its value at an accelerating rate as it approaches expiration.
7. Long and short options positions
In practice, once a call or put is bought, it is considered to be a long options position. I’m long
10, June 550 puts,’ you might say. Conversely, a call or put sold is considered to be a short
options position. I’m too short for my own good,’ means that you have sold too many calls or
puts, or both, for your peace of mind.
It may be helpful to think that when the terms ‘long’ and ‘short’ are applied to options, they
designate ownership. The same terms applied to a position in the underlying designate exposure
to market direction. To be short puts is to be long the market, i.e. you want the market to move
upward.
8. Writing Options: Covered and Uncovered
Whenever an option is written, an obligation is assumed. If the option writer is in a position to
fully meet this obligation, the option will be considered covered. Otherwise, it will be treated as
uncovered, or naked.
9. European versus American style
An option is European style if it cannot be exercised before expiration. The only way to close
this style of option before expiration is to make the opposing buy/sell transaction. More
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prevalent is the American-style option, which can be exercised at any time before expiration. If
such an option becomes so deeply in-the-money that it trades at parity with the underlying, then
it has served its purpose and represents cash tied up. As a result, it can be sold, or it can be
exercised to a position in the underlying stock or futures contract.
Because American-style options can be exercised before expiration, those in-the-money will
often contain an additional early exercise premium. This is not a significant amount for most
options on future contracts. It is more significant for puts on individual stocks because they can
be exercised to sell stock and as a result, interest is earned on the cash. Conversely, because of
the potential for early exercise, long out-of-the-money or at-the-money positions can profit
significantly. As these options become in-the-money, their early exercise premium increases
dramatically.
10. Volatility and pricing models
The most sophisticated and the most significant aspect of options pricing is that of volatility.
After all, the primary purpose of options is to hedge exposure to market volatility. Increased
market volatility leads to increased options premiums, while decreased market volatility has the
opposite effect.
There are two types of volatility used in the options markets: the historical volatility of the
underlying, and the implied volatility of the options on the underlying.
Historical Volatility
The historical volatility describes the range of price movement of the underlying over a given
time period. If, for a certain time period, an underlying’s daily settlement prices are three to five
points above or below its previous daily settlement prices, then it will have a greater historical
volatility than if its settlement prices are one to two points above or below. Historical volatility is
concerned with price movement, not with price direction.
As its name implies, historical volatility looks at the past and tells us how volatile a stock (or an
index) has been over a given time frame. There is little room for argument here. A stock has
traded the way it has traded, and that is. You can’t argue with the past.
The only place where there is room for debate is in deciding which period represents the stock’s
history. By comparing volatility at a longer period (maybe a year) and a short period (maybe a
month), we can see if the stock is currently more or less volatile than its average. It should be
noted that for the more established stocks (blue chips), the historical volatility tends to be
relatively constant.
Future Volatility
How volatile will a stock actually be over the next three months? Unfortunately, we do not know
this until the next three-month comes. Then, the volatility will become historical one, not future
one.
Implied volatility
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Although a theoretical value for an option can be determined by the historical volatility, an
option’s market price is determined by supply and demand. An options market accounts for past
price movement, but it also tries to anticipate future price movement. The market price of an
option, then, implies a range of expected price movements for the underlying through expiration.
If we insert the market price of the option into the pricing model, and if we decide the former
historical volatility, the model substitutes another volatility number, the implied volatility of the
option.
This implied volatility can then be used as the implied volatility to calculate market prices of
options at other strike prices within the same contract month.
With the classic option pricing equation, given stock price, exercise price, time to expiration,
interest rate, and volatility, it is possible to calculate the pricing of the option. Since four of the
five, except the volatility, are empirically verifiable; and it is possible to obtain the option pricing
from what price the option is trading on one of the exchanges, therefore it is possible to solve for
volatility using the option premium and the four observable variables. This is how to determine
the implied volatility.
Forecasted Volatility
Implied volatility can be viewed as the market’s volatility forecast. Of course, anyone can
forecast future volatility. Just remember that a forecast is only a best guess and therefore contains
an element of subjectivity.
11. Geometric Brownian Motion (Random Walk)
A stock price is said to be under “geometric Brownian motion” when the transition from the
price at one time, t, to the next, t+1, is:
Price(t+1) = Price(t) * exp ( mu + 0.5 * sigma * Z )
Where mu = “drift”, sigma = “volatility” and Z is the value of a (0,1) under normal
random variable.
12. Standard Wiener Process
A continuous-time stochastic process W(t) for t >= 0 with W(0) = 0 and such that the increment
W(t) – W(s) is Gaussian with mean 0 and variance t – s for any 0 <= s < t, and increments for
nonoverlapping time intervals are independent. Brownian Motion, i.e. random walk with random
step sizes) is the most common example of a Wiener process.
13. Monte Carlo Methods
It is a method solves a problem by generating suitable random numbers and observing that
fraction of the numbers obeying some property or properties. The method is useful for obtaining
numerical solutions to problems, which are too complicated to solve analytically. One
application of the Monte Carlo method is Stochastic Geometry.
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Current Modeling Method on Options Pricing
A.
Black-Scholes Pricing Models
Once of the historical volatility is know, it becomes an input for an options pricing model. The
primary model used in the options industry is the Black-Scholes models; which is the most
classical one introduced in 1973. This model has been revised over the past decades or so in
order to price options on different underlying, but it remains the foundation of the business.
Began in the early 1970’s, the Chicago Board of Options Exchange started the trading of options
in exchanges, although options had been regularly traded by financial institutions in the over-thecounter markets previously. In 1973, Black and Scholes (1973) and Merton (1973) published
their seminal papers on the theory of option pricing. Since then the growth of the field of
derivative securities has been phenomenal. The Black-Scholes general equilibrium formulation
of the option pricing theory is attractive since the final valuation formulas deduced from their
model is a function of a few observable variables (except one, which is the volatility parameter)
so that the accuracy of the model can be ascertained by direct empirical tests with market data.
When judged by its ability to explain the empirical data, the option pricing theory is widely
acclaimed to be the most successful theory not only in finance, but also in all areas of economics.
Consider a writer of a European call option on a stock, he is exposed to the risk of unlimited
liability if the stock price rises acutely above the strike price. To protect his short position in the
option, he should consider purchasing certain amount of stock so that the loss in the short
position in the option is offset by the long position in the stock. In this way, he is adopting the
hedging procedure. A hedge position combines an option with its underlying asset so as to
achieve the goal that either the stock protects the option against loss or vice versa. Practitioners
in financial markets have commonly used this risk-monitoring strategy. By adjusting the
proportion of the stock and option continuously in a portfolio, Black and Scholes demonstrated
that investors can create a riskless hedging portfolio where all market risks are eliminated. In an
efficient market with no riskless arbitrage opportunity, any portfolio with a zero market risk must
have an expected rate of return equal to the riskless interest rate. In such way, the Black-Scholes
formulation establishes the equilibrium condition between the expected return on the option, the
expected return on the stock and the riskless interest rate.
However, the following assumptions on the financial markets are made in order to derive the
governing partial differential equation for the price of a European call.
1.
2.
3.
4.
5.
6.
7.
trading takes place continuously in time;
the riskless interest rate r is known and constant over time;
the asset pays on dividend;
there are no transaction costs in buying or selling the asset or the option, and no taxes;
the assets are perfectly divisible;
there are no penalties to short selling and the full use of proceeds is permitted;
there are no riskless arbitrage opportunities
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The evolution of the asset price S at time t is assumed to follow the Geometric Brownian motion
dS / S = ρ dt + σ(S,t) dZ
where ρ is the expected rate of return
σ is the volatility
dZ is the standard Wiener process.
Both ρ and σ are assumed to be constant. Consider a portfolio that involves short selling of one
unit of a European call option and long holding of ∆ units of the underlying asset, the value of
the portfolio Π is given by
Π = - c + ∆ S,
where c = c(S,T) denotes the call price of the asset S at time t.
After some calculations and rearranging the terms, we obtain
( ∂c / ∂t )+ 0.5 σ2 S2 ( ∂2c / ∂S2 ) + rS ( ∂c / ∂S ) – rc = 0
The above parabolic partial differential equation is called the Black-Scholes equation. Note that
the parameter ρ, which is the expected rate of return of the asset, does not appear in the equation.
To complete the formulation of the option-pricing model, we need to prescribe the auxiliary
(terminal tradeoff) condition for the European call option. At expiry, the payoff of the European
call is given by
c(S,T) = max (S – X, 0)
where T is the time of expiration and X is the strike price. Since both the equation and the
auxiliary condition do not contain ρ, one can conclude that the risk preferences of the investors
do not affect the option price. This observation of risk neutrality is a major breakthrough in the
option pricing theory pioneered by Black and Scholes. The option pricing model involves five
parameters: S, T, X, r and σ; all except the volatility σ are observable parameters.
The governing equation for a European put option can be derived similarly and the same BlackScholes equation is obtained. Indeed, let V denote the price of a derivative security contingent on
S; it can be shown that V is governed by
( ∂V / ∂t )+ 0.5 σ2 S2 ( ∂2V / ∂S2 ) + r S ( ∂V / ∂S ) – r V = 0
The price of a particular derivative security is obtained by solving the above equation subject to
the appropriate auxiliary conditions for the corresponding derivative security. The solution of the
Black-Scholes equation with different auxiliary conditions then provides valuation formulas for
different types of derivative securities.
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With all the inputs the models yields an option price, which can become a basic from which to
trade. If we compare this option price to its current market price, however, we will probably find
a discrepancy. The reason for this is simply a difference between theory and practice.
Conclusion
It has been shown that a Black-Scholes type model can be derived from weaker assumptions.
The main attractions of the model are:
1. the derivation is based on the relatively weak condition of avoiding dominance;
2. the final formula is a function of “observable” variables;
3. the model can be extended in a straight-forward fashion to determine the rational
price of any type of option;
On the other hand, the drawbacks are:
1. it is a complete option-pricing model depending ONLY on observable variables was
derived. It works fine ONLY when all the variables are well-defined and observable;
2. lots of weaker assumptions needed to be made in order to sustain its completeness,
for example. the stock volatility is NOT constant and difficult to observe;
3. it is ONLY good to be a theoretical models, by which the argument of risk neutrality
is sustained, not quite practical in the sense to obtain those “observable” variables;
4. unfortunately, price volatility of most optionable securities varies considerably over
time and accurate prediction is far from easy;
B.
The Garch option Pricing Model
Introducing GARCH
GARCH stands for Generalized Autoregressive Conditional Heteroscedasticity. Loosely
speaking, you can think of heteroscedasticity as time-varying variance (i.e., volatility).
Conditional implies a dependence on the observations of the immediate past, and autoregressive
describes a feedback mechanism that incorporates past observations into the present. GARCH
then is a mechanism that includes past variances in the explanation of future variances. More
specifically, GARCH is a time-series technique that allows users to model the serial dependence
of volatility.
In this manual, whenever a time series is said to have GARCH effects, the series is
heteroscedastic, i.e., its variances vary with time. If its variances remain constant with time, the
series is homoscedastic.
In order to develop the Garch option-pricing model, the conventional risk-neutral valuation
relationship has to be generalized to accommodate heteroskedasticity of the asset return process.
Why Use GARCH?
GARCH Modelling builds on advances in the understanding and modelling of volatility in the
last decade. It takes into account excess kurtosis (i.e., fat tail behavior) and volatility clustering,
two important characteristics of financial time series. It provides accurate forecasts of variances
and covariances of asset returns through its ability to model time-varying conditional variances.
As a consequence, you can apply GARCH models to such diverse fields as risk management,
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portfolio management and asset allocation, option pricing, foreign exchange, and the term
structure of interest rates.
You can find highly significant GARCH effects in equity markets, not only for individual stocks,
but also for stock portfolios and indices, and equity futures markets as well. These effects are
important in such areas as value-at-risk (VaR) and other risk management applications that
concern the efficient allocation of capital. You can use GARCH models to examine the
relationship between long-term and short-term interest rates. As the uncertainty for rates over
various horizons changes through time, you can also apply GARCH models in the analysis of
time-varying risk premiums. Foreign exchange markets are particularly well suited for GARCH
modelling because of highly couple persistent periods of volatility and tranquillity with
significant fat tail behaviour
The other reason to use GRACH for volatility estimation is when the market prices are not
available. Practitioners construct an implied volatility matrix from the available market prices of
options with various strike prices and times to expiration. When there is need to price an option
for which on price is available, practitioners can interpolate between the implied volatilities in
the matrix to find the implied volatility corresponding to the option’s strike price and time to
maturity. This implied volatility can be used to price the option.
It is still undesirable whether implied volatility outperforms the GARCH or vice versa. Perhaps,
GRACH and implied volatility should be combined in some optimal way. But the empirical
success of GARCH-based methods in modeling volatility coupled with their ability to produce
profits in at least some options markets argues that GARCH methods are important optionpricing models.
GARCH Limitations
Although GARCH models are useful across a wide range of applications, they do have
limitations:
1.
2.
3.
GARCH models are only part of a solution. Although GARCH models are usually
applied to return series, financial decisions are rarely based solely on expected returns
and volatilities. It is common that GARCH is used to estimate volatility and then using
Black-Scholes Model to obtain the option pricing;
GARCH models are parametric specifications that operate best under relatively stable
market conditions. Although GARCH is explicitly designed to model time-varying
conditional variances, GARCH models often fail to capture highly irregular phenomena,
including wild market fluctuations (e.g., crashes and subsequent rebounds), and other
highly unanticipated events that can lead to significant structural change;
GARCH models often fail to fully capture the fat tails observed in asset return series.
Heteroscedasticity explains some of the fat tail behavior, but typically not all of it. To
compensate for this limitation, fat-tailed distributions such as Student's t have been
applied to GARCH modeling;
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C.
Implied Binomial Tree
Suppose we allow the volatility to vary with time and the underlying asset price. Thus, the
dynamics of the stock price can be described by
dS / S = ρ dt + σ(S,t) dZ
where ρ is the expected rate of return
σ is the volatility
dZ is the standard Wiener process.
The problem we face is that we do not know anything about the functional form of σ(S,t).
However, we do have a great deal of information from the smile: we know how implied
volatility varies by strike price and time to maturity. The idea behind an implied volatility tree is
to use the information from the smile to build up our knowledge of σ(S,t). A useful byproduct of
this technique is that we can use the implied tree to hedge and value options more effectively.
Theory of Binomial Trees
To understand how to construct an implied binomial tree, we first need to review binomial
option-pricing theory. Suppose we construct an approximation to the continuous-time process.
To do so, we divide the time to the options’ expiration into many small intervals. During each
interval, we assume that the stock price can move from its current value to one of two values.
With probability q it can move up from S, its current value, to Su. Or, with probability 1-q, it can
move down from S to Sd. Intuitively, we would expect that the magnitude and probabilities of the
up and down movements should vary with the local volatility during each time interval. Since the
volatility varies with S and t, we would expect the probabilities and magnitudes of the
movements to depend on the time and the current level of the stock price.
Let us assume that the per-period interest rate is r and that the per-period dividend rate is r*.
Consider a European call option with strike price K and current price C. Since S moves up and
down each period, the call price must do the same. So, the price moves up to Cu with probability
q from C, or down to Cd from C with probability 1-q.
Recall that the crucial idea behind option pricing is that we can find a replicating portfolio of the
stock plus riskless borrowing and lending that exactly matches the payoff of the option for each
time period. That is how we derived the Black-Scholes formula in the simpler case already
discussed. Now we will purchase the same idea here.
Suppose we are at the beginning of the period and that the current stock price is S. To from the
replicating portfolio, we borrow x dollars at interest rate r. At the same time, we invest y dollars
in the stock, which pays dividends at the rate r* over the period. The following table shows the
payoff of this portfolio at the end of the period.
Borrow x dollars
Invest y dollars in stock
Payoff when stock price = Su
-x(1+r)
y(1+r*) Su/S
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Payoff when stock price = Sd
-x(1+r)
y(1+r*)Sd/S
Guide Study 2003/2004 – AI Modes in Options Pricing
Now, let us match the payoffs of the portfolio for each of the two possible ending stock prices to
the two possible ending call option prices:
Cu = y (1+r*) Su / S – x (1+r)
Cd = y (1+r*) Sd / S – x (1+r)
We can solve these two equations for x and y to find out how much we must invest and borrow
to match the payoffs of the call. Since the portfolio’s payoffs are identical to the call’s, the initial
cost of the portfolio must be the same as the initial cost of the call. Thus.
C=y–x
Solving for x and y, we find that the value of the call C is
C = Cu p / (1+r) + Cd (1-p) / (1+r)
where p = ( F – Sd ) / (Su – Sd)
and
F = (1+r) S / (1+r*), the forward rate over the period
Notice that p is a number between 0 and 1, like a probability, and that Cu is weighted by p and Cd
is weighted by 1-p. C appears to be the expected value of the end-of-period call price discounted
by the interest rate r. We call p the risk-neutral probability.
We found the solution for C during this particular period. Now let us go to the next period. If the
stock price starts at Su, it can rise to Suu with probability pu or fall to Sud with probability 1-pu.
And it the price starts at Sd, it can rise to Sdu with probability pd or fall to Sdd with probability 1pd. pu and pd are the risk-neutral probabilities. We impose the condition Sud = Sdu, so that the tree
is recombining. (Cud = Cdu as well). Then we find
Cu = Cuu pu / (1+r) + Cud (1-pu) / (1+r)
Cd = Cdu pd / (1+r) + Cdd (1-pd) / (1+r)
and
where pu = (F – Sud) / (Suu – Sud)
pd = (F – Sdd) / (Sdu – Sdd)
λ1 to be ppu / (1+r)2
λ2 to be (p(1-pu) + (1-p)pd) / (1+r)2
λ3 to be (1-p)(1-pd) / (1+r)2
Then we can write the price of the call as
If we define
C = Σ i=1,2,3 λi Ci
where the Ci are the three ending call prices. The λi are called he “Arrow-Debreau” prices.
They are the discounted probabilities of observing each call price at the end of some period.
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We can then continue this reasoning. Suppose we divide the time to expiration into N periods. At
the end of the nth period, there are n+1 possible ending stock prices and n+1 possible call values.
Since the European call can be struck at the end of the period, we know all of its ending values:
MAX(Si-K, 0) for i between 0 and n. Let λi be the Arrow-Debreau price for each ending call
value. Then the value of the call is
C = Σ i=1,2, … n λi MAX(Si – K, 0)
where K is the strike price of the underlying
The analogous formula for a put is
P = Σ i=1,2, … n λi MAX(K – Si, 0)
D.
Stochastic-Volatility Option Pricing
The existence of the volatility smile shows that the Black-Scholes assumption of constant
volatility is incorrect. We can see this fact as well by looking at time-series evidence.
Practitioners are well aware that implied volatilities change frequently; moreover, as we saw in
our discussion of GARCH methods, actual volatility changes over time. We could explain these
facts by assuming that the volatility function varies with the current stock price and time, but a
more flexible and realistic model would allow volatility to vary stochastically. Stochasticvolatility models are much harder than Black-Scholes to estimate and implement, but they may
help to explain the biases in Black-Scholes option pricing such as the volatility smile.
Stochastic-Volatility Theory
Consider the following common stochastic-volatility option-pricing model:
and
dS / S = ρ dt + σ(S,t) dZ1
dln(σ2) = -β( ln(σ2) - α )dt + ψdZ2
where ρ is the expected rate of return
σ is the volatility
α is the long-run mean of the variance
β is the speed of the mean-reversion
ψ is the volatility of the volatility
dZ1, dZ2 the standard Wiener process.
It is a bivariate partial differential equation (PDE) that contains a risk premium, λ, whose form is
determined by the preferences of the representative investor. This equation must be solved for
the option price subject to the boundary conditions required by the option’s features.
When volatility is stochastic, we no longer have a preference-free solution. In the constant
volatility Black-Scholes world, we could find a portfolio of the stock and riskless borrowing or
lending that always replicated the value of the derivative. This was possible because the value of
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the stock and the derivative were both driven by the same random factor. Arbitrage
considerations allowed us to value the option in this case – the investors’ preferences were
irrelevant and we obtained the preference-free pricing result. However, stochastic volatility
introduces an additional source of uncertainty into the model. If we had a volatility asset, one
whose value was driven by the same random factor as the volatility, we could include the
volatility asset in our replicating portfolio and still price options using preference-free valuation.
But since there is no volatility asset, we must in general abandon preference-free pricing.
Solving the PDE Numerically
The stochastic-volatility equation is more complicated not only because it has more unknown
parameters but also because it is bivariate, requiring more complex solution techniques. Hull and
White (1987a) and Scott (1987) use Monte Carlo methods. It is significantly more computerintensive than a numerical solution of the Black-Scholes equation.
The critical factor in this argument and in any contingent claims valuation model is the precise
description of the stochastic process governing the behavior of the basic asset, and the
development of an approach to the option valuation problem that connects it directly to the
structure of the underlying stochastic process. It will be useful then, to give a brief and informal
discussion of the stochastic processes that have previously been used.
Summary and Conclusion
The type of stochastic process determining the movement of the underlying stock is of prime
importance in the option valuation. The Stochastic Processes model uses an economically
interpretable technique for solving option problems which has intuitive appeal and should
facilitate the solution of other problems in this field. This technique is used to find explicit option
valuation formulas, and solutions to some previously unsolved problems that involve the pricing
of securities with payouts and potential bankruptcy.
Biases Caused by Stochastic Volatility
Since volatility is actually stochastic, the continuously compounded returns are no longer normal;
these returns have fat tails, i.e. extreme returns are more likely to be seen than would be observed
in the constant volatility case.
Consider an out-of-the money European call option when volatility is stochastic. Its value
depends on the chance that there will be a large increase in the stock price. Since large price
increases are more likely under a fat-tailed distribution, the price of this option will be higher
than the Black-Scholes model would suggest. Thus, Black-Scholes will undervalue an out-ofthe-money call.
Conversely, Black-Scholes will also undervalue an in-the-money put. By put-call parity, when a
call is out-of-the-money, a put with the same strike price is in-the-money. Therefore, an in-themoney put must have the same pricing bias as an out-of-the-money call: Black-Scholes will
undervalue it. Similarly, an in-the-money call will have the same pricing bias as an out-of-themoney put. But an out-of-the-money put will be undervalued by Black-Scholes because
stochastic volatility makes extreme downward moves in the stock price relatively more likely.
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Black-Scholes, then, will undervalue in-the-money and out-of-the money puts and calls when the
true volatility is stochastic.
What about at-the-money options? Hull and White (1987a) and Ball and Roma (1994) show that
the true option price under stochastic volatility is less than the Black-Scholes price. For at-themoney options, Black-Scholes overvalues options.
Intelligent System in Business: The Key Features
Now, I will discuss five key features of intelligent systems, which make them particularly
attractive for solving financial, and business problems. These are: learning, adaptation, flexibility,
explanation and discovery. It should be noted that not all intelligent techniques exhibit all these
features. Each different intelligent technique has particular strengths and weaknesses and cannot
be applied universally to every type of problem.
A. Learning
The most important feature of intelligent systems in business is their ability to learn decisions or
the tasks they have to perform, directly from data. That is, they can derive a model of business
practices purely by trawling through hundreds or thousands of past transactions. Typically
personnel in organizations who have had many years of experience in performing particular
business tasks only hold such operational knowledge. Neural networks and genetic algorithms
have the capability to learn such models of business processes from past data. A learning
approach can also help to overcome the limitations that are inherent in human professionals
including the possible existence of gaps in an expert's knowledge and the correctness of
knowledge. Furthermore, 'objective' learning methods have advantages with respect to
consistency.
B. Adaptation
Business is constantly changing. A specific business process may become quickly outdated
because of a variety of reasons including changes in the macro-economy, changes due to new
competitive pressures, or changes due to government regulations. Intelligent systems used to
support decisions in business should therefore ideally have the capability to adapt to such
changes in the business environment. It is not sufficient for an intelligent system to learn the
initial knowledge needed to perform a task, it also has to monitor its performance constantly and
revise its knowledge according to changes in its operating environment.
C. Flexibility
When human make decisions there is an inherent flexibility. Humans can make decisions even
when the available information is imprecise and incomplete. Unfortunately, traditional computer
programs do not have such flexibility. Most programs work on yes/no, 'black and white' logic
which does not permit shades of grey. Therefore, traditional computing systems are not robust in
their operation – they fail to function even if a single condition is left unspecified or misspecified.
In contrast, intelligent systems such as neural networks and fuzzy systems have the capability to
make decisions in a flexible manner that is similar to human decision-making. They can reason
with incomplete information and recognize patterns in conditions that they have not encountered
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before. Neural network can be used to learn how to group similar customers together bases on
customer attributes, such as length of account, frequency of service usage and average revenue.
Once these clusters are learnt, new unseen customer records can be presented to the system, and
it will then make a decision as to which type of cluster (highly profitable, least profitable, etc.) a
new record belongs.
D. Explanation
Some intelligent systems such as expert systems provide explicit explanations; other techniques
such as neural networks have difficulties in explaining their decisions. There are still further
organizational reasons for having intelligent systems that can explain themselves. In portfolio
management where the decisions involve very large amounts of money, sometimes the life
savings of customers, reassurance as to the soundness of the decision-making procedure is
needed. The ability to cite the exact conditions and reasoning of a trading decision is therefore
often required by senior managers in fund management companies.
It is also important to have an understanding of the reasoning process in order to improve
intelligent systems. If an intelligent system ceases to produce correct decisions due to some
reason, it can only be corrected if the reasoning processes are understood by a human. On the
other hand, if an opaque or 'black-box' decision system ceases to make good decisions, then it
will be very difficult to understand what has caused the system to behave in that manner.
Finally, transparency of intelligent system is also important to allow interaction with human
experts. There is evidence to suggest that under certain conditions expert revisions to quantitative
decision models can improve the quality of their results. In a nutshell, intelligent systems should
provide access to their core knowledge and reasoning mechanisms in a format that humans can
understand.
E. Discovery
Knowledge discovery, or data mining as it is popularly known, can be defined as the 'nontrivial
extraction of implicit, previously unknown, and potentially useful information form data'. There
are several intelligent techniques that can trawl through large databases and find relationships
and business patterns that were previously unknown. Generic algorithms have been used to find
patterns in supermarket checkout data and they have found previously unknown purchase
patterns such as the relationship between weather fluctuations and the sales of fruit. However,
checks must be made to validate whether the discovered relationships are truly representative
and not merely statistical flukes. Therefore, it is essential that relationships discovered by an
intelligent system should be verified by a human expert before they are used in an operational
context.
Introduction to Intelligent Techniques (mainly on Neural Networks)
A. Neural Networks
Neural Network (Beale & Jackson, 1990; Aleksander & Morton, 1990) are computing devices
inspired by the function of nerve cells in the brain. They are composed of many parallel,
interconnected computing units. Each of these performs a few simple operations and
communicates the results to its neighboring units. In contrast to conventional computer programs
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where step-by-step instructions are provided to perform a particular task, neural networks can
learn to perform tasks by a process of training on many different examples.
Typically the nodes of a neural network (denoted Σ for processing element) are organized into
layers with each node in one layer having a connection to each node in the next layer. Associated
with each connection is a weight and each node has an activation value. During pattern
recognition, each node operates as a simple threshold device. A node sums all the weighted
inputs (multiplying the connection weight by the state of the previous layer node) and then
applies a (typically non-linear) threshold function.
Multilayer neural network
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A typical artificial neuron
It is the values of weights that determine the types of patterns a neural network can recognize. A
learning algorithm is a procedure used to find the values of these weights for a given task. A
popular neural network learning algorithm is the back-propagation algorithm (Aleksander &
Morton, 1990; Wasserman, 1989). The back-propagation algorithm adjusts weights by
presenting example training pairs of input-target patterns (e.g. a handwritten A with a perfect A).
An input pattern is presented at the input layer and is propagated through all the processing
elements in the network to produce outputs at the output layer. This output pattern is then
compared with the 'ideal target pattern, and an error is propagated back through the network.
The propagated error is used to adjust the weights of the connections. This training process is
then repeated with a new training pair, and a new error is propagated backwards. This process is
repeated many times with many example pairs of patterns until the error is small, at which time
the network has been trained.
The relationships learnt should be truly representative of the business task in general and not
merely reflect properties contained in the training data which may be statistically
unrepresentative. If a neural network is allowed to 'overtrain', it would only be able to recognize
the patterns in the training data – it would not be able to recognize patterns outside the training
set which means it would not have the flexibility or generalization capabilities that business
problems demand. In order to avoid this situation all neural networks (and other learning systems
such as generic algorithms) should be thoroughly validated on 'out-of-sample' data – data outside
the training set. There are several methods to determine when a learning system has the 'correct'
level of training, and Weiss & Kullikowshi (1991) introduce an excellent principle in this area.
Strength and Limitations
Neural network provide a relatively easy way to model and forecast non-linear systems. This
gives them an advantage over many current statistical methods used in business and finance
which are primarily linear. They are also very effective in learning patterns in data that are noisy,
incomplete and which may even contain contradictory examples. The ability to learn and the
capability to handle imprecise data makes them very effective in financial and business
information processing. A main limitation of neural networks is that they lack explanation
capabilities. They do not provide users with details of how they reason with data to arrive at
particular conclusions.
Neural networks are therefore best suited for applications requiring pattern recognition in noisy,
incomplete data, and for tasks where experts are either unavailable or where clear rules cannot be
easily formulated. They are not suitable for applications where explanation of reasoning is
critical.
B. Other Intelligent Techniques
Other intelligent techniques include genetic algorithms, fuzzy systems and expert systems; and
below gives only a brief introduction of each :Genetic Algorithms
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Genetic algorithms are efficient problem-solving mechanisms that are inspired by the
mechanisms of biological evolution. They reward candidate solutions that contribute towards
solving a problem at hand and penalize solutions that appear unsuccessful. GAs have produced
very good solutions for complex optimization problems that have large numbers of parameters.
Areas where these have been applied include electronic circuit layout, gas pipeline control, and
job shop scheduling (Davis, 1991). The main idea of a genetic algorithm is to start with a
population of solutions to a problem, and then attempt to produce new generations of solutions
which are better then the previous ones. This is a direct analogue of the Darwinian principle of
the "survival of the fittest". GA has proved to be very effective at efficiently searching very large
data sets. This search process also has another advantage in being highly suitable for parallel
computer implementations. The limitation of genetic algorithm is that the setting of parameters
such as the crossover and mutation rates is problem dependent and is a time-consuming "trial and
error" process.
Fuzzy Systems
Fuzzy Logic is designed to handle imprecise 'linguistic' concepts such as small, big, young, old,
high or low. Systems based on fuzzy logic exhibit an inherent flexibility and have proven to be
successful in a variety of industrial control and pattern-recognition tasks ranging from
handwriting recognition to credit evolution. There are now several consumer products including
washing machines, microwave ovens and autofocus cameras that use fuzzy logic in their control
mechanisms. One of the main strengths of fuzzy logic compared with other schemes to deal with
imprecise data, such as neural networks, is that their knowledge bases, which are in a rule format,
are easy to examine and understand. This rule format also makes it easy to update and maintain
the knowledge base. For the limitations of fuzzy logic, the main shortcoming is that the
membership functions and rules have to be specified manually. Determining membership
function can be a time-consuming, trial-and-error process. Further, the elicitation of rules from
human experts can be an expensive, error-prone procedure. Additionally, they cannot adapt
automatically to changes in the operating environment – new rules have to be manually altered if
business conditions change.
Expert Systems
Expert systems represent the earliest and most established type of intelligent systems. There are
many hundreds of operational expert systems in domains ranging from fault diagnosis to
commodity trading. They attempt to embody the 'knowledge' of a human expert in a computer
program. The process of acquiring the knowledge from an expert – knowledge elicitationtypically involves a series of interviews and the careful recording of observations when the
expert is performing tasks. A great strength of expert systems is the explicit representation of
knowledge, so that the knowledge contained in the programs is relatively easy to read and
understand. Also, expert systems can generate explanation of how they arrived at a particular
conclusion. The limitation of expert systems is that they have no mechanisms for automatic
learning of the rules they use. Further, they cannot adapt or learn from changes in the business
environment in which they operate.
C.
Intelligent Hybrid Systems
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Since each intelligent technique has particular strengths and limitations that make it suitable for
particular applications and not for others. The following table lists the comparison of intelligent
techniques.
Techniques
Neural networks
Genetic algorithms
Fuzzy systems
Expert systems
Learning
Flexibility
Adaptation
Explanation
Discovery
Intelligent hybrid systems cover not only the combination of different intelligent techniques but
also the integration of intelligent techniques with conventional computing systems such as
spreadsheets and databases. For intelligent systems to add value to organizational decisions they
must be able to extract and use information from a wide variety of sources. In addition, the
decisions of results produced by the intelligent system should be disseminated to existing
applications or other systems for further processing. Intelligent hybrid systems are a very
powerful class of computational methods that can provide solutions to problems that are not
solvable by an individual intelligent technique alone. The limitations is that the development and
application of hybrid systems is still relatively new, there is not the same availability of tools and
development environments compared with more established techniques.
Comparisons between Neural Nets and Other Time Series Methods
Underlying the use of neural networks for financial time series analysis is the idea that neural
networks out-perform traditional modeling techniques. Over the last few years a number of
empirical studies have been conducted to investigate this claim, many of which have been based
on data from the M-competition. The M-competition set out to compare different forecasting
techniques by running a competition. The competitors were asked to make forecasts over the
withheld data and the results were compared. Although this competition pre-dates neural
network time series modeling, the experiment has since been re-run using neural networks on a
number of occasions.
Below tabulates the neural network forecasting comparisons
Study
Experiments
Conclusions
(Hill et al, 1992)
111 time series
Neural nets superior to classical models
(Sastri et al, 1990)
92 simulated 1 real series
Backpropagation best model
(Sharda and Patel,
75 time series
Neural networks compatible to auto1990a)
box
(Sharda and Patel,
111 time series
Neural networks out-perform
1990b)
Box_Jenkins
(Lapedes and Farber,
3 chaotic time series
Neural networks better than regression
1987)
models
(Tang et al, 1990)
3 time series
Neural networks better long term
forecasting then Box-Jenkins
(Foster et al, 1991)
111 time series
Neural networks inferior to classical
methods
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Their results, summarized above, vindicate neural network techniques. They maintain that in
most instances neural networks were usually better, and at worst were equivalent to classical
techniques.
One factor that strongly supports the empirical evidence in favor of neural networks as compared
to other modeling techniques is the fact that feed-forward neural networks have been shown to
be universal functional approximations. This fact ensures that in principle it is always possible to
find a feed-forward network capable of approximating the functional behavior of all other forms
of modeling technique. In this sense neural networks should always be capable of matching the
performance of other modeling styles, and that the real issue is not so much whether a neural
network can out-perform a given techniques but rather how easy it is to construct a good model.
To some extent the power of neural network modeling can hinder the construction process in that
a network is always capable of finding a spurious model that happens to match the training data.
Such models do not guarantee good generalization.
The Proposed AI Modeling on Options Pricing
The Proposed Modeling is a neural network which will consists of a number of interconnected
homogeneous processing units, neurons. Each unit is a simple computation device. Its behavior
can be modeled by simple mathematical functions. A unit i receives input signals from other
units, aggregates these signals based on an input function Ii and generates an output signal based
on an output function Oi (sometimes called a transfer function). The output signal is then routed
to other units as directed by the topology of the network. Although no assumption is imposed on
the form of input/output functions at each node other than to be continuous and differentiable, we
will use the following functions as suggested in Rumelhart et al.
Ii = Σ j Wij Oj + ϕi
where Ii
Oj
Wij
ϕi
and
Oi = 1/ 1+exp (Ii)
= input of unit i,
= output of unit i,
= connection weight between unit i and j,
= bias of unit i
Transfer Function
The neuron then transforms the combined signal to an output numerical value via some
differentiable function called transfer or activation function. Several kinds of common transfer
function are shown below:
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The hard limit transfer function showed above limits the output of the neuron to either 0, if the
net input argument n is less than 0, or 1, if n is greater than or equal to 0.
This transfer function is commonly used in back-propagation networks, in part because it is
differentiable. The function logsig generates outputs between 0 and 1 as the neuron’s net input
goes from negative to positive infinity.
Alternatively, multilayer networks may use the tan-sigmoid transfer function tansig.
Occasionally, the linear transfer function purelin is used in back propagation networks.
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We can get some intuition regarding the usefulness of this function in decision-making and
classification. An input signal does not have to be exactly "1" to make a match and be recognized
as a "1". Thus a neuron is a good candidate for recognition systems.
Feed forward Networks
Suppose a neural network:
1.
2.
3.
4.
Each neuron is linked only to neurons in next layer.
No other linkages between neurons in the same layer.
No backward linkages to the previous layer.
No skip layers, i.e. all neurons within a layer are linked by other neurons.
If a network satisfies all conditions above, it is said to be a feed forward network. Layers
between the input layer and output layer are called ‘hidden layer’. A feed forward neural network
with the configuration of 3-3-2 is shown below:
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For multiple-layer networks, transfer functions tansig and purelin are always used. A 2 layer
tansig/purelin network 2 inputs, 1 hidden layer and 1 output layer is shown.
2 layer tansig/purelin network
where W is a weight matrix and b is a bias, both of them are scalar parameters and adjustable.
Back-propagation Learning Algorithm
A human learns by having similar experiences repeated. For instance, when I see one human,
then another, and another, and store their main features in my memory. In other words, I train my
neural network, my brain, to recognize patterns in humans and classify them as such. When I
meet a completely new human, I can immediately say it is a human I am looking at. In ANN
parlance, neural network has been trained to generalize well, or to make predictions outside the
familiar set on which it was trained. The objective of training is to find weights such that the
performance functions let say the sum of mean squared error (MSE) is minimized. After training,
the network has learnt a specific pattern from a series of input data. In this project, the pattern of
movement in stock prices is learnt from a set of historical data.
How does a neural network be trained? It can be achieved by different kinds of training
algorithm, say back propagation. All of these algorithms use the gradient of the performance
function to determine how to adjust the weights to minimize performance. The gradient is
determined using a technique called back propagation, which involves performing computations
backwards through the network. The back propagation computation is derived using the chain
rule of calculus. The basic back propagation training algorithm, in which the weights are moved
in the direction of the negative gradient.
Back propagation Algorithm
There are a number of variations on the basic algorithm which are based on other standard
optimisation techniques, such as conjugate gradient and Newton methods. The simplest
implementation of back propagation learning updates the network weights and biases in the
direction in which the performance function decreases most rapidly – the negative of the gradient.
One iteration of this algorithm can be written as:
x k +1 = x k - α k g k
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where x k is a vector of current weights and biases, g k is the current gradient, and α k is the
learning rate.
When will the training come to the end? A performance goal is set in the neural network. When
this goal is met, the training stops since further training will not improve the overall performance
and the pattern learnt so far is a good fit.
Properly trained back propagation networks tend to give reasonable answers when presented
with inputs that they have never seen. Typically, a new input will lead to an output similar to the
correct output for input vectors used in raining that are similar to the new input being presented.
This generalization property makes it possible to train a network on a representative set of
input/target pairs and get good results without training the network on all possible input/output
pairs.
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Moving Simulation
For prediction of an financial system, such as stock prices, in which the prediction rules are
changing continuously, learning and prediction must follow the changes.
I suggested a prediction method called moving simulation. In this system, prediction is done by
simulation while moving the objective learning and prediction periods. The moving simulation
predicts as follows.
As shown below, the system learns data for the past M months, then predicts for the next L
months. The system advances while repeating this.
Testing
After training ends, we can use the estimated function to see how much error it makes if applied
on completely new data, such as data out of sample in time series forecasting. This is crucial
because the network may literally "memorize" the training data, so that, when it "sees" new data
it cannot recognize them (it gives large prediction errors). To avoid this problem, it is good to
take a section of the training data out and use it for validation. Validation is the process where
the user checks how the network performs in data it was not trained on. Testing can check the
network's generalizing ability.
Difficulties in Training
A. Convergence
The graph below shows an error function with a global minimum and a local one:
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Local
Minimum
Global
Minimum
Error function
If you start the algorithm from a point in the bottom of a local minimum, the network will
converge immediately, but you will find a bad approximation and you never reach the global
minimum. The way out may be to try a variety of initial values. However, this still raises the
question of replicability, especially if there are many local minima, difficult to distinguish. We
cannot replicate the work of another researcher, unless we start exactly with the same initial
conditions and use exactly the same algorithm.
B. Optimal Stopping Point
Two unsatisfactory situations can occur when the training stops at wrong place:
•
•
Under training. Training the network too little causes the estimated function to pick up
too few of the features of a function.
Over training. Training the network too much, may lead to complete "memorization"
of the training set, with very weak generalization ability.
C. Number of hidden nodes
Adding more nodes helps in approximating the true mapping between two variables. However, if
you care about generalization, adding too many nodes may not help very much. The idea is that
each of these nodes finally becomes in a sense "dedicated" to each input data point, so
generalization will not be successful.
Conclusion
Refer to the research paper 'Critical Assessment of Option Pricing Methods Using Artificial
Neural Networks (ANNs)", ANNs outperform the Black-Scholes Formula (BSF) in forecasting
Option Pricing. The season is that BSF formula suffers from systematic biases when compared to
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options market prices. To avoid the parametric models deficiencies we can address our attention
to market data driven models and not to depend on models that spring from the theoretical
concepts of the options pricing field. Furthermore, The ANNs performance improves even more
when a hybrid ANN model is utilized.
With using the Black-Scholes Models, it is also possible to perform a forecast on the volatility
with using the ANNs. Using ANNs to obtain the implied volatility is found to be having more
promising results. Furthermore, wavelets can be used to predict the volatility by turning it into a
stochastic variable. Wavelets have been applied to various financial engineering problems, for
example pricing of financial derivatives or iterative long-term forecasting of financial time series.
The main advantage of wavelets is their ability to model jumps and discontinuities present in the
financial time series. Besides, performing a signal/noise decomposition of the underlying time
series becomes a straightforward operation in the nonlinear wavelet domain.
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Reference
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
p. ix, xiii, 3 – 62
Options plain & simple – successful strategies without rocket science
ISBN: 0-273-63878-5
CityU: HG6024.A3 J67 2000
p. 1 – 11
The Options Strategist
ISBN: 0-07-140895-9
CityU: HG 6042.V45 2003
p5–7
The Options Primer
IBSN: 1-57718-071-2
CityU: HG 6024.A3 K652 199
The Garch option Pricing Model
http://www.mathworks.com/access/helpdesk/help/toolbox/garch/garch.shtml
Wiener Process
http://mathworld.wolfram.com/WienerProcess.html
Monte Carlo Method
http://mathworld.wolfram.com/MonteCarloMethod.html
p.47 – 51
p.58 – 70
Volatility in the Capital Markets - state-of-the-Art Techniques for Modeling, Managing and
Trading Volatility
ISBN: 1-888998-05-9
CityU: HG 4523.V65 1997
p.22 – 35
Mathematical Models of Financial Derivatives
ISBN: 981-3083-255
HKU: 332.645 K98
Mathematics of Financial Markets
ISBN: 3-540-76266-3
HKU: 332.60151 E4
p.59 – 108
p.117 – 128
p.325 – 339
Volatility - New Estimation Techniques for Pricing Derivatives
ISBN: 1-899332-46-4
CityU: HG 4637.V64 1998
p.1 – 100
p.141 – 194
Understanding Options
ISBN: 0-471-08554-5
CityU: HG6024.A3 K653 1995
p.1 – 27
p.75 – 96
Intelligent Systems for Finance and Business
ISBN: 0-471-94404-1
HKU: 658.0563 I6
p. 26 – 54
Intelligent systems and Financial Forecasting
ISBN: 3-540-76098-9
HKU: 332.028563 K54
p. 1 – 83
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15.
16.
17.
18.
19.
20.
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p. 65 Parameters for Back-Propagation Networks
p. 127 Comparison of Neural Network and Expert Systems
p. 198 – 201
p. 343 – 356 Stock Market Prediction System with Modular Neural Networks
p. 357 – 369 Stock Price Pattern Recognition: A Recurrent Neural Network Approach
Neural Networks in Finance and Investing – Using Artificial Intelligence to Improve Real-World
Performance
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Financial Calculus – An introduction to derivative pricing
ISBN: 0-521-55289-3
CityU: HG6024.A3 B39 1997
p. 81 – 132
p. 179 – 201
p. 233 – 248
Options - Classic Approaches to Pricing and Modeling
ISBN: 1-899-332-66-9
CityU: HG6024.A3 O648 1999
Garch SDK
http://www.mathworks.com/products/garch/
Dissertation Writing
http://www.ai.mit.edu/people/shivers/diss-advice.html
http://www.cs.purdue.edu/homes/dec/essay.dissertation.html
http://www.cs.gatech.edu/student.services/phd/phd-advice/thesis.html
The Back propagation Algorithm
http://www.speech.sri.com/people/anand/771/html/node37.html
Options Data Collection
http://www.ivolatility.com/
Stochastic volatility options pricing with wavelets and artificial neural networks (pdf)
Critical Assessment of Option Pricing Methods Using Artificial Neural Networks (pdf)
MATLAB Used to Build and Test a New Option-Pricing Method (pdf)
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Appendix
Comparison of Neural Network and Expert Systems
Neural Network
Expert System
Example Based
Rule Based
Domain Free
Domain specific
Finds rules
Needs Rules
Little programming needed
Much programming needed
Easy to maintain
Difficult to maintain
Fault tolerant
Not fault tolerant
Needs (only) a database
Needs a human expert
Fuzzy Logic
Rigid Logic
Adaptive system
Requires reprogramming
Parameters for Back-Propagation Networks
Network Decisions:
Transfer Functions:
Sigmoid, Hyperbolic Tangent, Sine
Learning Rules:
Delta Rule, Cumulative Delta Rule,
Normalized Cumulative Delta Rule
Topology:
Number of Hidden Layers, Number of neurons
per layer, Functional Link Layer (if any),
Connection to Prior Layers
Learning Rates
Learning Rates for Each Layer
Problem Specific Parameters:
Number of Input neurons:
Number of Inputs to Network
Number of Output neurons:
Number of Outputs from the Network
Min-Max Table:
Required to Normalize Data
Instruments:
RMS Error, Confusion Matrix
Steps in building Neural Network
1. Picking an architecture
2. Need training data set
3. Test data set
4. Avoid over-fitting
Principal Options Exchanges in the United States
Chicago Board Options Exchange (CBOE)
Options on individual stocks, options on stock
indexes, and options on Treasury securities
Philadelphia Stock Exchange (PHLX)
Stocks, futures, and options on individual
stocks, currencies, and stock indexes
American Stock Exchange (AMEX)
Stocks, options on individual stocks, and
options on stock indexes
Pacific Stock Exchange (PSE)
Options on individual stocks and a stock index
New York Stock Exchange (NYSE)
Stocks and options on individual stocks and a
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Chicago Board of Trade (CBOT)
Chicago Mercantile Exchange (CME)
Coffee, Sugar and Cocoa Exchange (CSCE)
Commodity Exchange (COMEX)
Kansas City Board of Trade (KCBT)
MidAmerica Commodity Exchange (MIDAM)
Minneapolis Grain Exchange (MGE)
New York Cotton Exchange (NYCE)
New York Futures Exchange (MYFE)
New York Mercantile Exchange (NYME)
stock index
Futures, options on futures for agricultural
goods, precious metals, stock indexes, and debt
instruments
Futures, options on futures for agricultural
goods, stock indexes, debt instruments, and
currencies
Futures and options on agricultural futures
Futures and options on futures for metals
Futures and options on agricultural futures
Futures and options on futures for agricultural
goods and precious metals
Futures and options on agricultural futures
Futures and options on agricultural, currency,
and debt instrument futures
Futures and options on stock indexes
Futures and options on energy futures
Acronyms/Glossary
CBOT
Chicago Board of Trade
CBOE
Chicago Board Options Exchange
OEX
Options Exchange Index or Standard and Poor’s 100 index, the options on
which are traded at the CBOE
CME
Chicago Mercantile Exchange
FTSE-100
Financial Times Stock Exchange 100 index
LIFFE
London International Financial Futures and Options Exchange
OTC
Over the Counter
NYMEX
New York Mercantile Exchange
DJIA
Dow Jones Industrial Average
S&P5000
Standard and Poor’s 500 index
SPX
Options on the S&P500 also traded at the CBOE
Aarch
Augmented autoregressive conditional heteroskedasticity
ARCD
Autoregressive conditional density
Arch
Autoregressive conditional heteroskedasticity
Arch-M
Autoregressive conditional heteroskedasticity in the mean
Garch
Generalised autoregressive conditional heteroskedasticity
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