Topic 8 INVESTMENT ANALYSIS

INTRODUCTION

Among the most innovative and most rapidly growing markets to be developed in recent years are the markets for financial futures and options. Futures and options trading are designed to protect the investor against interest rate risks, exchange rate risks and price risks. In the financial futures and options markets, the risk of future changes in the market prices or yields of securities are transferred to someone (an individual or an institution) who is willing to bear that risk. Financial futures and options are used in both short-term money markets and long-term capital markets to protect both borrowers and lenders against the risks involved. Although financial futures and options are relatively new in the field of finance, risk protection through futures and options trading is an old concept in marketing commodities. As far back as the Middle Ages, traders in farm commodities developed contracts calling for futures delivery of farm products at

TTooppiicc

88 Derivatives

Market

LEARNING OUTCOMES

By the end of this topic, you should be able to:

1. Explain the nature and characteristics of forward contract, futurescontract and options;

2. Analyse the factors that influence the price of options;

3. Differentiate between options and futures contract;

4. Justify the importance of hedging in managing risk; and

5. Evaluate how derivatives are priced using the basic Binomial Pricingtheorem and the Black-Scholes pricing model.

TOPIC 8 DERIVATIVES MARKET

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a guaranteed price. Trading in rice futures began in Japan in 1697. In the United States, the Chicago Board of Trade established a futures market in grain in 1848. More recently, the Chicago Board developed futures and options markets for financial instruments. The Malaysian first derivatives exchange was established in July 1993 under the name Kuala Lumpur Options and Financial Futures Exchange (KLOFFE). In October 1995, the Malaysian Derivatives Clearing House (MDCH) was established for both KLOFFE and the Commodity and Monetary Exchange of Malaysia (COMMEX), and the Malaysian Derivative Exchange (MDEX) is the only derivative exchange in Malaysia. MDEX in now known as Bursa Malaysia Derivatives Berhad (BMD) and it operates under the supervision of Securities Commission.

GENERAL DESCRIPTION OF DERIVATIVES

The underlying asset may be a share, Treasury Bill, foreign currency or even another derivative security. For example:

(a) The value of a share option depends upon the value of the share on which it is written.

(b) The value of a Treasury Bill futures contract depends upon the price of the underlying Treasury Bill.

(c) The value of a foreign currency forward contract depends upon the foreign currency forward rates.

(d) The value of a swap depends upon the value of the underlying swap contract.

Two types of derivative security, futures and options, are actively traded on organised exchanges. These contracts are sstandardised with regard to a description of the underlying asset, the right of the owner, and the maturity date. Forward contracts, on the other hand, are nnot standardised; each contract is customised to its owner, and they are traded in what is called over-the-counter. Options can be found embedded in other securities, convertible bonds and extendible bonds being two such examples. A convertible bond contains a provision that gives an option to convert the security into common share. As extendible bond contains a provision that gives an option to extend the maturity of the bond.

A dderivative security is a financial contract written on an underlying asset.

8.1

TOPIC 8 DERIVATIVES MARKET 128

Once we understand simple derivative securities, then it will be easy to understand the complicated examples of derivative securities such as these embedded options. This topic explains derivative securities: forward contracts, futures, and both call and put options.

FORWARD CONTRACT

The specified price will be referred to as the delivery price. At the time the contract is written, the delivery price is set such that the value ff of the forward contract is zero. The party that agrees to sell the underlying asset is said to have a short position. The party that agrees to buy the underlying asset is said to have a long position. A forward contract is settled at the delivery date, sometimes called the maturity date. The holder of the short position delivers the specified quantity of the assets at the specified place and in return receives from the holder of the long position a cash payment equal to the delivery price. No cash exchange occurs prior to the delivery date. For example, suppose that a company enters into a forward contract today, at date t. The forward contract matures at date TT. Let ff(t, T) denote the forward price. There are two arguments in this price. The first argument, t, denotes the date that forward price is quoted, and the second argument, TT, denotes the delivery date of the contract. When the contract is initiated, by definition the forward price equals the delivery price, denoted by KK(t). The delivery price is determined so that no cash is exchanged at this time; the delivery price is fixed over the life of the contract. f(t, T) K(t)

t T Initial date Delivery date Let SS(t) denote the spot exchange rate (RM/USD) at date tt. When the contract matures at date TT, the spot exchange rate is denoted by SS(T). This spot exchange rate is unknown when the forward contract is initiated. It is called a random

8.2

A fforward contract is an agreement to buy or sell a specified quantity of asset at a specified price, with delivery at a specified time and place.


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variable. The value of the forward contract at the delivery date, date TT, to the long position initiated at date tt for one currency is:

S(T) K(t) Note that the argument t in the delivery price refers to the date the contract was initiated. The value of the forward contract at delivery equals the value of the foreign currency, SS(T), less the delivery price paid, KK(t). As illustrated in the above example, the value of the forward contract at delivery can be either positive or negative. A graph of the possible values is shown below.

Figure 8.1: Possible value of a forward contract

The delivery price KK(t) equals the forward price ff(t, T) when the contract is initiated. If the spot exchange rate at the delivery date is less than the delivery price SS(T) < K(t), then the value of the forward contract is negative; otherwise, it is zero or positive. The delivery price equals the prevailing forward price, KK(t) = f(t, T), when the contract is initiated. Once the contract is written, the delivery price is fixed over the life of the contract. The forward price, which represents the delivery price of newly written contracts, of course can change. If you contracted with a financial institution tomorrow, date tt + 1, about buying USD for delivery date TT, in general there would be a new delivery price or forward price, KK(t + 1) = f(t + 1, T). This completes the institutional description of a forward contract. We will return to these contracts when we analyse pricing and hedging.


FUTURES CONTRACT

This part of the definition of a futures contract is identical to that of a forward contract. But futures contracts differ from forward contracts in four important ways. The differences are:

(a) Futures contracts allow participants to realise gains or losses on a daily basis, while forward contracts are cash settled only at delivery.

(b) Futures contracts are standardised with respect to the quality and the quantity of the asset underlying the contract, the delivery date or period, and the delivery place if there is physical delivery. In contrast, forward contracts are customised on all these dimensions to meet the needs of the two counterparties.

(c) Futures contracts are settled through a clearing house. The clearing house acts as a middleman. This minimises credit risk as the second party to a futures contract is always the clearing house.

(d) Futures markets are regulated, while forward contracts are unregulated. Now let us look at an example of a futures contract. Consider that A wants to sell futures (that means he wants to sell the underlying asset) that matures in three months. The price of the underlying asset now is RM2.30 and the risk-free interest rate is 4% per year. What A could do is: (1) borrow RM2.30 from a bank for three months, and (2) use the borrowed money to buy the asset at RM2.30. After three months, A will sell the asset at the agreed price f and the proceeds will be used to pay the loan and he can keep the balance. This strategy is called the ccash-and-carry strategy.

8.3

A ffutures contract is an agreement to buy or sell a specified quantity of an asset at a specified price, and at a specified time and place.

List at least three examples of forward contracts available in theMalaysian market. You can refer to the Malaysian Investor website athttp://www.min.com.my under products, and other resourcematerials such as business newspapers and magazines. Also list theadvantages and disadvantages of a forward contract.

ACTIVITY 8.1


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The payoff from this strategy is:

Stock investment 2.30 Futures contract [2.30 ff ] Repay borrowing 2.30 (1 + 0.04 0.25) Net payoff ff 2.30(1 + 0.04 0.25)

8.3.1 Clearing House

This intervention of the clearing house means that the futures market has no counterparty risk. If A plans to buy futures and B plans to sell futures, both parties will refer to the clearing house to fulfil their intentions. The clearing house is thus the counter party to every contract. In this case, B is not the counter party to A. The clearing house must have reserves to guarantee that its contracts are executed and are considered risk-free. It actually accepts that the counterparty may default on the contract and for this it charges a small fee for each contract executed. Further, the clearing house only accepts contracts from recognised traders and sets a mmargin account. In margin accounts, investors are required to maintain or keep some amount of money. In Malaysia, the clearing house derivatives is the Malaysian Derivative Exchange (MDEX). Please note that if the net payoff is not equal to zero, then arbitrage profit is possible. The functions of the Clearing House can be summarised as shown in Figure 8.2.

Figure 8.2: Clearing House functions


8.3.2 Settlement Price

A futures contract is marked-to-market each day. When each trading is closed, the exchange will establish the closing price, which is the ssettlement price. This settlement price is used to compute the investors position, whether at a loss or a gain compared with the initial settlement price agreed upon at the inception of the contract. This means that the change in the futures price over the day is credited (debited) to the account of the long (short) if the change is positive. If the change is negative, the account of the long (short) is debited (credited).

8.3.3 Daily Margin

When a person enters into a futures contract, the individual is required to deposit funds in an account with the broker. This account is called the mmargin account. The exchange sets the minimum amount of margin required, but brokers can increase the margin if they feel that the risk of the investors default is increased. This margin account may or may not earn interest. The economic role of the margin account is to act as collateral to minimise the risk of default by either party in the futures contract. Since the price of futures is marked-to-market, the gain or loss on the futures contract will also change on a daily basis. Thus the margin needs to be adjusted to reflect these changes. These changes are reflected in what is known as the maintenance margin. This will ensure that a minimum amount is kept in the margin account with respect to the changes in gain or loss. Lets look at an example of the initial margin and the maintenance margin. Consider A, who on 3 March enters into a futures contract to buy 100 troy ounces of gold at the futures price of RM365 per troy ounce. The initial margin for the contract is set at RM2,000 and the maintenance margin is set at 75% of the initial margin, or RM1,500. Table 8.1 below shows the calculation that reflects the changes in the margin when the daily price changes. We observe that on 5 March, the price drops to RM359 and a cash flow of RM300. This reduces the initial margin to RM1,400 which is below the maintenance margin of RM1,500. At this point, A has to top-up the margin account back to the original RM2,000. Thus, A has to withdraw or put in RM600 into the margin account. We assume that A will withdraw any excess margin (in excess of RM2,000).


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Table 8.1: Margin and Marking to Market

Date Futures Price Cash Flow Cash Deposit/ Withdrawal Ending Margin

3/3 365 0 2000 2000 4/3 362 300 0 1700 5/3 359 300 600 2000 6/3 364 +500 500 2000 7/3 365 +100 100 2000 8/3 367 +200 200 2000

8.3.4 Basis

Let FF(t, T) denote the futures price at date tt, for delivery at time TT. Let the contract be written on an underlying asset, with spot price SS(t). In a well-functioning and efficient market, the futures price equals the price of the underlying asset at the delivery date. Figure 8.3 shows that FF(t, T) = S(T). s(t) S(T) tt F(t, T) TT

Figure 8.3: F(t, T) = S(T) In fact, the futures price at T for immediate delivery should be equal to SS(T). That is FF(T, T) = S(T). The difference between the futures price and the spot price is known as the bbasis. Thus, the basis at time tt is;

Basist = F(t, T) S(t). In the forward or futures contracts investment, the risk involved is represented by the change in the value of the basis. Thus, when the change in the basis is relatively large, the risk involved in that investment is also large. This is important when we consider the use of forward or futures contracts to hedge, for instance, our investment in the cash market. We will consider this idea later in the topic. A typical graph of the basis with respect to maturity is shown in Figure 8.4.


Figure 8.4: Basis with respect to maturity

In the graph above, we see that the basis is positive. However, this is not always true. Basis can also be negative. What is important is that the graphs should converge.

8.3.5 Using Futures for Hedging

Futures are usually used to hedge our investment or lock the price of the underlying asset. Thus with hedging, we can construct a portfolio consisting of assets on both the spot and the derivatives markets. It is important to understand that in the spot market, we are dealing with the price risk whereas in the futures market, we are faced with the basis risk. If bbasis0 = basisT, then we should be able to use perfect hedging, which means the value of the underlying asset at time 00 and time TT are equal. Hedging is easier to understand by using examples. Lets look at examples where futures are used for hedging. Be sure to follow the example and understand what happens to the value of our portfolio when bbasis0 basisT.

We will assume that as the price in the spot market increases, the price in the futures market will also increase. This is to say that the two prices are closely


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correlated, which is what we usually observe. Suppose you sell one unit of futures and buy one unit of the underlying asset. Thus, your portfolio is (S F). At time tt, your portfolio is worth ((S(t) F(t)), and at time TT, it is worth ((S(T) F(T)). Thus, the change in spot price is ((S(T) S(t)) and the change in the futures price is (F(T) F(t)). To see how the value of our portfolio changes, refer to the following formula.

(S(t) F(t)) = S(t) F(t) = (S(T) S(t)) (F(T) F(t)) = (S(T) F(T)) (S(t) F(t)) = BasisT Basist = (Basis) If there is no change in the basis, then the value of our portfolio remains the same, thus we have locked in the value of the portfolio. The result would be different if the values of the basis are not constant. Now, lets look at the mechanism in using futures for hedging using two companies, Gold Mining Company and Jewellery Company. The Gold Mining Company expects to sell 1,000 ounces of gold next month and the Jewellery Company expects to buy 1,000 ounces of gold next month. However, to hedge against the risk of changes in the price of gold in a months time, both companies want to lock in todays price. Assume that todays price for gold is $352.40 per ounce and the current futures price is $397.80. Each futures contract is for 100 ounces. You are advised to carefully study Tables 8.2 8.5 on the next page to understand the mechanism of using futures in hedging. Attention should be given to what happens to hedging when the basis at the time the contract is initiated is not equal to the basis when the futures mature. This will explain the basis risk which will ultimately influence the gain or loss when futures are used. We also note that since the Gold Mining Company wishes to sell gold in the future, then it will short or sell futures when the contract is initiated. Similarly, the Jewellery Company will long or buy futures since it wishes to buy gold in the future.


Table 8.2: Hedging Price That Locks Gold Spot Price: Spot Price Decrease

Assumption

Spot price when hedging is made $352.40 per oz. Futures price when hedging is made 397.80 per oz. Spot price when hedging expires 304.20 per oz. Futures price when hedging expires 349.60 per oz. Number of ounces hedged 1000 Number of ounces in one futures contract 100 Number of futures contract used 10

Short (sell) hedging by Gold Mining Company

Cash market Futures market Basis When hedging is made Value of 1,000 ounces: Sell 10 contracts: 1,000 x $352.40 = $352,400 10 x 100 x $397.80 = $397,800 $45.40 per oz. When hedging expires Value of 1,000 ounces: Buy 10 contracts: 1,000 x $304.20 = $304,200 10 x 100 x $349.60 = $349,600 $45.40 per oz. Loss in cash market Gain in futures market = $48,200 = $48,200

Overall loss or gain = $$0

Long (buy) hedging by Jewellery Company

Cash market Futures market Basis When hedging is made Value of 1,000 ounces: Buy 10 contracts: 1,000 x $352.40 = $352,400 10 x 100 x $397.80 = $397,800 $45.40 per oz. When hedging expires Value of 1,000 ounces: Sell 10 contracts: 1,000 x $304.20 = $304,200 10 x 100 x $349.60 =$349,600 $45.40 per oz. Gain in cash market Loss in futures market = $48,200 = $48,200 Overall loss or gain = $$0


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Table 8.3: Hedging Price That Locks Gold Spot Price: Spot Price Increase

Assumption



Cash market Futures market Basis When hedging is made

Value of 1,000 ounces: Sell 10 contracts: 1,000 x $352.40 = $352,400 10 x 100 x $397.80 = $397,800 --$45.40 per oz.

When hedging expires Value of 1,000 ounces: Buy 10 contracts: 1,000 x $392.50 = $392,500 10 x 100 x $437.90 = $437,900 --$45.40 per oz. Gain in cash market Loss in futures market = $40,100 = $40,100

Overall loss or gain = $$0



Value of 1,000 ounces: Buy 10 contracts: 1,000 x $352.40 = $352,400 10 x 100 x $397.80 = $397,800 --$45.40 per oz.

When hedging expires Value of 1,000 ounces: Sell 10 contracts: 1,000 x $392.50 = $392,500 10 x 100 x $437.90 = $437,900 --$45.40 per oz. Loss in cash market Gain in futures market = $40,100 = $40,100 Overall loss or gain = $$0


Table 8.4: Hedging Price That Locks Gold Spot Price: Spot Price Decrease and Basis Increase

Assumption






Overall loss or gain = $$36,200




When hedging expires Value of 1,000 ounces: Sell 10 contracts: 1,000 x $304.20 = $304,200 10 x 100 x $437.90 = $385,800 --$81.60 per oz. Loss in cash market Gain in futures market = $48,200 = $12,000



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Table 8.5: Hedging Price That Locks Gold Spot Price: Spot Price Increase Basis Increase

Assumption


Shhort (sell) hedging by Gold Mining Company








When hedging expires Value of 1,000 ounces: Sell 10 contracts: 1,000 x $392.50 = $392,500 10 x 100 x $474.10 = $474,100 --$81.60 per oz. Loss in cash market Gain in futures market = $40,100 = $76,300 Overall loss or gain = $$36,200


OPTIONS

Before we proceed further with options, an important concept that needs to be stressed is the difference between options and futures contracts, how the pricing of options are determined, and how investors can reduce their investment risk or reach their investment objectives by using options contracts.

By giving the right, the writer will receive a fee called the price of the options or the options premium.

8.4

An options contract is a contract where the writer (or seller) of the options gives to the buyer of the options the right, but not an obligation, to buy (call options) or sell (put options) to the writer something (or the underlying) at a specified price, during a specified period (or specific time).

1. Suppose you bought a share at time t at RM4.70, and the price of the futures on the share is RM4.60. At time T, the price of theshare drops to RM3.90 and the price of the futures drop toRM4.00. If you short (sell) the futures at time t, what is the valueof your share at time T?

2. Consider a forward contract written on a non-dividend paying

asset. The current spot price is RM65. The maturity of thecontract is 90 days and the simple interest rate for this period is4.50% per annum.

(a) Determine the forward price. What is the value of thecontract?

(b) A corporate client wants a 90-day forward contract with the delivery price set at RM60. What is the value of thecontract?

(Assume a 365-day year).

EXERCISE 8.1


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Elements in an options contract consist of the following:

(a) Either the buyer of the call options has the right to buy something or the buyer of put options has the right to sell something.

(b) The underlying is usually a financial instrument, index or commodity that could be traded.

(c) The strike price or the exercise price.

(d) Expiration date.

(e) Either the buyer of the options can exercise anytime during a specified duration (for American options) or at a specific time (European options).

Now lets look at an example of European call options.

Price = cc Exercise price = KK Expiration date = three months Initial price of share (the underlying) = SS0

This means that the buyer of the call options can buy the share in three months time at the price XX, no matter what the price of the share will be.

Options can be traded in an organised market or over-the-counter. There are three advantages when options are traded in an organised market:

(i) The exercise price and the expiration date can be standardised.

(ii) Clearing house can function in the options market similar to that in the futures market.

(iii) The transaction cost is much lower compared with the options traded over-the-counter.

Usually institutional options buyers need a specific or a tailor-made option that matches their needs. This usually happens in portfolio management of fixed income securities. In fact, in portfolio management of fixed income securities like bonds, over-the counter trading is more popular because the risk in the cash market can be hedged by using the options.


8.4.1 Options Moneyness

The transaction can be immediately profitable or not. Moneyness is always viewed from the buyers or the long position viewpoint and not from the sellers viewpoint. Furthermore, moneyness is obtained by comparing the exercise price with the spot value of the underlying asset. There are three types of moneyness in options. They are shown in Table 8.6:

Table 8.6: Types of Moneyness

Types of Moneyness

Remarks However, the remarks are true only when:

For call options For put options

In-the-money (ITM)

An option is said to be in-the-money if it is profitable to immediately exercise.

Exercise price (K) < Spot price of underlying asset

Exercise price (K) > Spot price of

underlying asset

At-the-money (ATM)

An option is said to be at-the-money if it does not matter if immediately exercised.

Exercise price (K) = Spot price of

underlying asset

Exercise price (K) = Spot price of

underlying asset

Out-of-the -money (OTM)

An option is said to be out-of-the-money if it is not profitable to immediately exercise.

Exercise price (K) > Spot price of underlying asset

Exercise price (K) < Spot price of underlying asset

8.4.2 Difference between Options and Futures Contracts

There are some fundamental differences between futures and options. They are:

(a) Only the sellers (not the buyers) are obliged to buy or sell. The buyers of options need not buy or sell the underlying. In the futures contract, the buyers and sellers are obliged to buy or sell. Thus, the main advantage of options is where the holder or the buyer of options will benefit from an upside benefit while limiting a downside loss.

Options moneyness describes the relationship between the options stated price and the price of the underlying asset and determines if it is a profitable transaction.


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(b) The buyer of options must pay a fee or the price of the options to the seller to get the right. In the futures contract, there is no exchange of money when the contract is initiated.

(c) The buyer of the options will decide on the price of the options to buy (for call options) or to sell (for put options) but can take the opportunity if the price of the options is low. In the futures contract, the price is already fixed and the parties to the contract cannot obtain profit or suffer losses from any price movement.

8.4.3 Characteristics of Returns and Risk in Options

There are four basis positions in options contracts. Most other options contracts are some combinations of the four basis positions. The four basic options contracts are:

(a) Buy call options

(b) Sell call options

(c) Buy put options

(d) Sell put options The buyer of an option can seize opportunities to make a profit from the movement of the price of the underlying asset. However, they must pay a fee for this. The maximum profit for the seller of options is the price of the options itself. At the same time, the seller of options is also exposed to the risk of loss from the movement of the price of the underlying asset. Figures 8.6 8.9 show the gain or loss incurred by the seller (writer) or the buyer of the respective options with the following assumptions:

In the earlier section, we discussed the definition of options and futures contracts. We also discussed the characteristics of each security. Based on your understanding, what is the main difference between options and futures contracts?

SELF-CHECK 8.1


Price of options = $3 Exercise price ((x) = $100 C = sell call options + C = buy call options P = sell put options + P = buy put options S = sell share + S = buy share The following graphs show what happens to the profit or value of options when the price of the underlying asset changes. A full understanding of the graphs is very important as they are used in more exotic options portfolios, since the concept is similar. For instance, in understanding put-call parity, all we need to do is add the relevant linear graphs to look at the profit or the value of our options portfolio.

Figure 8.5: Profit/loss for seller (writer)

of call options

Figure 8.6: Profit/loss for buyer of call options


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Figure 8.7: Profit/loss for Seller (writer) of put

options

Figure 8.8: Profit/loss for buyer of put options

Figure 8.9: Value of asset (S)


To understand the options transactions (buy and sell) better, let us look at the following example. Suppose that todays price of a MAS share is RM5.00. A European three-month call option on the MAS share is quoted as RM4.90 MAS @ 0.30. (i.e. the price of this call option is RM0.30). This means that in three months time the buyer of the call option can exercise or buy a MAS share at RM4.90, at whatever the price MAS shares will be. Based on this information:

Price of MAS in 3 months time Exercise Profit/Loss

Price of MAS share goes up to RM5.30 Exercise (buy) 5.30 4.90 0.30 = 0.10

Price of MAS share goes up to RM5.70 Exercise (buy) 5.70 4.90 0.30 = 0.50

Price of MAS share drops to RM4.30 Dont exercise 0.30

Using the same information given above, except that instead of a call option, the option is a put option. Then we will have:

Price of MAS in 3 months time Exercise Profit/Loss

Price of MAS share goes up to RM5.30 Dont exercise 0.30

Price of MAS share goes up to RM5.70 Dont exercise 0.30

Price of MAS share drops to RM4.30 Exercise (sell) 4.90 4.30 0.30 = 0.30

8.4.4 Put-Call-Parity

Graphs can also be used to look at the profit/loss of some combinations of call, put options and the underlying asset. One of the most important portfolios of options is the portfolio (SS + P C). This means when we buy one share, we buy one put on the share and sell one call on the share. This portfolio is an interesting portfolio as it gives what is known as put-call parity. The graph for the portfolio is shown in Figure 8.10:


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Figure 8.10: Graph for portfolio S + P C To calculate the profit for portfolio (SS + P C), lets look at the following example. Assume the exercise price for the options is XX. We consider what happens to the portfolio when the price of the underlying asset is less than XX and when it is greater than XX. Further assume that the price of the underlying asset is S when the option is exercised.

Price < X Price > X

Share S S

Put (X S) 0

Call 0 (S X)

Profit for S + P C X X

We observe that whatever the value of the underlying asset is, it always equals to X, the exercise price. It should be clear that the graph in Figure 8.10 gives the same result. Thus, we can plot the profit of a portfolio of options by using either graphs or calculations similar to the above. The portfolio (SS + P C) is a riskless portfolio and should earn a riskless rate of return (Rf ).


We can value the portfolio (SS + P C) by using the formula below:

S + P - C =

1+ Tf

XR

This formula is known as the pput-call parity and it explains the relationship between the price of call and put options.

8.4.5 Factors Affecting the Price of an Option

The diagram below shows the time diagram between time 0 and T, where we include the relevant variables that affect the price of options. Then we will tabulate to see how they affect the price of options.

where:

c : European call option price C : American call option price p : European put option price P : American put option price S0 : Share price today ST : Share price at option maturity K : Strike price D : Present value of dividends during options life T : Life of option r : Risk-free rate for maturity T with continous compounding : Volatility of stock price We summarise how the variables affect the price of options below:

Variable c p C P S0 + + K + + T ? ? + +

+ + + + r + + D + +


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Where:

++ means the variable and the option move in the same direction (for example if rr increases then cc will also increase); means the variable and the option move in opposite directions (for example if DD increases then c will decrease); and ?? means it depends on our expectations regarding the price of the underlying asset.

THEORIES IN PRICING OF OPTIONS

We will look at how options are priced by considering the Binomial Pricing Model and the Black-Scholes model. First, we will explain the Binomial Pricing Model, followed by the Black-Scholes model, for which we will only state the formula. The derivations of Black-Scholes model is beyond the scope of this module.

8.5.1 The Binomial Pricing Model

In this model, we use a rriskless portfolio (SS C) where we buy a unit of the underlying asset and sell a call option on the asset. We further assume that there are only two possible states market goes up or market goes down. The assumptions used in this model are as follows:

(a) There are no market frictions.

(b) Market participants entail no counterparty risk.

(c) Markets are competitive.

(d) Market participants prefer more wealth to less.

(e) There are no arbitrage opportunities. We will now consider both the cash and the options markets. Assume that if the market goes up, the price will increase by 20% and if the market goes down, it will decrease by 20%. Thus, if the initial price of the share is SS0, it will be S0(1 + 0.20) if the market goes up and SS0(1 0.20) if the market goes down. Here, we assume that we only need one call option (in reality, we need to first calculate the number of options needed).

8.5


Figure 8.11: Stock market

Figure 8.12: Option market

If we combine the two markets, we will have:

Figure 8.13: Combination of stock market and option market

Since the portfolio is a riskless portfolio, 1.20S0 fu = 0.80S0 fd. Thus we can find the value of C by considering the formula below (assuming the maturity of the portfolio is T);


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0 00 0

1.20 0.8S C = or

1+ 1u u

T Tf f

S f S fS C

R R

From the formula above, we obtain:

00

1.20

1u

Tf

S fC S

R

It should be noted that S0, Rf and T are known when the contract is initiated. Thus, we are left with the prices of call and put. If we know the price of the call, we can find the price of the put, and vice-versa.

8.5.2 The Black-Scholes Model

The Black-Scholes model for pricing put (pp) and call (cc) options is as follows:

where:

0 1 2

2 0 1

20

1

20

2 1

2

2

rT

rT

c S N d K e N d

p K e N d S N d

In S IK r I Td

TIn S IK r I T

d d TT


N(d1), N(d2), = the cumulative probability density. The value for N(.) is obtained from a normal distribution that is tabulated in most statistics textbooks.

c = European call option price p P = European put option price S0 = Share price today ST = Share price at option maturity K = Strike price T = Life of option R = Risk-free rate for maturity T with continous compounding = Volatility of stock price

1. Consider a European call option on an asset with price S(t), strikeprice K, and maturity T.

(a) What is the payoff to this call option at date T? Consider aEuropean put option on the same asset S(t) with strike priceK and maturity T.

(b) What is the payoff to this put option at date T?

(c) Is the payoff to the call option exactly opposite to the payoffto the put option? Explain.

2. The current share price is RM50. The value of a European call

option with a strike price of RM47 and maturity 100 days is RM4.The 100-day default-free discount rate is 5%, assuming a 360-dayyear.

(a) For a put option with a strike price of RM47 and maturity100 days, you are quoted a price of RM2. Is this consistentwith the absence of arbitrage? Please justify your answer.

(b) If your answer to (a) is that arbitrage is possible, howwould you construct an arbitrage portfolio to takeadvantage of the situation?

EXERCISE 8.2


153

Derivatives are usually used to hedge the risk that exists in the cash markets. It should be understood that when we discuss hedging, we are concerned with reducing or transferring risk in the cash market.

The common derivatives used are the forward, the futures and the options. The two types of options include call and put options. Factors influencing the options are the price of the underlying asset, the

maturity, the exercise price, interest rate, dividend and the variance of the price of the underlying.

The prices of call and put options are related through the put-call parity. This

means that once we know the price of call options, we can theoretically find the price of put options.

Pricing of options can be complicated compared with pricing of futures. The

basic idea of options pricing can be seen from the binomial options pricing. This method discusses the pricing in the two-state model and can be extended into a multi-period model.

3. Suppose you construct the following portfolio:

(a) Long 1 call, strike price 40

(b) Short 1 call, strike price 50

(c) Short 1 call, strike price 70

(d) Long 1 call, strike price 80

All options are written on the same share and mature at the sametime. Without the aid of diagrams, describe the total value of theportfolio when the options mature.

Topic 8 INVESTMENT ANALYSIS

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