Intro to Derivatives 1C- Introduction to Financial Derivatives PDF

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Lecture Notes 1C: Derivatives Markets and Instruments Lecture Notes 1C: Derivatives Markets and Instruments In this section we define all the derivatives covered in this course, discuss in detail the differences between them, and describe the markets in which they trade. We will also explain how derivatives became such a popular financial asset and their application by both firms and investors. Finally, we give some quantitative values to the payoffs to three important derivatives: the forward contract, and the call and put options. Derivatives are created by contract between two parties, a buyer and a seller of the derivative, and they have a life span as specified by the contract. There is an additional asset referred to in the contract, known as the underlying asset. Most derivatives involve buying or selling (or at least the right to buy or sell) the underlying asset on (or sometimes before) a specified date at a specified price. Some other derivatives define other contractual payments that could change hands depending on the performance of a specified asset, currency exchange, interest rate, or some other variable. Similar to most other financial assets, derivatives are risky but it is also possible to use this risky asset to actually reduce the overall risk of either the buyer or the seller of the derivative, depending on what other risk exposure the buyer or seller holds. At this point it makes sense to actually define each derivative we will look at, so these statements will not sound so mysterious. Derivatives in some way are mysterious (I have often called them magical but perhaps that is a stretch), but my desire is for full clarity in these notes. One category of derivatives known as forward commitments includes the forward contract, the futures contract, and the swap contract. In each case the commitment for some future transaction is firm, and each party must follow through with their obligations otherwise they are considered to have defaulted on the contract. Forward Contract

A contract between buyer and seller which gives the buyer the right and the obligation to buy an asset at a specified future date at a price agreed upon today.

Example:

Two parties, agree to exchange one ounce of gold one year from today at a price of $480 per ounce.

Suppose this contract is made today, which we can define as t = 0. Notice that there is a specified forward price determined at t = 0, in this case F0 = $480, and a specified settlement date (expiration date) 1 one year from today. The quantity of gold exchanged is also specified as one ounce. We generally call the time of settlement t = T, so for the sake of this example, if t is counted in years, t = T = 1 at settlement. The price for trading the underlying asset immediately is known as the cash price or the spot price. This is just the regular price of gold that you are familiar with and it fluctuates constantly,

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For forwards as well as futures, the agreed date for the future transactions is sometimes referred to as the settlement date, sometimes the expiration date, and it can even be called the maturity date. To get you used to all the different ways used in practice, I will mix it up in my notes.

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Lecture Notes 1C: Derivatives Markets and Instruments like most financial assets. In this example, let us assume that the spot price of gold at t = 0 is $400. The buyer agrees to buy gold at t = T for F0 and the seller agrees to sell at that price. The forward price can be negotiated between parties, or if there are a lot of buyers and sellers, it is determined in equilibrium. Either way, it can change from minute to minute, same as the spot price, but once the contract is initiated, the negotiated forward price for delivery of the underlying at t = T will be fixed for this particular buyer and seller. One of the defining features of a forward contract is that the transaction will take place unless one of the parties defaults. If the spot price at the expiration of the forward is greater than $480 the buyer wants to go through with it, and if the spot price at expiration is less than $480, the seller wants to go through with it, but either way, the contract specifies that the gold must be exchanged. With a forward contract, the “buyer” of the forward becomes the buyer of the underlying at expiration, while the “seller” of the forward becomes the seller of the underlying. Value of the Forward at Settlement The value of a forward at settlement is very easy to determine once you are at settlement. You just need to think about which participant, buyer or seller of the contract, is entitled to what cash flows and what they also must give up. The buyer has both the right as well as the obligation to buy, so no matter where the future spot price, ST, ends up at settlement: Value of the forward to the buyer at t = T:

ST − F0

Value to the forward to the seller at t = T:

F0 − ST

Money coming to you is positive, while money you give up is negative. In the above, when you are at settlement, both ST and F0 are known quantities, but any time before settlement only F0 is known; ST, the spot price at settlement remains a random variable right up to the exact date of the expiration of the forward contract. Example:

Suppose that we are discussing the contract above, where F0 = $480. Now further suppose at settlement the spot price ST = 500.

Value of the forward to the buyer at t = T:

ST − F0 = 500 – 480 = 20

Value to the forward to the seller at t = T:

F0 − ST = 480 – 500 = −20

The buyer of the contract will gain while the seller suffers the loss. Also notice that for the purpose of determining the value of the contract at expiration, the spot price at t = 0 plays no role. Now suppose that ST = $460 at settlement.

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Lecture Notes 1C: Derivatives Markets and Instruments Value of the forward to the buyer at t = T:

ST − F0 = 460 – 480 = −20

Value to the forward to the seller at t = T:

F0 − ST = 480 – 460 = 20

Now the seller gains while the buyer suffers the loss. With a forward contract no money needs to be exchanged at t = 0, because both buyers and sellers will not initiate the contract unless they both feel that the forward price is fair to both of them. The forward price is not a fixed number at any point in time, but is allowed to fluctuate. In the microeconomic sense, the forward price brings buyers and sellers of the forward contract together in equilibrium. It is a bit like a price of a house, and the sale goes through when the price is right; no side payment needs to be made to make the sale happen. As you will see below, this is different from the options markets. Purpose of Forward Markets (as well as Futures Markets) It is now time to discuss the purposes of future markets and why they exist. First, you could be a speculator, and think gold will be much higher than $480 a year from today, therefore you would buy a contract (or multiples of this contract) or perhaps you think it will fall in which case you would sell. You might wonder why the speculator would transact in the forward market instead of the spot underlying commodity. One reason could be the increased leverage obtained from the forward market. You can control a much larger amount at risk with a small investment. In fact, in a theoretical sense, you can enter the forward market with absolutely no cost since no money needs to change hands at t = 0. But if we measure expected returns as a rate of return rather than a dollar return, speculating in the forward market is very risky although expected returns can also be quite large. Perhaps you are a gold mine operator, and you want to avoid an unknown selling price for the gold that is still in the ground. You would then be a hedger, and you want to sell a forward contract. You could also be a jewelry manufacturer and need a supply of gold in the future. You would still be a hedger, but now you want buy the forward contract. Already you should see that the speculator could be taking on a risk, while hedgers could actually be reducing their risk. It turns out that while derivatives are inherently risky, they can reduce the risk to certain participants. For example, the gold mine operator already has the price risk of not knowing what he can sell his gold for and by selling in the forward market, the two risks can cancel out. The jewelry manufacturer also has a risk for his future buying price and he can hedge by buying in the forward market. They each will know they are going to transact at $480 per ounce. Incidentally, futures markets (a form of forwards) got their start in the U.S. as a way for farmers to hedge their future products in the Midwest on the Chicago Board of Trade. There is one more player in this game, and he is quite important at determining the forward price. This is the arbitrageur. One way to motivate this discussion is to ask if there is another way that the jewelry manufacturer could have hedged in the discussion

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Lecture Notes 1C: Derivatives Markets and Instruments above. He or she could have purchased the gold at t = 0 for $400 but since he only needed the gold at t = 1, he would need to store it for a year, and more important, he would need to buy a year earlier than needed, so funds need to be borrowed for the purchase or at least the opportunity use of the cash is lost for a year. Which way the jewelry manufacture would act (buy a forward or buy the gold today at the spot price) should depend on which way is cheaper.

Arbitrage and the Pricing of Derivatives The above discussion leads to our first arbitrage valuation formula: The cost of carry formula for forwards: F0 = S0 + θ where θ is the cost of carry. The cost of carry should include the opportunity cost of the cash (interest expenses) as well as storage and insurance cost. The interesting fact about this model is that if it does not hold, under certain conditions anybody could make money, and the action of doing so will eventually drive the forward price to comply with the formula. For example, suppose the interest rate to borrow is 10%, and the storage and insurance is $20 per ounce in advance. If you borrow enough to buy both the gold and the cost of storage and insurance, which then is $420 altogether, your interest expense is 420 x .10 = 42, and your total cost of carry is 20 + 42 = 62. This means that F0 = 480 > S0 + θ = 400 + 62 = 462. The higher asset is the forward and the correct arbitrage prescription here is to borrow 420, buy the gold and store it, and simultaneously sell a forward. The cash flows to the arbitrageur is: at t = 0 buy gold pay storage borrow sell forward

−400 −20 +420 0 0

t=T value of gold at t = T and pay back (with interest) deliver gold and get forward price

ST −462 −ST 480 18

There are several important points to be made here. 1) The cash flows to the seller of the forward is F0 − ST. Just think who gives up what and gets what at settlement. The buyer of the contract gets the underlying asset and gives up the agreed upon forward price. The seller delivers the gold and gets the forward price. 2) At t = 0, there is no cash flow to buyer or seller due to initiating the contract because they both agree that the forward price is a fair price. This is actually set by all the players buying and selling contracts.

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Lecture Notes 1C: Derivatives Markets and Instruments

3) The arbitrageur notices that the forward price in this case is actually too high according to the cost of carry model so he becomes a seller of the forward. The reason the arbitrageur can do this easily because it is just as easy to sell a forward as to buy it, and you need not own anything to do it. 4) Notice that the arbitrageur has no interest in buying gold, he is just out to get the $18 per contract without risk at t = T (he does need to wait to get his money). He is neither hedging nor speculating initially. In some sense, once he buys the gold and sells the forward contract he is perfectly hedged, and that is why this is a riskless arbitrage. 5) Nobody knows the actual spot price of gold at t = T. ST is a random variable, but the arbitrageur is not at risk because he has the gold to pay off the forward contract. 6) If enough arbitrageurs keep buying gold, eventually it will go up and the forward price will go down, until the cost of carry model holds.2 We will discuss the opposite condition when F0 < S0 + θ in next week’s lecture. The purpose here was to demonstrate arbitrage as well as further explain the forward contract. Arbitrage is one of the most important concepts to come out of derivatives, although arbitrage forces work for many financial assets. The concept is also known as the law of one price. Two assets that lead to identical cash flows should cost the same price. Of course sometimes you need to factor in various transactions costs. We often value securities such as derivatives using arbitrage principles, which imply that we do not think the arbitrage opportunity continuous. As shown above, the very act of doing the arbitrage will bring prices back in line, but if nobody is doing the arbitrage, if nobody even bothers checking the prices to see if arbitrage is possible, prices might again get to the point where arbitrage is possible. If you studied efficient markets in finance, in another course, there are some similarities. For example, we might say that the work of many analysts make the stock market efficient. But if nobody did any analysis, what would keep stock market prices efficient? An efficient stock market is where the information available is already in the price, but this is due to analysts making buy/sell recommendations. Another derivative, very similar to the forwards is the futures contract. In fact these two, forwards and futures are almost the same but forwards trade in an over-the-counter market, where two private parties usually brought together by an intermediary, often a bank or investment firm, and the deal is brokered leading often to a unique contract. The market is unregulated and transactions are usually hidden from the public. There is a greater risk of default by either party, although parties like to trade with known counterparties. 2

Hedgers that need gold in the future should also buy the spot commodity at t = 0 and not use a forward contract. They will also contribute to the price pressure that changes prices until the cost of carry model will again hold.

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Lecture Notes 1C: Derivatives Markets and Instruments Futures Contracts A futures contract is an organized market’s version of a forward contract, where procedures are in place to reduce the risk of default and contracts are standardized. If you are familiar with the margin accounts that are used when you either borrow money to buy shares or to sell shares short (selling shares that you borrow and do not own), then you would be close to the procedures which reduce default in futures. Futures trade on organized exchanges such as the Chicago Board of Trade, and a clearinghouse steps inbetween each trade, but then imposes requirements on each trader. In particular, traders put up funds on deposit from which withdrawals or additions are made daily, depending on participants’ positions and the movement of the futures price. This is known as marking to market or daily settlement. 3 If the balance in any account moves below a prescribed limit, the participant must deposit more funds or the position is closed out. A further difference between futures and forward markets is the specification of the contract. As mentioned above, forward markets can be tailored to individual participants, but this makes the contract much less liquid. Futures, which trade on organized exchanges, have a very standardized contract which increases the participants in each contact making futures much more liquid. In fact, a futures market can be closed out anytime, by simply reversing the initial trade. If you initial sold a contract, you now buy it, and if the forward price went down in between the sale and the purchase, you would have made money. The reverse holds, if you purchased initially and later you sell. If the futures price went down you lose money. Our final forward commitment contract is a swap. In a swap contract, two parties exchange payments, but typically, only the difference between the two payments is made to one party. The direction of the payment is linked either to an interest rate or a currency rate, or even a stock or commodity price. Because it is a series of payments, it is sometimes described as a series of forward contracts. Because it is again an asset that has a market in over the counter markets, it is not as tightly regulated and mostly stays private. Options We now turn to contingent claims that are also known as options. There are two kinds: a call option and a put option. Call Option Contract between the buyer and seller of the call, giving the call buyer the right, but not the obligation, to buy some specified asset from the call seller at a specified future date but at a price agreed upon today. Put Option

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Similar to above, but giving the put buyer the right, but not the obligation, to sell the agreed upon asset to the put seller at a specified future date but at a price agreed upon today.

We will describe the marking to the market in the lecture on futures.

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Lecture Notes 1C: Derivatives Markets and Instruments There was only a single type of forward contract, with a seller and buyer for this forward, but there are two types of options, a call and a put; since most people in the industry know that calls and puts are options, they often omit the word “option”. The difference between options and forwards is the “obligation” in a forward to carry out the contract. With a forward contract, the buyer must buy and the seller must sell at the agreed upon future date. With a call, the buyer of the option has the right to buy the underlying asset but does not have to, and with a put, again the buyer does not have to sell. In fact, you could say the buyer of the call or the put has an “option”! Since in both cases, the seller of either the call or the put has no “option” at all and only the obligations to go through with the contract (and you can bet that only happens when he is on the losing end) a payment has to be made from the buyer to the option seller at the initiation of the contract. This is again a big difference between options and forwards. It will be more clear when we start to write down the value of an option at expiration (term used by options markets instead of “settlement”), but the only benefit the seller of an option (call or put) will ever have is the payment they receives for initiating the contract on the sell side. Before we turn to more differences, let us turn to some definitions related to options. Unfortunately, sometimes there exists two definitions for the same concept, and you will have to learn both of them since you are bound to run into both. Underlying Asset

The asset stated in the contract. In the examples below, the underlying asset will be a share of stock.

Strike Price or Exercise Price

The stated price in the contract.

Exercising

Taking advantage of the right conveyed by the option and buying (for the case of a call) or selling (for the case of the put) the underlying asset in exchange for the exercise price.

Expiration Date or Maturity Date

The last day on which the option may be exercised.

American Option

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