TABLE OF CONTENTS Chapter 1: Introduction 4 The use of financial derivatives and the importance of options between a buyer and a seller 5 The scope of the work 6 Chapter 2: Derivatives 7 2.1 Introduction 7 2.2 Definition 8 2.3 Forwards Contracts 8 2.4 Futures Contracts 11 2.5 Options 14 2.6 Swaps 18 2.7 Usage 21 2.8 Hedging 21 2.9 Arbitrage and Speculation 22 Chapter 3: Asian Options 25 3.1 Introduction 25 3.2 Definition 26 3.3 Types of Asian Option 27 3.4 Asian Approximation formula 27 3.4.1 Arithmetic Asian options 27 3.4.2 Geometric Asian options 28 3.5 Average of Asian Option 29 3.6 Black-Scholes formula 32 3.7 Monte Carlo method 33 Chapter 4: European and American Options 37 4.1 Introduction 37 4.2 Call Options 38 4.2.1 Definition 38 4.2.2 Analyze and present examples of call options 38 4.2.3 Trading Options Strategies 42 4.2.4 Long Call Options 43 4.2.5. In the money Call Option 44 1
4.3 Put Options 44 4.3.1 Definition 44 4.3.2 Analyze and present examples of put options 45 4.3.3 Long Put Options 47 4.3.4 In-the-money Put Option 48 4.4 Further Cases of Options 49 4.4.1 Stock Option 49 4.4.2 Exchange Traded Options 50 4.4.3 Option Expiration 50 4.4.4 Weekly Options 52 4.4.5 Binary Option 53 4.4.6 Leap Option 54 4.4.7 Index Option 55 4.4.8 Stop Order 56 4.4.9 Exercising Options 57 4.4.10 Option Value and Option Pricing 58 4.5 European Options 59 4.6 American Options 60 4.7 Differences between European and American Options 60 4.8 The Binomial Option Pricing Model 62 Binomial tree simulation 68 Chapter 5: Final Conclusion 73 BIBLIOGRAPHY 74 2
List of Plots Plot of a long forward payoff 10 Plot of a short forward payoff 10 Plot of payoff from buying a call (long call) 16 Plot of payoff from buying a put (long put) 16 Plot of payoff from writing a call (short call) 17 Plot of payoff writing a put (short put) 18 Graph of weekly s option 52 Binomial tree 71 List of tables Table of equity derivatives 20 Table of Microsoft Corp Com share (ticker symbol MSFT) 40 Table of AAPLE share (ticker symbol AAPL) 41 Table of AAPLE share with use of expiration date 51 3
Chapter 1 Introduction The derivatives market is the financial market for derivatives, financial instruments like futures contracts or options, which are derived from other forms of assets. The market can be divided into two, that for exchange-traded derivatives and that for over-the-counter derivatives. The legal nature of these products is very different, as well as the way they are traded, though many market participants are active in both. A financial market is a market in which people trade financial securities, commodities, and other fungible items of value at low transaction costs and at prices that reflect supply and demand. Securities include stocks and bonds, and commodities include precious metals or agricultural products. In economics, typically, the term market means the aggregate of possible buyers and sellers of a certain good or service and the transactions between them. The term "market" is sometimes used for what are more strictly exchanges, organizations that facilitate the trade in financial securities, e.g., a stock exchange or commodity exchange. This may be a physical location or an electronic system (like NASDAQ). Much trading of stocks takes place on an exchange; still, corporate actions are outside an exchange, while any two companies or people, for whatever reason, may agree to sell stock from the one to the other without using an exchange. Trading of currencies and bonds is largely on a bilateral basis, although some bonds trade on a stock exchange, and people are building electronic systems for these as well, similar to stock exchanges. Also, it s worth to be a mention at the importance of derivatives in the economy of the country. A derivative, in finance, is a contract that derives its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often called the "underlying". Derivatives can be used for a number of purposes, including insuring against price movements (hedging), increasing exposure to price movements for speculation or getting access to otherwise hard to trade assets or markets. Some of the more common derivatives include forwards, futures, options, swaps, and variations of these such as synthetic collateralized debt obligations and credit default swaps. Most derivatives are traded over the counter or on an exchange such as the Chicago Mercantile Exchange, while most insurance contracts have developed into a separate industry. Derivatives are one of the three main categories of financial instruments, the other two being stocks, like shares and debt. 4
The usage of financial derivatives and the importance of options Derivatives are used for the following: 1) Hedge or mitigate risk in the underlying, by entering into a derivative contract whose value moves in the opposite direction to their underlying position and cancels part or all of it out. 2) Create option ability where the value of the derivative is linked to a specific condition or event. 3) Provide leverage (or gearing), such that a small movement in the underlying value can cause a large difference in the value of the derivative 4) Obtain exposure to the underlying where it is not possible to trade in the underlying. 5) Speculate and make a profit if the value of the underlying asset moves the way they expect. 6) Switch asset allocations between different asset classes without disturbing the underlying assets, as part of transition management. 7) Avoid paying taxes. For example, an equity swap allows an investor to receive steady payments, while avoiding paying capital gains tax and keeping the stock. Options have two main uses: speculating and hedging (see in paragraph 1.7). Investing in options has a leverage compared to investing directly in the corresponding underlying assets for the speculators. For instance, if you believe that Ericsson shares are due to increase then you may speculate by becoming the holder of a suitable call option. Typically, you can make a greater profit relative to your original payout than you would do by simply purchasing the shares. That is why the options become more and more popular in the financial market. On the other hand, it is a very useful tool for hedging. Especially in the view of the option writers, they may not be able to afford the huge potential risk just as the speculators could. They write options to gain some certain and less risky profits. The mechanism is like this: the writer can sell an option for more than it is worth and then hedge away all the risk she or he might be possible to take, and then make a locked gain. This idea is central to the theory and practice of option pricing. 5
Scope of the work This thesis is introduced and studied the importance of usage of derivatives in finance. The most of economic activities become with the use of derivatives and of course contracts. Finance is a field that deals with the study of investments. Which includes the dynamics of assets and liabilities over time under conditions of different degrees of uncertainty and risk. Finance can also be defined as the science of money management. A key point in finance is the time value of money, which states that purchasing power of one unit of currency can vary over time. Finance aims to price assets based on their risk level and their expected rate of return. Finance can be broken into three different sub-categories: public finance, corporate finance and personal finance. In the first chart we define derivatives and we analyze the main types of derivatives, such as forwards, futures, options, and swaps. We explain them with examples and also explain the importance operation of Arbitrage. In the second chart we define Asian Options and we give the four types of Asian options, such as Continuous arithmetic average Asian call or put option, Discrete arithmetic average Asian call or put option, Continuous geometric average Asian call or put option and Discrete geometric average Asian call or put option. Furthermore we explain the significant use of Black-Scholes formula and Monte Carlo method. In the third chart we give a detailed reporting of European and American option. Firstly, we define Call and Put options and we analyze other types of options. Also, we define, give examples and explain the difference between a European and an American option. Finally, we analyze a Binomial option pricing model, write an algorithm in R programming language and explain the result of this. 6
Chapter 2 Derivatives 2.1 Introduction Derivatives are contracts between two parties that specify conditions, especially the dates, resulting values and definitions of the underlying variables and the notional amount under which payments are to be made between the parties. The most common underlying assets include commodities, stocks, bonds, interest rates and currencies, but they can also be other derivatives, which adds another layer of complexity to proper valuation. The components of a firm's capital structure, like bonds and stock, can also be considered derivatives, more exactly options, with the underlying being the firm's assets, but this is unusual outside of technical contexts. However, financial derivatives, from the economic point of view, are cash flows that are conditionally stochastic and discounted to present value. The market risk here in the underlying asset is connected to the financial derivative through contractual agreements and therefore can be traded separately. The underlying asset does not have to be acquired. Also, derivatives allow the breakup of ownership and participation in the market value of an asset. Furthermore, this provides a considerable amount of freedom regarding the contract design. That contractual freedom allows to modify the participation in the performance of the underlying asset almost arbitrarily. So, the participation in the market value of the underlying can be effectively weaker, stronger or implemented as inverse. Specifically, the market price risk of the underlying asset can be controlled in almost every situation. Before, it will be given definition of derivatives, it is important to be a report of the size of the derivative market. An economist has reported that as of June 2011, the over the counter (OTC) derivatives market amounted to approximately $700 trillion, and the size of the market traded on exchanges totaled an additional $83 trillion. However, these are "notional" values, and some economists say that this value greatly overstates the market value and the true credit risk faced by the parties involved. For example, in 2010, while the aggregate of OTC derivatives exceeded $600 trillion, the value of the market was estimated much lower, at $21 trillion. The credit risk equivalent of the derivative contracts was estimated at $3.3 trillion. Still, even these scaled down figures represent huge amounts of money. The budget for total expenditure of the United States government during 2012 was $3.5 trillion, and the total current value of the U.S. stock market is an estimated $23 trillion. The world annual Gross Domestic Product is about $65 trillion. And for one type of derivative 7
at least, Credit Default Swaps (CDS), for which the natural risk is considered high, the higher, nominal value, remains relevant. It was this type of derivative that investment magnate Warren Buffett referred to in his famous 2002 speech in which he warned against "weapons of financial mass destruction." CDS notional value in early 2012 amounted to $25.5 trillion, down from $55 trillion in 2008. That all report to size of market is important to be understanding the important use of derivatives at economy. 2.2 Definition Derivatives are financial contracts, or financial instruments, whose values are derived from the value of something else which is known as the underlying. The underlying value on which a derivative is based can be a traded asset, such as a stock, an index portfolio, a futures price, a commercial real estate or some measurable state variable, such as the weather condition at some location. The payoff can involve various patterns of cash flows. Payments can be spread evenly through time, occur at specific dates, or a combination of the two. Derivatives are also referred to as contingent claims. The main types of derivatives are forwards, futures, options, and swaps. 2.3 Forwards Contracts A forward contract, in finance, is a non-standardized contract between two parties to buy or to sell an asset at a specified future time at a price agreed onto today, making it a type of derivative instrument. This is in contrast to a point contract, which is an agreement to buy or sell an asset on its spot date, which may vary depending on the instrument. The party agreeing to buy the underlying asset in the future assumes a long position, and the party agreeing to sell the asset in the future assumes a short position. The price agreed upon is called the delivery price, which is equal to the forward price at the time the contract is entered into. The price of the underlying instrument is paid before control of the instrument changes. This is one of the many forms of buy or sell orders where the time and date of trade is not the same as the value date where the securities themselves are exchanged. The forward price of a contract is commonly contrasted with the spot price, which is the price at which the asset changes hands on the spot date. The forward 8
premium or forward discount by the purchasing party is the difference between the spot and the forward price. Forwards, like other derivative securities, can be used to hedge risk, typically currency or exchange rate risk, as a means of speculation, or to allow a party to take advantage of a quality of the underlying instrument which is time sensitive. However, a closely related contract is a futures contract, so they differ in certain respects. Forward contracts are very similar to futures contracts, except they are not exchange-traded, or defined on standardized assets. Also, forwards typically have no temporary partial settlements in margin requirements like futures, such that the parties do not exchange additional property securing the party at gain and the entire unrealized gain or loss builds up while the contract is open. Therefore, forward contracts specification can be customized and may include mark-to-market and daily margin calls. So, a forward contract arrangement might call for the loss party to promise collateral or additional collateral to better secure the party at gain. In other words, the terms of the forward contract will determine the collateral calls based upon certain "trigger" events relative to a particular counterparty such as among other things, credit ratings, value of assets under management or redemptions over a specific time frame, for example quarterly or annually. Analyze a Forward Contract The value of a forward position on expiration date depends on the relationship between the delivery price (K) and the underlying price (ST) at that time. This payoff two different types: 1) For a long position, it is: f = K ST, and 2) For a short position, it is: f = ST K On expiration date, the final value of a forward position depends on the mark price which will then be dominated, so this contract can be viewed. 9
Plot of a long forward payoff Plot of a short forward payoff 10
Example of a forward contract works Assume that an investor, in our example his name is Tom, wants to buy a field a year from now. At the same time, suppose that somebody else (in our example his name is Peter) currently owns a $50,000 field that he wishes to sell a year from now. Both parties could enter into a forward contract with each other. Suppose that they both agree on the sale price in one year's time of $52,000. Tom and Peter have entered into a forward contract. Tom, because he is buying the underlying, is said to have entered a long forward contract. On the other hand, Peter will have the short forward contract. At the end of one year, suppose that the current market valuation of Andy's field is $550,000. Then, because Peter is obliged to sell to Tom for only $52,000, Tom will make a profit of $3,000. To see why this is so, one needs only to recognize that Tom can buy from Peter for $52,000 and immediately sell to the market for $55,000. Tom has made the difference in profit. In contrast, Andy has made a potential loss of $3,000 and a profit of $2,000. Therefore, it s easy to understand how a forward contract works. One party opens a forward contract to buy or sell a currency (in our example a contract to buy euros) to expire or settle at a future date, as they do not wish to be exposed to exchange rate or a currency risk over a period of time. As the exchange rate between dollars and euros fluctuates between the trade date and the earlier of the date at which the contract is closed or the expiration date, one party gains and the loss as one currency strengthens against the other. A lot of times, the buy forward is opened because the investor will need euros at a future date such as to pay a debt owed that is denominated in euros. Other times, the party opening a forward does so, not because they need euros nor because they are hedging (see in paragraph 1.8) currency risk, but because they are speculating on the currency, expecting the exchange rate to move favorably to generate a profit on closing the contract. 2.4 Futures Contracts A future contract is a standardized contract between two parties to buy or sell a specified asset of standardized quantity and quality for a price agreed upon today, i.e. the futures price, with delivery and payment occurring at a specified future date, the delivery date, making it a derivative product, i.e. a financial product that is derived from an underlying asset. The contracts are negotiated at a futures exchange, which acts as an intermediary between buyer and seller. The party agreeing to buy the underlying asset in the future, the "buyer" of the contract, is said to be "long", 11
and the party agreeing to sell the asset in the future, the "seller" of the contract, is said to be "short". While the futures contract specifies a trade taking place in the future, the purpose of the futures exchange is to practice as intermediary and minimize the risk of default by either party in the intervening period. For this reason the futures exchange requires both parties to put up an initial amount of cash the margin. Margins, sometimes set as a percentage of the value of the futures contract, need to be maintained at all times during the life of the contract to support this reduction because the price of the contract will vary in keeping with supply and demand and will change daily and thus one party or the other will theoretically be making or losing money. To minimize risk and the possibility of default by either party, the product is marked to market on a daily basis where the difference between the prior agreed-upon price and the actual daily futures price is determined by a daily basis. This is sometimes known as the variation margin where the futures exchange will draw money out of the losing party's margin account and put it into the other party's thus ensuring that the correct daily loss or profit is reflected in the respective account. If the margin account goes below a certain value set by the exchange, then a margin call is made and the account owner must refill the margin account. This process is known as "marking to market". On the delivery date, the amount exchanged is not the specified price on the contract but the spot value. In the market, the strike price is often reached and creates lots of income for the "caller". A forward contract is a closely related contract. So, a forward is like a futures in that it specifies the exchange of goods for a specified price at a specified future date. However, a forward is not traded on an exchange and thus does not have the temporary partial payments due to marking to market. Nor is the contract standardized, as on the exchange. Unlike an option, both parties of a futures contract must fulfill the contract on the delivery date. The seller delivers the underlying asset to the buyer, or, if it is a cash-settled futures contract, then cash is transferred from the futures trader who sustained a loss to the one who made a profit. To exit the obligation before to the settlement date, the holder of a futures position can close out its contract obligations by taking the opposite position on another futures contract on the same asset and settlement date. Therefore, a profit or loss is the difference in futures prices. Analyze futures contracts Main purpose of futures contracts is to minimize risk. For that, the product is marked to market on a daily basis where the difference between the initial agreedupon price and the actual daily futures price is reevaluated daily. This is sometimes known as the variation margin (margin is a performance bond or risk minimization), where the Futures Exchange will draw money out of the losing party's margin 12
account and put it into that of the other party, ensuring the correct loss or profit is reflected daily. Furthermore, it s important to be a report at kinds of futures contracts. Some of them are Foreign exchange market, Money market, Bond market, Equity market and Soft commodities market. Also, a futures exchange or futures market is a central financial exchange where people can trade standardized futures contracts. That is, a contract to buy specific quantities of a commodity or financial instrument at a specified price with delivery set at a specified time in the future. These types of contracts fall into the category of derivatives. In additional, futures traders are important for the operation of futures contracts. Futures traders are traditionally placed in one of two groups: hedgers, who have an interest in the underlying asset (which could include an intangible such as an index or interest rate) and are seeking to hedge out the risk of price changes and speculators, who seek to make a profit by predicting market moves and opening a derivative contract related to the asset "on paper", while they have no practical use for or intent to actually take or make delivery of the underlying asset. In other words, the investor is seeking report to the asset in a long futures or the opposite effect for a short futures contract. Finally, it will be a report at arbitrage arguments. So, arbitrage arguments apply when the deliverable asset exists in plentiful supply. The forward price represents the expected future value of the underlying discounted at the risk free rate. We define the forward price to be the strike K such that the contract has 0 value at the present time. Assuming interest rates are constant the forward price of the futures is equal to the forward price of the forward contract with the same strike and maturity. It is also the same if the underlying asset is uncorrelated with interest rates. On the other hand, the difference between the forward price on the futures (futures price) and forward price on the asset, is proportional to the covariance between the underlying asset price and interest rates. Thus, assuming constant rates, for a simple, non-dividend paying asset, the value of the futures/forward price, F(t, T), will be found by compounding the present value S(t) at time to maturity T by the rate of risk-free return r. It is given by the type: and, if it is with continuous compounding. Also, this relationship may be modified for storage costs, dividends, dividend yields, and convenience yields. In a perfect market the relationship between futures and spot prices depends only on the above variables. 13
In practice there are various market imperfections, like transaction costs, differential borrowing and lending rates, restrictions on short selling, that prevent complete arbitrage. Thus, the futures price in fact varies within arbitrage boundaries around the theoretical price. Example of a future contract Assume a future contract with a $100 price. Let's say that on day 50, a futures contract with a $100 delivery price (on the same underlying asset as the future) costs $88. On day 51, that futures contract costs $90. This means that the "mark-tomarket" calculation would requires the holder of one side of the future to pay $2 on day 51 to track the changes of the forward price ("post $2 of margin"). This money goes to the holder of the other side of the future. So, the loss party wires cash to the other party. However, a forward-holder, may pay nothing until settlement on the final day, potentially building up a large balance. This may be reflected in the mark by an allowance for credit risk. So, except for tiny effects of convexity bias, due to earning or paying interest on margin, futures and forwards with equal delivery prices result in the same total loss or gain, but holders of futures experience that loss or gain in daily increments which track the forward's daily price changes, while the forward's spot price converges to the settlement price. Therefore, for a futures this gain or loss is realized daily, while for a forward contract the gain or loss remains unrealized until expiry. 2.5 Options In finance, an option is a contract which gives the buyer (the owner) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on or before a specified date. The seller has the corresponding obligation to fulfill the transaction that is to sell or buy, if the buyer "exercises" the option. The buyer pays a premium to the seller for this right. An option that conveys to the owner the right to buy something at a certain price is a call option (see in paragraph 3.2), an option that transfers the right of the owner to sell something at a certain price is a put option (see in paragraph 3.3). Both are commonly traded, but the call option is more frequently discussed. Options valuation is a topic of ongoing research in academic and practical finance. In basic terms, the value of an option is usually separates into two parts: 1) The first part is the "intrinsic value", defined as the difference between the market value of the underlying and the strike price of the given option. 14
2) The second part is the "time value", which depends on a set of other factors which, through a multivariable, non-linear interrelationship, reflect the discounted expected value of that difference at expiration. Options contracts have been known for many centuries, however both trading activity and academic interest increased when, as from 1973, options were issued with standardized terms and traded through a guaranteed clearing house at the Chicago Board Options Exchange. Today many options are created in a standardized form and traded through clearing houses on regulated options exchanges, customized contracts between a single buyer and seller, one or both of which may be a dealer or market-maker. Options are part of a larger class of financial instruments known as derivative products or simply derivatives. Subsequently, it is important to have a report at basic trades. These trades are described from the point of view of a speculator. If they are combined with other positions, they can also be used in hedging. An option contract usually represents 100 shares of the underlying security. Also, inhere, it s necessary to observe 4 payoffs for long call option, for long put option, for short call option and for short put option: 1) Long call option. A trader who expects a stock's price to increase can buy a call option to purchase the stock at a fixed price ( strike price ) at a later date, rather than just purchase the stock itself immediately. The cash cost on the option is the premium, which is much lower than what would be required for a stock purchase. The trader would have no obligation to buy the stock, and only has the right to do so at the expiration date. The risk of loss would be limited to the premium paid, unlike the possible loss had the stock been bought outright. The owner, for example of an American call option (see in paragraph 3.6), can sell his option holding at any time until the expiration date, and would consider doing so when the stock's spot price is above the exercise price, especially if he expects the price of the option to drop. By selling the option early in that situation, the trader can realize with an immediate profit. Alternatively, he can exercise the option and then sell the stock, realizing with a profit. A trader would make a profit if the spot price of the shares raises by more than the premium. For example, if exercise price is 100 and premium paid is 10, then if the spot price of 100 raises to only 110 the transaction is break-even and an increase in stock price above 110 produces a profit. So, if the stock price at expiration is lower than the exercise price, the holder of the options at that time will let the call contract expire worthless, and only lose the amount of the premium. 15
Plot of payoff from buying a call 0 premium profit Share price at Maturity Strike price 2) Long put option. A trader who expects a stock's price to decrease can buy a put option to sell the stock at a fixed price ("strike price") at a later date. The trader will be under no obligation to sell the stock, and only has the right to do so at the expiration date. If the stock price at expiration is below the exercise price by more than the premium paid, he will make a profit. If the stock price at expiration is above the exercise price, he will let the put contract expire worthless and only lose the premium paid. In the transaction, the premium also plays a major role as it enhances the break-even point. For example, if exercise price is 100, premium paid is 10, then a spot price of 100 to 90 is not profitable. He would make a profit if the spot price is below 90. Plot of payoff from buying a put 0 premium profit Share price at Maturity Strike price 16
3) Short call option. A trader who expects a stock's price to decrease can sell the stock short or instead sell, or "write", a call. The trader selling a call has an obligation to sell the stock to the call buyer at a fixed price ("strike price"). If the seller does not own the stock when the option is exercised, he is obligated to purchase the stock from the market at the then market price. If the stock price decreases, the seller of the call (call writer) will make a profit in the amount of the premium. If the stock price increases over the strike price by more than the amount of the premium, the seller will lose money, with the potential loss being unlimited. Plot of payoff from writing a call 0 <- Payoff premium profit Share price at Maturity Strike Price 4) Short put option. A trader who expects a stock's price to increase can buy the stock or instead sell, or "write", a put. The trader selling a put has an obligation to buy the stock from the put buyer at a fixed price ("strike price"). If the stock price at expiration is above the strike price, the seller of the put (put writer) will make a profit in the amount of the premium. If the stock price at expiration is below the strike price by more than the amount of the premium, the trader will lose money, with the potential loss being up to the strike price minus the premium. 17
Plot of payoff writing a put 0 premium Payoff -> profit Share price at Maturity Strike Price 2.6 Swaps A swap is a derivative in which two counterparties exchange cash flows of one party's financial instrument for those of the other party's financial instrument. The benefits in question depend on the type of financial instruments involved. For example, in the case of a swap involving two bonds, the benefits in question can be the periodic interest payments associated with such bonds. Specifically, two counterparties agree to exchange one stream of cash flows against another stream. These streams are called the swap's legs. The swap agreement defines the dates when the cash flows are to be paid and the way they are accrued and calculated. Usually at the time when the contract is initiated, at least one of these series of cash flows is determined by an uncertain variable such as a floating interest rate, foreign exchange rate, equity price, or commodity price. The cash flows are calculated over 18
a notional principal amount. Contrary to a future, a forward or an option, the notional amount is usually not exchanged between counterparties. Consequently, swaps can be in cash or collateral. Swaps can be used to hedge certain risks such as interest rate risk, or to speculate on changes in the expected direction of underlying prices. Swaps were first introduced to the public in 1981 when the World Bank entered into a swap agreement. Today, swaps are among the most heavily traded financial contracts in the world. The total amount of interest rates and currency swaps outstanding is more than $348 trillion in 2010, according to the Bank for International Settlements (BIS). The five generic types of swaps are interest rate swaps, currency swaps, credit swaps, commodity swaps and equity swaps. Furthermore there are many other types. So, let s analyze some generic types of swaps: 1) An interest rate swap (IRS) is a liquid financial derivative instrument in which two parties agree to exchange interest rate cash flows, based on a specified notional amount from a fixed rate to a floating rate (or vice versa) or from one floating rate to another. Interest rate swaps can be used for both hedging and speculating. 2) A currency swap (or a cross currency swap) is a foreign exchange derivative between two institutions to exchange the principal or interest payments of a loan in one currency for equivalent amounts, in net present value terms. Currency swaps are motivated by comparative advantage. A currency swap should be distinguished from interest rate swap, for in currency swap, both principal and interest of loan is exchanged from one party to another party for mutual benefits. For example, currency swaps are over-the-counter (OTC) derivatives. 3) A commodity swap is an agreement where a market or spot price based on an underlying commodity is traded for a fixed price over a specified period. No Commodities are exchanged during the trade. In this swap, the user of a commodity would secure a maximum price and agree to pay a financial institution this fixed price. Then, the user would get payments based on the market price for the commodity involved. On the other side, a producer wishes to fix his income and would agree to pay the market price to a financial institution, in return for receiving fixed payments for the commodity. 4) An equity swap is a financial derivative contract, like as a swap where a set of future cash flows are agreed to be exchanged between two counterparties at set dates in the future. Also, an equity swap involves a notional principal, a specified duration and predetermined payment intervals. Let s observe in the text below a chart from an equity swap. 19
In addition to text above, it s necessary to be a report in a kind of Swaps. It is called Credit default swap (CDS). A CDS is a financial swap agreement that the seller of the CDS will compensate the buyer (usually the creditor of the reference loan) in the event of a loan default or other credit event. This is to say that the seller of the CDS insures the buyer against some reference loan defaulting. The buyer of the CDS makes a series of payments, in other words the CDS fee or spread to the seller and receives a payoff if the loan defaults. In the event of default the buyer of the CDS receives compensation and the seller of the CDS takes possession of the defaulted loan. However, anyone can purchase a CDS, even buyers who do not hold the loan instrument and who have no direct insurable interest in the loan. If there are more CDS contracts outstanding than bonds in existence, a protocol exists to hold a credit event auction. So, the payment received is usually substantially less than the face value of the loan. Furthermore, CDS data can be used by financial professionals, regulators and the media to monitor how the market views credit risk of any entity on which a CDS is available. Finally, some claim that derivatives such as CDS are potentially dangerous in that they combine priority in failure. A CDS can be unsecured, without collateral, and be at higher risk for a default. 20
2.7 The usage of financial derivatives It s fine to write and discuss each other for economy and especially for financial derivatives, but it s necessary to learn the usage of them. In a few words, derivatives are used for the following: a) Hedge or mitigate the risk in the underlying, by entering into a derivative contract whose value moves in the opposite direction to their underlying position and cancels part or all of it out, b) Create option ability where the value of the derivative is linked to a specific condition or event, c) Provide leverage or gearing, such that a small movement in the underlying value can cause a large difference in the value of the derivative, d) Speculate and make a profit if the value of the underlying asset moves the way they expect, e) Switch asset allocations between different asset classes without disturbing the underlying assets, as part of transition management, f) Avoid paying taxes, and g) Create option ability where the value of the derivative is linked to a specific condition or event 2.8 Hedging A key feature of derivatives is hedging. A hedge is an investment position intended to offset potential losses or gains that may be incurred by a companion investment. In simple language, a hedge is used to reduce any substantial losses or gains suffered by an individual or an organization. Also, a hedge can be constructed from many types of financial instruments, including stocks, exchange-traded funds, insurance, forward contracts, swaps, options, many types of over-thecounter and derivative products, and futures contracts. Furthermore, public futures markets were established in the 19th century to allow clear, standardized, and efficient hedging of agricultural commodity prices. So, since then they have expanded to include futures contracts for hedging the values of energy, precious metals, foreign currency, and interest rate fluctuations. Derivatives allow risk related to the price of the underlying asset to be transferred from one party to another. We will explain hedging with an example below. Assume a wheat farmer and a miller could sign a futures contract to exchange a specified amount of cash for a specified amount of wheat in the future. Both parties have reduced a future risk. For the wheat farmer, the uncertainty of the price, and 21
for the miller, the availability of wheat. However, there is still the risk that no wheat will be available because of events unspecified by the contract, such as the weather, or not that one party will renege on the contract. Although a third party, called a clearing house, insures a futures contract, not all derivatives are insured against counter-party risk. From another view, the farmer and the miller both reduce a risk and acquire a risk when they sign the futures contract. The farmer reduces the risk that the price of wheat will fall below the price specified in the contract and acquires the risk that the price of wheat will rise above the price specified in the contract. On the other hand, the miller acquires the risk that the price of wheat will fall below the price specified in the contract and reduces the risk that the price of wheat will rise above the price specified in the contract. So, for one type of risk, one party is the insurer (risk taker) and for another type of risk the counter-party is the insurer (risk taker). In addition to the text above, hedging also occurs when an individual or institution buys an asset and sells it using a futures contract. The individual or institution has access to the asset for a specified amount of time and can then sell it in the future at a specified price according to the futures contract. Of course, this allows the individual or institution the benefit of holding the asset, while reducing the risk that the future selling price will deviate unexpectedly from the market's current assessment of the future value of the asset. Finally, we should say that derivatives trading of this kind may serve the financial interests of certain particular businesses. 2.9 Arbitrage and Speculation At this point, it s appropriate to be a report in the meaning so arbitrage as speculation. In finance, arbitrage is the market activity of buying and selling of same security on exchanges or between spot prices of a security and its future contract. Amazing a combination of matching deals that capitalize up on the imbalance, the profit being the difference between the market prices. When used by academics, an arbitrage is a transaction that involves no negative cash flow at any probabilistic or temporal state and a positive cash flow in at least one state; in simple terms, it is the possibility of a risk-free profit after transaction costs. An arbitrage, for instance, is present when there is the opportunity to buy low and sell high. In academic use, an arbitrage is risk-free. An arbitrage involves taking advantage of differences in price of a single asset or identical cash-flows. In common use, as in statistical arbitrage, it may refer to expected profit, though losses may occur, and in practice, there are always risks in arbitrage, some major and some minor. It is also used to refer to 22
differences between similar assets. So, People who engage in arbitrage are called arbitrageurs. Furthermore, derivatives can be used to acquire risk, rather than to hedge against risk. Thus, some individuals and institutions will enter into a derivative contract to speculate on the value of the underlying asset, betting that the party seeking insurance will be wrong about the future value of the underlying asset. Speculators look to buy an asset in the future at a low price according to a derivative contract when the future market price is high, or to sell an asset in the future at a high price according to a derivative contract when the future market price is less. Individuals and institutions may also look for arbitrage opportunities, as when the current buying price of an asset falls below the price specified in a futures contract to sell the asset. Subsequently, we will report the meaning of speculation. Speculation is the practice of engaging in risky financial transactions in an attempt to profit from fluctuations in the market value of a good trade such as a financial instrument, rather than attempting to profit from the underlying financial attributes embodied in the instrument such as capital gains, interest, or dividends. Many speculators pay little attention to the fundamental value of a security and instead focus purely on price movements. Also, speculation can in principle involve any tradable good or financial instrument. Speculators are particularly common in the markets for stocks, bonds, commodity futures, currencies, collectibles, fine art, real estate and of course derivatives. 23
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Chapter 3 Asian Options 3.1 Introduction Exotic options are options that do not share one or more of the characteristics of the plain vanilla options. There are two main types of exotic options. Correlation options and Path dependent options. Correlation options are options whose payoffs are affected by more than one underlying asset. Path dependent options are options whose payoffs are affected by how the price of the underlying asset at maturity was reached, the price path of the underlying asset. One particular path dependent option, called Asian option. It will be explained in the text below. So, an Asian option is a path-depending exotic option, which means that either the settlement price or the strike of the option is formed by some aggregation of underlying asset prices during the option lifetime. This thesis will focus on European style Arithmetic Asian options where the settlement price at maturity is formed by the arithmetic average price of the last seven days of the underlying asset. For this type of option it does not exist any closed form analytical formula for calculating the theoretical option value. There exist closed form approximation formulas for valuing this kind of option. One such approximate the value of an Arithmetic Asian option by conditioning the valuation on the geometric mean price. To evaluate the accuracy in this approximation and to see if it is possible to use the Black-Scholes formula for valuing Asian options. In general, the Asian approximation formula is helpful for valuing Asian options. In addition to the text below, let s see a small description of Asian Options. Asian options are one of the most popular path dependent options and are also called average-price options. The characteristic of an Asian option is that the payoff is dependent of the average price of the underlying asset, over some prespecified period and prespecified frequency, during the lifetime of the option. The average price of the underlying asset can either determine the underlying settlement price (average-price Asian options) or the option strike price (average-strike Asian options). Furthermore, the average prices can be calculated using either the arithmetic mean or the geometric mean. The type of Asian options that will be examined throughout this thesis is arithmetic price Asian options. As will be explained later, the difference between arithmetic and geometric price Asian options is very important. 25
Also, regarding to the Black-Scholes formula, in general, this formula overestimate the Asian option value. This is expected since the Black-Scholes formula applies to standard European options, which considers the underlying asset price at maturity of the option as settlement price. This price is in average higher than the Asian option settlement price when the underlying asset price has a positive drift. However, a useful formula is too Monte-Carlo formula. We will analyze all them in the text below. 3.2 Definition The payoff of an Asian style option (or average price option) depends on the difference between the average price of the underlying asset over a certain time period, and the strike price. Such options allow the investor to buy or sell the underlying asset at the average price instead of at the spot price. They are prevalent in commodity markets where a party may have regular and ongoing transactions in a particular underlying asset and hence a desire to hedge itself against price fluctuations. Asian options are also used in situations where the purchaser wants to cover many spot transactions using only one hedging instrument or in situations where it is prudent to reduce the dependence of an option on the spot price of the underlying on a single date. In general, Asian options are less expensive than their European counterparts, since the volatility of the average price will be less than the volatility of the spot price. As an example, consider regular consumers of crude oil whose supply price is not fixed, but is set weekly from a particular benchmark. They are concerned that there may be a spike in oil prices over the next few months and want to hedge themselves using options. They require that the payoff of the hedge reflects the weekly purchases made over a specified time period. An Asian style option can be tailored to meet this requirement through the use of weekly price fixings over the applicable period. The option captures changes in the commodity over the averaging period and is significantly less expensive than the alternative of purchasing a basket of European options each maturing on a given fixing date. Most Asian style options use an arithmetic average and sample at discrete and regular time intervals (daily, weekly or monthly closing prices). In addition, there are options that use a geometric averaging procedure. 26
3.3 Types of Asian options There are some different types of Asian options: a) Continuous arithmetic average Asian call or put defined by the type T 1 ( S) ( S( t) dt K) T or 0 1 Φ(S)=(K- S(t)dt) T b) Discrete arithmetic average Asian call or put defined by the type m 1 it ( S) ( S( ) K) m 1 m or ( S) (K S( )) i0 m 1 i0 m c) Continuous geometric average Asian call or put defined by the type T T 1 1 (S) (exp log( S( t) dt K) T or (S) (K exp log( S( t) dt ) 0 T 0 d) Discrete geometric average Asian call or put defined by the type m 1 it (S) (exp log S( K) m1 i0 m or T 0 m 1 it m 1 it (S) (K exp log S( ) m1 i0 m Denote the price of the arithmetic average Asian call and put options at time 0 by Ka, Ka, C ( S T ) and p ( S T ), respectively and denote the price of the O, geometric average Asian call and put options at time 0 by p K,g ( S T ), respectively. O, O, C K,g ( S T ) and O, 3.4 Asian approximation formula 3.4.1 Arithmetic Asian options The arithmetic mean doesn t follow a lognormal distribution and therefore it is not possible to obtain a closed form formula to price Arithmetic Asian option. However, since it is possible to approximate the arithmetic mean using the geometric mean, it is possible to derive an approximation of the price of an Arithmetic Asian option. One such way is to value the Arithmetic Asian option by conditioning on the geometric mean price of the underlying asset, c exprt E E max( A K,0) / G and p exprt E Emax( K A,0 / G, 27
where A is the arithmetic and G is the geometric mean of the underlying asset price. In the case when the averaging period has not yet started, the price for a non-dividend paying Arithmetic Asian option can be approximated by n i 2 ln(k) xi ln( K) c exp rt ( exp / 2 ( )) KN( ) n i1 x x x and n ln( K) i 2 ln(k) xi p exprt KN( ) ( exp / 2 ( )) x n i1 x x where 2 i ln( S) ( r / 2)( t1 ( i 1) ) Where points. i ( t1 ( i 1) t) 2 xi ( t1 t(( i 1) i( i 1) / 2 ) 2 ln( S) ( r / 2)( t (n1) / 2) x t1 t(n1)(2n 1) / 6n n 1 (ln( K) ) / R 2K n t 1 i1 i 1 2 2 2 xi i xi x 2 x 2 is the time to the first average point, t is the time between averaging 3.4.2 Geometric Asian options The geometric average share is the lognormal distribution, which was showed for the stock price earlier. The lognormal distribution of the geometric mean will not be proved here, but the pricing formula for a non dividend-paying stock can be expressed as, c S A N( d T ) K exp rt N( d ) where o j n j 2, n j n j 2, p K exp N( d ) S A N( d T ) d n j n j o j n j n j 2 So ln( ) ( r ) T1, n j ln(b j ) K 2 T 2, n j A exp rt( T T ) ( T T ) / 2 B T 2 j 1, n j 1, n j 2, n j j 1, n j n j ( n j 1) h ( T ) n 2 28