Bab 1

 Trading

Topic 7 Strategies invloving Options LEARNING OUTCOMES By the end of this topic, you should be able to: 1. Apply the option knowledge on option strategies ; 2. Evaluate the a single option and a stock strategies; 3. Comprehend the option spreads strategies; 4. Appreciate the flexibility of options;

 INTRODUCTION One of the main advantages of options is their flexibility. This flexibility arises from the fact that options may or may not be exercised. The flexibility enables an investor to establish positions that may not be possible with other instruments. The inherent flexibility also enables options to be combined with positions in the underlying asset and in other derivative instruments. An option strategy is established with an objective in mind or for a given market outlook (Bacha, 2001). The flexibility in usage of option and possible combinations with other assets results in infinite number of possible option strategies. For ease of clarity the strategies are characterized into four broad categories:    

7.1

Uncovered/naked positions Hedge positions Spreads Combination strategies

Uncovered/naked positions

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TRADING STRATEGIES INVLOVING OPTIONS

An uncovered or ‘naked’ position is where one takes a position in an asset without establishing an offsetting position. Using calls, put and stocks (underlying asset), there are six possible uncovered positions.      

Long call Short call Long put Short put Long stock Short stock

7.2 Hedge Strategies A hedge strategy combines an option with the underlying asset in such a way that the overall position either reduces or eliminates risk. A fully hedged position is riskless. The combination is such that price movements offset each other. Three common hedge strategies: 1. Portfolio Insurance (Hedging exposure to a long stock position) 2. Hedging exposure to a short stock position 3. Conversion Strategy (Locking-in a fixed value of underlying asset) In next sections, we are going to explain each of them one by one.

7.2.1 Portfolio Insurance - Long Stock Position A long stock position exposes the investor to downward movements in the stock’s price. Thus an investor with a long stock position would want to protect downside risk while keeping as much as possible the upside profit potential. The investor should combine the long stock position with an option position that would profit if the stock goes down in value. Two option positions gain when the underlying asset goes down in value:  Short call position (receive only premium);  Long put position (will gain if underlying asset price is down). As we seen, the short call position only provides downside protection to the extent of the premium received while the upside potential on the stock cannot be realized by the investor. Therefore, the logical hedge choice would be to use the long put position which provides upside potential if the underlying asset price is down. This strategy is often commonly referred to as portfolio insurance. Long stock position- Illustration

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Suppose you had just gone long one lot of Company ABC stock at a price of RM 10.00 each, for a total investment of RM 10,000. You wish to protect yourself from any short term downside movement in price. Suppose 3-month, at-the-money put options on Company ABC stock are being quoted at RM 0.20 or 20 cents each. The appropriate option strategy to hedge the long stock position would be: Combined position

Long 1 lot Company ABC stock @ RM 10 Long 1, 3-month Company ABC put @ RM 0.15

Long stock Long Put

Payoff to hedged long stock position: Table 7.1: Payoff to hedged long stock position Stock price Value of long Profit/loss to long put Value at maturity stock positions position at 20 cents position (RM) (RM) (RM) (RM) 6 6000 3800 8 8000 1800 10 10,000 (200) 12 12,000 (200) 15 15,000 (200)

Payoff profile to portfolio insurance will be:

of combined at maturity 9800 9800 9800 11,800 14,800

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Figure 7.1: Payoff profile to portfolio insurance Source: Module writer The combined position: Long position in a stock and Long position in a put Portfolio Insurance (Long stock position) – Key Features Table 7.2: Strategies of Portfolio Insurance When to use Risk profile Break-even point Desired objective

When one needs protection against falling value of the underlying asset in which one has a long position. Limited downside risk, unlimited upside potential. Since overall position is that of a long call; exercise price + premium = RM 10.00 + .20 = RM 10.20 To gain from potential upside rally while limiting downside risk.

7.2.2 Hedging exposed to a Short Stock Position With a short stock position, the investor needs protection from a rise in the underlying stock price. The hedge position should combine the short stock position with an option that rises when the underlying stock price increases. The logical strategy would be to long a call option on the stock.  Long call position (will gain if underlying asset price is up). Short stock position – Illustration Suppose you shorted Company DEF stock at RM 20.00. You would like to hedge yourself from potential large losses if the stock price rises. How can you hedge? Let say: 3-month, RM 20.00 call option on the stock is being sold at a premium of RM 0.30 or 30 cents. The appropriate option strategy to hedge the short stock position would be: Combined position

Short 1 lot of Company DEF stock at RM 20.00 Long 1, 3-month RM 20.00 call @ RM 0.30

Short stock Long call

Payoff to hedged short stock position: Stock price Value of long Profit/loss to long put Value of combined at maturity stock positions position at 20 cents position at maturity

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(RM) 16 18 20 22 25

(RM)


(RM)

4000 2000 0 (2,000) (5000)

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(RM) (300) (300) (300) 1700 4700

3700 1700 (300)

(300) (300)

Figure 7.2: Short stock position Source: Module writer The combined position: Short position in a stock and Long position in a call Hedging a Short Stock Position – Key Features Table 7.3: Hedging a Short Stock Position – Key Features When to use Risk profile Break-even point Desired objective

When one needs protection against rising values of the underlying asset of which one is short. Limited downside risk, unlimited upside potential. Since overall position is that of a long put; exercise price – premium = RM 20.00 – RM 0.30 = RM 19.70 To gain from falling prices while limiting risk associated with rising prices.

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Spread Strategies

A spread strategy can be thought of as a speculative position with a safety net. A spread position typically involves the establishment of offsetting positions in the same asset but across different markets, at different maturities or different exercise prices. Spread positions are established to profit from expected marginal price movements while protecting downside risk. Two types of spreads are common, bull spreads and bear spreads. Both these spreads can be established using either calls or puts. Table 7.4: Types of Spreads Types of spread

Name of strategy Using calls

bull call spread

Using puts

bull put spread

Using calls

bear call spread

Using puts

bear put spread

Bull spread

Spread

Put Spread

7.3.1 Bull call spread Illustration Suppose you are moderately bullish about the performance of Gombak Berhad stock over the next 60 days. You want to apply options to benefit from the expected moderate stock price appreciation same time you want to minimise your downside risk. In this case, the appropriate option strategy would be to create a bull spread which could be set up using call options. Assume the following two call options are available:  Gombak Berhad, RM 19.50 call @ RM 0.40  Gombak Berhad, RM 20.50 call @ RM 0.15 The bull call spread would require the purchase of the call with the lower exercise price and sale of the call with the higher exercise price.

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Value combined position (0.25)

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Long RM 19.50 call @ RM 0.40 Short RM 20.50 call @ RM 0.15 Table 7.5: Payoff of bull call spread Stock (RM) 15 17 19 19.6 21 23 25

price

at

maturity Profit/loss to long RM 19.50 call @ RM 0.40 (0.40) (0.40) (0.40) (0.30) 1.1 3.1 5.1 (4.35)

Profit/loss to Short RM 20.50 call @ RM 0.15 0.15 0.15 0.15 0.15 (0.35) (2.35) 0.75

(0.25) (0.25) (0.15) 0.75 0.75

Figure 7.3: Illustration of Bull call spread Source: Module writer

7.3.2 Bull Put Spread A bull put spread is the use of put options instead of calls to establish the spread. Suppose in our previous example. The following 60-day put options are available:  Gombak Berhad, RM 19.50 put @ .20

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 Gombak Berhad, RM 20.00 put @ .60 The bull put spread strategy would be: Long 19.50 put @ .20 Short 20.00 put @ .60 Table 7.6: Payoff of bull put spread Stock price at Profit/loss to Long Profit/loss to maturity (RM) 19.50 put @ .20 Short 20.00 put @ .60 15 4.3 (4.4) 17 2.3 (2.4) 19 0.30 (0.40) 19.6 0.20 (0.20) 21 0.60 (0.20) 23 0.60 (0.20) 25 0.60 (0.20)

Figure 7.4: Illustration of Bull put spread Source: Module writer

Value combined position (0.10)

(0.10) (0.10) 0 0.40

0.40 0.40

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7.3.3 Summary of Bull Spreads A Bull Call/Put Spread - Key Features Table 7.7: Summary of Bull Spreads When to use Risk profile Break-even point Desired objective

Strategy Bull call spread

Bull put spread

When one is neutral to bullish or moderately bullish about the underlying asset price. Limited loss, limited profit. As described above. To take advantage of expected marginal up movement while also limiting downside risk. Break-even point Maximum loss Maximum Profit Lower exercise Difference in Difference in price + difference premium exercise price – in premium difference in premium Higher exercise Difference in Difference in price - difference exercise price – premium in premium difference in premium

7.3.4 A Bear Call Spread A bear spread is an appropriate strategy when one is mildly bearish or neutral to bearish. Illustration: Suppose you are neutral to bearish about Gombak Corporation stock. You want to make some money without exposing yourself to large potential losses if the stock price in fact goes up. The following 60-day call options on Gombak Corporation stock are available:  Gombak Corporation, RM 9.50 call @ .90  Gombak Corporation, RM 10.50 call @ .15 Appropriate Bear Call Spread would be established by following strategy:

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Short RM 9.50 call @ .90 Long RM 10.50 call @ .15 Table 7.8: Payoff of bear call spread Stock price at Profit/loss to Short Profit/loss to Value maturity (RM) RM 9.50 call @ .90 Long RM 10.50 combined position call @ .15 7 0.90 (0.15) 0.75 8 0.90 (0.15) 0.75 10 0.40 (0.15) 0.25 10.25 0.15 (0.15) 0 11 0.35 (0.25) (0.60) 13 2.35 (0.25) (2.60) 15 4.35 (4.60) (0.25)

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Figure 7.5: Illustration of bear call spread Source: Module writer

7.3.5 A Bear Put Spread The strategy is established by going long the higher exercise put and shorting the lower exercise one. Illustration:

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Using the earlier example of Gombak Corporation stock, suppose the following puts are available: Gombak Corporation, RM 9.50 put @ .50 Gombak Corporation, RM 10.50 put @ .80 Appropriate Bear Put Spread would be established by following strategy: Short RM 9.50 put @ .50 Long RM 10.50 put @ .80 Table 7.9: Payoff of bear put spread Stock price at Profit/loss to Short Profit/loss to Value maturity (RM) Long RM 10.50 combined RM 9.50 put @ .50 put @ .80 position 7 (2.00) 2.70 0.70 8 (1.00) 1.70 0.70 9 0 0.70 0.70 10.2 0.50 (0.50) 0 11 (0.80) (0.30) 0.50 13 0.50 (0.80) (0.30) 15 0.50 (0.80) (0.30)

Figure 7.6: Illustration of bear put spread Source: Module writer

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7.3.6 Summary of Bear Spreads Bear Call/Bear Put Spread – Key Features Table 7.10: Summary of bear spreads When to use Risk profile Break-even point Desired objective

Strategy Bear call spread

Bear put spread

When one is neutral to bearish or moderately bearish about underlying asset price. Limited loss, limited profit. As described above. To take advantage of expected marginal fall in underlying asset price while limiting loss potential

Break-even point Lower exercise price + difference in premium

Maximum loss Maximum Profit Difference in Difference in exercise price – premium difference in premium Higher exercise Difference in Difference in price - difference premium exercise price – in premium difference in premium

ACTIVITY 7.1 Suppose you had just gone long one lot of Company XYZ stock at a price of RM 20.00 each, for a total investment of RM 200,000. You wish to protect yourself from any short term downside movement in price. Suppose 6-month, at-the-money put options on Company XYZ stock are being quoted at RM 0.30 or 30 cents each. What is the appropriate option strategy to hedge this position? Show the payoff profile of the appropriate strategy.

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SELF CHECK 7.1 (a) Diffecntiate the diffrent positions in option contracts; (b) Explain both hedge strategies such as portfolio insurance and hedging exposure to a short stock position with help of diagrams; (c) Explain both spread strategies such as bull and bear spreads with help of diagrams;

7.4

Combination Strategies

A combination involves the use of both types of options, calls and puts as part of the strategy. The difference between the spread and a combination is that a spread uses only one type of options, either a call or a put, whereas combinations use both call and put options. Though combinations can be in several variants, the two most common combination strategies are:  The straddle  The strangle Both these strategies are designed for extremes of volatility.

7.4.1 Straddle Strategy The straddle strategy comes in two variants, the long straddle and short straddle. Table 7.11: Variants of straddle strategy

Straddle Strategy

Type of straddle Long straddle

Short straddle

Purpose Involves the purchase (long) of both a call and a put option on the same underlying asset, at the same exercise price and of the same maturity. Involves shorting a call and a put of the same underlying asset, exercise price and maturity.

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A long straddle strategy is designed to profit from extreme volatility, while the short straddle to profit from minimal volatility.

7.4.2 Long Straddle Illustration: Siti Bhd., the country’s largest oil refiner, has just been subject to a hostile takeover by Melati Bhd. You realize that this is a potentially volatile situation. Suppose 60-day at-the-money calls and puts are priced as follows:  Siti Bhd., RM 10.00 call @ .20  Siti Bhd., RM 10.00 put @ .10 To benefit from the underlying stock volatility, you could establish a long straddle position as follows: Long RM 10.00 call @ .20 Long RM 10.00 put @ .10 Table 7.12: Payoff of long straddle Stock price at Profit/loss to Long Profit/loss to Value maturity (RM) Long RM 10.00 combined RM 10.00 call @ .20 put @ .10 position 7 (0.20) 2.90 2.70 8 (0.20) 1.90 1.7 9 (0.20) 0.9 0.70 10.2 0 (0.10) (0.10) 11 0.7 0.8 (0.10) 13 2.8 (0.10) 2.7 15 4.8 (0.10) 4.7

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Figure 7.7: Illustration of Long Straddle Source: Key features of long straddle: Table 7.13: Features of long straddle Scenario Risk profile Break-even point Maximum loss Desired objective

Underlying asset likely to undergo extreme volatility. Limited loss, unlimited profit. Call exercise + Total premium; Put exercise – Total premium Total premiums To take advantage of potential large price swings.

7.4.3 Short Straddle Illustration: A short straddle position can be established as follows: Short RM 12.00 call @ .40 Short RM 12.00 put @ .30 Table 7.14: Payoff of short straddle Stock price at Profit/loss to Short Profit/loss to Value maturity (RM) Short RM 12.00 combined RM 12.00 call @ .40 put @ .30 position 8 0.40 (3.70) (3.30)

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9 10 11 11.3 12 12.7 13 14 15


0.40 0.40 0.40

(2.70) (1.70) (0.70)

(2.30) (1.30) (0.30)

0.40 0.40 (0.30) (0.60) (1.60) (3.60)

(0.40) 0.30 0.30 0.30 0.30 0.30

0

0.70 0 (0.30) (1.30) (3.30)

Figure 7.8: Illustration of short straddle Source: Module writer Key features of short straddle: Table 7.15: Features of short straddle Scenario Risk profile Break-even point Maximum profit Desired objective

When minimal price movement is expected. Limited profit, unlimited loss. Call exercise + Total premium; Put exercise – Total premium Total premiums To profit from unchanged underlying asset price.

7.4.4 Strangle Strategy Strangles is used for largely the same objective and market/asset price expectation as straddles. The difference between a straddle and strangle is that in a strangle, the options are bought (sold) at different exercise prices.

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Table 7.16: Variants of strangle Strategy

Strangle strategy

Type of strangle Long strangle

Short strangle

Purpose Appropriate when extreme volatility is expected to cause the underlying asset to break-out of its trading range. Appropriate when minimal volatility would mean continued range trading.

7.4.5 Long Strangle Illustration: Taman Bhd. is a large Oil & Gas firm whose stock has traditionally been trading in the RM 9.50 to RM 10.50 range. You believe that given the company’s problems with due to oil price fluctuations, the stock price could come under turbulence. Taman Bhd RM 10.5 and RM 9.50 of 60-day calls and puts are quoted as follows:  Taman Bhd., RM 10.50 call @ .30  Taman Bhd., RM 9.50 put @ .50 The long strangle position is established by following strategy: Long RM 10.50 call @ .30 Long RM 9.50 put @ .50 Table 7.17: Payoff of strangle Strategy Stock price at Profit/loss to Long Profit/loss to maturity (RM) RM 10.50 call @ .30 Long RM 9.50 put @ .50 7 (0.30) 2.00 8 (0.30) 1.00 8.7 (0.30) 0.30 9 (0.30) 0 10 (0.30) 0 11 0.20 (0.50)

Value combined position 1.70 0.70 0

(0.30) (0.30) (0.30)

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11.3 12 13 15


0.50 1.20 2.20 4.20

(0.50) (0.50) (0.50) (0.50)

0 0.70

1.70 2.70

Figure 7.9: Illustration of strangle Strategy Source: Module writer Long Strangle – Key Features Table 7.18: Features of strangle Strategy Scenario Risk profile Break-even point Maximum loss Desired objective

Underlying asset price expected to breakout of current trading range. Limited loss, unlimited profit. Call exercise + Total premium; Put exercise – Total premium Total premiums To take advantage of extreme volatility causing a price breakout from range.

7.4.5 Short Strangle Illustration: You have just read that given current inventory, expected supply and demand, palm oil prices are expected to remain stable over at least the next 3 months.

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Given this this information, you believe that the Wangsa Group (KLK), a large oil palm plantation, to continue trading in its current RM 9.50 to RM 10.50 range. The 60-day, RM 9.50 put and RM 10.50 call are quoted as follows:  Wangsa, RM 10.50 call @ .30  Wangsa, RM 9.50 put @ .50 The short strangle is established by the following strategy: Short RM 10.50 call @ .30 Short RM 9.50 put @ .50 Table 7.19: Payoff of short strangle Stock price at Profit/loss to Short Profit/loss to maturity (RM) RM 10.50 call @ .30 Short RM 9.50 put @ .50 7 0.30 (2.00) 8 0.30 (1.00) 8.70 0.30 (0.30) 9 0.30 0 10 0.30 0 11.30 (0.50) 0.50 12 (1.20) 0.50 13 (2.20) 0.50 15 (4.20) 0.50

Figure 7.10: Illustration of short strangle Source: Module writer

Value combined position (1.70) (0.70) 0

0.30 0.30 0 (0.70)

(1.70) (3.70)

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Short Strangle – Key Features: Table 7.20: Features of short strangle Scenario Risk profile Break-even point Maximum profit Desired objective

7.5

Underlying asset expected to continue trading in its current range. Limited profit, unlimited loss. Call exercise + Total premium; Put exercise – Total premium Total premiums To take advantage of underlying asset price remaining within range.

Strategies by Market Outlook

This subtopic will demonstrate different strategies by outlook by graphical forms so that readers can easily understand the strategies. Table 7.21: Strategies by market outlook Bullish

Neutral bullish

to

Bearish

Neutral bearish

to

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Neutral (min volatility)

Extreme volatility

Source: Bacha, 2012

SELF CHECK 7.2 (a) What is a combination strategy? (b) Explain both combination strategies such as Straddle and strangle strategies with help of diagrams;

 As one of the longest topic in the module, we have learnt different strategies which associated with options such as uncovered/naked positions, hedge positions, spreads and combination strategies.  We have explained each strategy with appropriate examples.  Now the students should appreciate the flexibility of options and its associated strategies.

Uncovered/naked positions Hedge strategies Portfolio Insurance Hedging exposure to a short Stock position Conversion Strategy Spread strategies Bull spread Bull call spread Bull put spread

Bear spread Bear call spread Bear put spread Combination strategies Straddle Long Straddle Short Straddle Strangle Long Strangle Short Strangle

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 Bacha, O. I. (2012). Financial Derivatives: Markets and Applications in Malaysia. 3rd Edition, McGraw-Hill (Malaysia).  Hull, J. C. (2012). Options, futures, and other derivatives. Pearson Education, Inc., publishing as Prentice Hall, United States of America.

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 Options

Topic 8 Pricing Model LEARNING OUTCOMES By the end of this topic, you should be able to: 5. Calculate the option prices by suing one and two steps binomial model; 6. Comprehend on American options 7. Interpret and calculate the option prices by applying the Black-Scholes pricing model;

 INTRODUCTION As another important aspect of options, in this chapter, we are going to look at how to price options instruments. There are two common methods to price options such as binominal model and the Black –Scholes pricing model. We will look into each of them one by one.

8.1

Binomial Option Pricing Model (BOMP) – Call Option

The binomial option pricing model is according to the logic that the current value of the option is equal to the present value of the possible payoffs to the option at maturity (Bacha, 2012, Hull, 2012). The BOMP is a discrete time model, in that underlying asset price changes at a given fixed time interval. Illustration: Suppose we want to find the value of a European style call option on an underlying stock which is currently selling at RM 20.00 with the following assumptions: Table 8.1: Illustration of a European style call option Option types Price changes Price movement

The call option on the stock has a RM 20 exercise price and one year maturity Only change in price once during the one year The percentage change in the stock’s price is 20%, that is, it can either go up or down by a fixed 20%

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Probability Risk-free interest rate

The probability of an up or down movement is an equal 50% 10% per annum

The possible stock and call values at maturity would be:

Figure 8.1: Single Period Binomial Option Pricing Model for call option Source: Module writer At maturity in one year, the stock’s price could either be 20% higher or lower. The probabilities of each outcome occurring is 50% or 0.5. Given this the call, denoted Ct now, would have a payoff of RM 4.00 (Stock price – Exercise price) or RM 0. The call is only valuable if it ends in-the-money. Since the probability of the stock going up is 50%, the RM 4.00 payoff from the call has a 50% probability. Using Single Period Binomial Option Pricing Model the value of call is RM 1.82. Formula

Illustration

= RM 1.82

8.1.1 Two-step binominal trees Holding all other assumptions are same, suppose we now allow the stock’s price to change twice within a year, i.e. once every 6 months. The stock price movement and corresponding payoff would be:

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Figure 8.2: Two-step binominal trees for call option Source: Module writer

The value of call using BOPM would be: RM 1.9955.

8.1.1 Three-step binominal trees Suppose we now relax the periodic price change assumption and allow the stock price to change at 3 times per year.

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Figure 8.3: Three-step binominal trees for call option Source: Module writer

How to get 0.125? We can follow the tree: 0.5*0.5*0.5=0.125. The second component we need to multiply by three: the trees will be three such as UP UP DOWN, UP DOWN UP and DOWN UP UP. Therefore, the value of call would be RM 2.68 by using BOPM.

ACTIVITY 8.1

Find the value of a European style call option on an underlying stock which is currently selling at RM 10.00 with the following assumptions:  The call option on the stock has a RM 10 exercise price and one year maturity;  change in price three times during the one year;  The percentage change in the stock’s price is 10%, that is, it can either go up or down by a fixed 10%;  The probability of an up is 60% and down movement is an equal 40%;  Interest rate is 8% per annum;

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Probabilities and Volatility

In the last topic, we have looked at only the price changes during the given time period. This subtopic will discuss the probability and volatility changes during the given time period.

8.2.1 Probability changes In scenarios till now, probability has been assumed homogenous across all possibilities. This may not be the case always, as a bullish opinion, would lead to higher weightage assigned to positive outcomes. For example:

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Figure 8.4: Three-step binominal trees with probability changes Source: Module writer In a 3 period scenario where probabilities for the alternative paths are not equal anymore the value of the call is different than equal probability weighted scenario. Therefore, the value of the call would be RM4.04 in this case.

8.2.2 Volatility Changes In earlier scenarios we had assumed that the underlying stock price would change by 20%. Suppose we now increase the volatility to 30%. How would the call value change?

Figure 8.5: Three-step binominal trees with volatility changes Source: Module writer

With 10% increase in volatility from 20% to 30%, the call option value goes from RM 2.68 to RM 3.956. This is how volatility impacts on the value of call option.

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Binomial Option Pricing Model (BOMP) – Put Option

The way pricing put options is exactly the same as that of the valuation of calls. In our earlier illustration, we assume a put option now instead of a call option.

Figure 8.6: Three-step binominal trees for put option Source: Module writer Volatility and Probability would impact the put option price in similar fashion as impact on call options. We are not going elaborate on these scenarios. However, students are encouraged to do so in order to better understand the impacts on probability and volatility on put options.

8.4

The Black-Scholes Pricing Model (BSOPM)

The BSOPM has applications in various areas beyond option pricing alone. Its authors, Fisher Black and Myron Scholes, were awarded the Nobel Prize for Economics in 1995. The biggest advantage of the BSOPM over other models is

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that the BSOPM provides a closed-form solution to option pricing. This model is in continuous time form as it is opposite to Binominal pricing model where the model based on discrete time form, meaning, the time interval between underlying asset price change is as small as to approach zero.

8.4.1 Underlying Assumptions of the BSOPM Table 8.2: Underlying assumptions of BSOMP 1 2 3 4 5 6 7

Efficient markets with frictionless trading. No transaction costs (the model ignores bid-ask spread, commissions etc.). Option has European style exercise. The underlying stock will pay no dividends during the maturity of the option. Underlying stock returns are log normally distributed (this means that the logarithmic stock returns are normally distributed). The risk-free interest rate remains unchanged over option maturity. Underlying stock volatility is constant over option maturity.

Of these assumptions the last two of unchanged interest rates and constant volatility of the underlying asset until option maturity are considered the most restrictive.

8.4.2 BSOPM for a call option The Black-Scholes option pricing model for a call option is as follows:

How to get d1 and d2? We can use the following formula:

d1

d2

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Where: S K T r N(.)

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Spot price of underlying asset Exercise price of call option Time to expiration Risk free interest rate Exponential function of rf interest rate and time Cumulative standard normal distribution (SND) function Volatility of underlying asset as measured by standard deviation Natural logarithm of S/K.

From the above illustrations, we could see that there are three steps to calculate the price of call options: Table 8.3: Steps in calculating price of call options First step Second step Third step

Calculate d1 and d2. Using the cumulative normal distribution table, find the values of N(d1) and N(d2). Plug the values into the model and solve.

Illustration Suppose: Stock price, S Exercise price, K Interest rate, r Maturity, T = 180 days Standard deviation, σ What is the correct price of the call? Step 1:

= RM 21 = RM 20 = 0.08 = 0.5 = 0.5

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d1

d2

= 0.43 Step 2: Look for N(d1) and N(d2) from cumulative standard normal distribution table: N(d1) = N(0.43) = 0.6664 N(d2) = N(0.08) = 0.5319 Step 3:

= RM13.99 – RM 19.22*0.5319 = RM 13.99- RM 10.22= RM3.77 Decomposing the above call value of RM 3.77 into intrinsic and time values, the intrinsic value here is RM 1.00 while the remainder RM 2.77 would constitute time value.

8.4.3 BSOPM for a Put Options Though the BSOPM was developed to price European call options, the model can just as easily be used in valuing an European style put option. We can employ the following formula:

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The steps involved in valuation are similar to earlier steps for call options. Illustration – Continuing with the earlier example we computed the price of the call option in above example to be RM 3.77. The first two steps, (i) calculating d1, d2, and (ii) finding N(d1) and N(d2) are the same. However, a small adjustment has to be made before we plug-in and solve for option value in step (iii). The small adjustment is to convert N(d1) and N(d2) to N(–d1) and N(–d2). As N(d1) was calculated earlier for the call to be 0.6664. Therefore: N(–d1) = 1- N(d1) = 1-0.6664= 0.3336 In a same way, since N(d2) was 0.5319: N(–d2) = 1- N(d2) = 1-0.5319= 0.4681

= RM 19.22*0.4681 – RM7.01= RM 8.00- RM 7.01= RM 0.99 The Put Option price is RM 0.99 or 99 cents.

8.5

Determinants of Option prices

In BSOPM there are 5 input variables: 1. Stock/Underlying Asset Price

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The change in stock or underlying asset price is positively correlated to call values and negatively to puts. This relationship between underlying asset price and option values are known as deltas.

2. Exercise Price The exercise price has a negative correlation with call options and a positive one with put options. Raising the exercise price benefits put options but works against calls. 3. Volatility Underlying asset price volatility has a positive correlation with both option prices. The relationship between option value and underlying asset volatility known as vega.

4. Interest Rates The impact of interest rates on option values can be seen directly from the intrinsic value equations. An increase in interest rates would increase call intrinsic value since the present value of the exercise price would be lower. In the case of put options, the effect is opposite; higher interest rates reduce the present value of exercise price. This relationship between interest rates and option values is termed rho.

5. Time to Maturity

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The relationship between time to maturity and option values is termed theta. Time to maturity is positively correlated with calls but has an ambiguous relationship to put values. This is due to the opposite impact of time to maturity on the put intrinsic and time values.

8.6

Implied Volatilities

Implied volatility is the volatility implied in an option price. As illustrated earlier, there are five (5) parameters go into the BSOPM to determine the option value. It is possible to work out the sixth variable given any five variables. Thus, given the four other input variables (s, k, r and T) and the call value, we can derive the volatility estimate that justifies the given call value. This would be the implied volatility. There are two common uses of implied volatility in option trading. First, traders can use implied volatility estimates to determine ‘expensiveness’ of an option relative to other options.

the

Second common use of implied volatility is in determining option mispricing. A rule of thumb use by traders is to compare implied volatility with actual or historical volatility.

SELF CHECK 8.1 (a) (b) (c) (d)

What are two common methods to price options? Explain the binominal option pricing model with examples; Explain the Black –Scholes options pricing model with examples; What are the determinants of option prices?

SELF CHECK 1.2

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In this chapter, we demonstrated how to price options instruments.

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We have illustrated the two common methods to price options such as binominal model and the Black –Scholes pricing model for both call and put options. Also, we have discussed the determinants of option prices and implied volatilities.

Binominal option pricing model One step binominal trees Two-step binominal trees Three-step binominal trees Probability changes

Volatility changes Black –Scholes pricing model Determinants of option prices Implied volatilities

Bacha, O. I. (2012). Financial Derivatives: Markets and Applications in Malaysia. 3rd Edition, McGraw-Hill (Malaysia). Hull, J. C. (2012). Options, futures, and other derivatives. Pearson Education, Inc., publishing as Prentice Hall, United States of America.