Laurence Booth Sean Cleary. LEARNING OBJECTIVES Options 12 12.1 Describe the basic nature of call options and the factors that influence their value.

Laurence Booth

Sean Cleary

LEARNING OBJECTIVESLEARNING OBJECTIVES

Options1212.1 Describe the basic nature of call options and the factors that influence their

value.

12.2 Describe the basic nature of put options and the payoffs associated with long and short positions in put options.

12.3 Explain how to use put-call parity to estimate call and put prices, and explain how it can be used to synthetically create call, put, and underlying positions.

12.4 Explain how to use the Black-Scholes option pricing model to price call options.

12.5 Explain how options are traded and what is meant by implied volatility.12.6 Understand by means of the simplest option pricing model what factors

affect the value of a call option and provide a guide to their order of magnitude.

12.1 CALL OPTIONS

• Call options give the holder the right, but not the obligation, to buy an underlying asset at a fixed price for a specified time

• The price at which an investor can buy the underlying asset is called the exercise or strike price and the

3Booth • Cleary – 3rd Edition

last date the option can be converted or exercised is the expiration date• Example: A call option on an underlying asset has a $50 exercise price; the

payoff for the buyer of the call is depicted in Figure 12-1

• Example: A call option on an underlying asset has a $50 exercise price. The payoff for the buyer of the call is depicted in Figure 12-1.

• This option is in the money, or would generate a


positive payoff if exercised today, for any underlying asset price above $50• This option is out of the money for any underlying asset price below $50• When the underlying asset price equals the strike price an option is at the

money

12.1 CALL OPTIONS

• An investor who buys a call option takes the long position and has the right but not the obligation to exercise the option if it is in the money

• The counterparty is the option writer who takes the short position; if the


option owner exercised the option, it would be exercised against the writer who would have to sell the underlying asset to the holder of the call option for the strike price (even though the strike would be less than the market value of the underlying asset if the option were in the money)

• The payoff diagram for an option writer is given in Figure 12-2

12.1 CALL OPTIONS

• Example: A call option on an underlying asset has a $50 exercise price. • Consider the following table showing the payoffs to both the holder and

the writer of the option and different asset prices

6

Asset Price ($) 30 40 50 55 60 70

Call holder’s payoff ($) 0 0 0 5 10 20

Call writer’s payoff ($) 0 0 0 -5 -10 -20

• Notice that when the underlying asset price < the strike price, the option is not exercised

• Notice also that the net payoff between the holder and the writer is zero in aggregate because options are an example of a zero sum game

Booth • Cleary – 3rd Edition

12.1 CALL OPTIONS

Call Option Values• The intrinsic value (IV) is the value of the option at expiration and

is positive when the option is in the money and zero when it is out of the money

• At expiration, when the call is in the money, the value of the option is the asset price minus the exercise price, S – X

• At expiration, when the call is out of the money, the value is zero because it will expire unexercised


12.1 CALL OPTIONS

Call Option Values• Equation 12-1: IV(Call) = Max (S – X, 0)• The option premium is the market value of the option, and the

time value (TV) is the difference between the option premium and intrinsic value

• Equation 12-2: Option premium = IV + TV• Equation 12-3: TV = Option premium – IV• Note: options that are deep are so far in (or out of) the money

that they are almost certain to be (or not to be) exercised


12.1 CALL OPTIONS

• Figure 12-3 shows that the call option’s value is non-linearly related to the underlying asset price

• Table 12-1 shows how various factors affect call and put prices


12.1 CALL OPTIONS

12.2 PUT OPTIONS• Put options give the holder the right, but not the obligation,

to sell an underlying asset at a fixed price for a specified time• The price at which an investor can buy the underlying asset

is called the exercise or strike price and the last date the option can be converted or exercised is the expiration date

• Example: A put option on an underlying asset has a $50 exercise price; the payoff for the buyer of the put is depicted in Figure 12-4 and the payoff for the writer of the put is depicted in Figure 12-5



12.2 PUT OPTIONS

• Example: A put option on an underlying asset has a $50 exercise price. • Consider the following table showing the payoffs to both the holder and

the writer of the option and different asset prices

12

Asset Price ($) 30 40 50 55 60 70

Put holder’s payoff ($) 20 10 0 0 0 0

Put writer’s payoff ($) -20 -10 0 0 0 0

• Notice that when the underlying asset price > the strike price, the option is not exercised

• Notice also that the net payoff between the holder and the writer is zero because in aggregate options are an example of a zero-sum game


12.2 PUT OPTIONS

• The intrinsic value of a put option is the strike price minus the underlying asset price X – S when it is in the money, and zero when it is out of the money

• Equation 12-4: IV(Put) = Max(X – S, 0)• The time value and option premium for put options is calculated

in the same manner as for call options (see above)


12.2 PUT OPTIONS

• There is a major difference between put and call options that can only be exercised on the expiration date and those that can be exercised at any time before the expiration date

• European options can only be exercised at maturity• American options can be exercised at any time up to and

including the expiration date• The distinction was originally geographic, but no longer describes

anything but the presence of absence of the ability to exercise the option prior to the expiration date


12.2 PUT OPTIONS

12.3 PUT-CALL PARITY• There are four basic option positions: long call, short call, long

put, short put• These can be combined as shown in Figure 12-6:


• Put-call parity is the relationship between the price of a call option and a put option that have the same strike price and expiry dates; it assumes that the options are not exercised before expiration

• Consider two portfolios:– Portfolio A has a long put (P) with X = $50 and the underlying

asset (S)– Portfolio B has a long call (C) with X = $50 and has invested

the present value of the exercise price, PV(X) in the risk free asset paying interest at RF

– Assume all options are European (cannot be exercised prior to maturity), and that the underlying asset does not pay dividends or other income


12.3 PUT-CALL PARITY

Table 12-2 shows the payoffs from these two portfolios if the underlying asset price is either $55 or $45. Do you notice anything?



Table 12-3 shows that the payoff from either strategy will always be the same.



• If the put premium is P, the call premium is C, the strike price is X and the underlying asset price is S, then equation 12-5 gives put-call parity:

P + S = C + PV(X)• Rearranging equation 12-5 gives equation 12-6, the basic put-call

relationship:C – P = S – PV(X)

• We can also use equations 12-7,12-8 and 12-9 to show how put-call parity can be used to solve for the call and put premiums, and the underlying asset price, respectively:

C = P + S – PV(X)P = C – S + PV(X)S = C – P + PV(X)



• Equation 12-9 shows that a long position in the underlying asset gives the same payoffs as a portfolio with a long call, a short put and an investment of PV(X) in the risk-free asset

S = C – P + PV(X)• Equivalently, a short position in the underlying asset gives the same

payoffs as a portfolio with a short call, a long put and borrowing PV(X) at the risk-free rate

– S = P – C – PV(X)• We will consider the construction of three types of synthetic positions:

– Protective Puts– Covered Calls– Collars



• A protective put is a long position in both a put option and the underlying asset

• The investor benefits from the underlying asset if its price appreciates, and


the put option establishes a price floor for the underlying asset limiting potential loss

• The net payoff position of a protective put resembles a long call position, as shown in Figure 12-7


• A covered call is a long position in the underlying asset and a short position (writing) in a call option

• The investor benefits from the underlying asset if its price appreciates, and also


gets to pocket the call premium• The investor loses if the underlying asset falls in price.• The net payoff position of a covered call resembles a short put

position, as shown in Figure 12-8


• A collar is a position between a floor and ceiling price established by buying both the underlying asset and a put option and selling a call option to finance the purchase of the put option

• Collars are a type of self-financing portfolio insurance• The net payoff position of a collar is shown in Figure 12-9:



12.4 OPTION PRICING• Equation 12-10 is the Black-Scholes model for valuing European

call options on non-dividend paying stocks:

• Based on put-call parity, the price of a corresponding put option is:

24

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rtXSd

dNXedSNC rt

12

2

1

21

)2/()/ln(

][][

][][ 12 dSNdNXeP rt


• Use software of a table of cumulative normal probabilities, such as Table 12-4 to calculate N[d1], which is the cumulative probability of the option being in the money at expiration

• Notice that the more risky the underlying asset (the larger the standard deviation), the larger N[d1] and the moneyness of the call


12.4 OPTION PRICING

• Example: Find the value of a three-month call option with a $20 strike price if the price of the non-dividend paying underlying asset is $20.50 and its standard deviation is 25%. Assume the risk-free rate is 5%.

• S = $20.50, X = $20, r = 5%, t = 3/12 = 0.25, σ = 25%

26

5929.0][

2350.025.025.03600.0

6406.0][

3600.025.025.0

)2/25.005.0(25.0)20/50.20ln()2/()/ln(

42.1$)5929.0(20)6406.0(50.20][][

2

12

1

22

1

)25.0(05.021

dN

tdd

dNt

rtXSd

edNXedSNC rt


12.4 OPTION PRICING

• The Greeks indicate the sensitivity of the option price to changes in the underlying parameters

• Delta (Δ) is the change in the price of an option for a given change in the price of the underlying asset

• Theta (θ) is the change in the option value with time• Gamma (γ) is the change in delta with respect to a change in the

underlying asset; i.e., the rate of change in the rate of change in the price of an option for a given change in the price of the underlying asset

• Rho (ρ) is the change in the option value with respect to a change in the interest rate

• Vega is the change in the option value with respect to a change in the volatility of the underlying asset


12.4 OPTION PRICING

• Options are traded over the counter, mainly by major financial institutions, and on organized exchanges like the Montreal Exchange (MX) in Canada.

• Options on all stocks comprising the S&P/TSX 60 Index and the S&P/TX MidCap Index trade on the MX.

• Table 12-5 shows quotations for options (c = call, p = put) on Teck common shares. The ask is the price a seller wants, while the bid is what a buyer is prepared to pay. The last price is the value of the last transaction. The volume shows the number of contracts traded and open (the open interest) is the total number of contracts outstanding.


12.5 OPTIONS MARKET

• Implied volatility is an estimate of the price volatility of the underlying asset based on observed option prices.

• The other determinants of option prices are observable, but forecasts of price volatility are not and so must be implied by option prices.

• Figure 12-11 shows the implied volatilities of the S&P/TSX 60 Index over time. Implied volatility tends to increase when investors are less certain about the future of the underlying asset’s price.


12.5 OPTIONS MARKET

Appendix 12A BINOMIAL OPTION PRICING AND RISK-NEUTRAL PROBABILITIES

• While the Black-Scholes model is a continuous time model, the binomial model is an option pricing model that uses discrete time and only two future states of the world (up and down)

• Example: Figure 12A-1 shows a call option with a $50 strike and $50 current underlying asset value. There is a 60% probability the underlying asset’s price increases to $55 next period, and a 40% chance it falls to $45. If the price increases to $55, the call option will have a $5 payoff. If the price falls, the option will expire worthless.


• The hedge ratio (h) is the number of calls an investor must sell to hedge a long position in the underlying asset, and is given by equation 12A-1:

wherePU = price up or the price in the up-statePD = price down or the price in the down-stateX = strike price

• Example: In the previous example, X = $50, PU = $55, PD = $45:

• Therefore an investor must sell two calls to hedge every long position in the underlying asset

31

XPU

PDPUh

25055

4555

XPU

PDPUh



• Equation 12A-2 gives the value of the call option, where h is the hedge ratio, S is the underlying asset price , PD is the price in the down-state and r is the risk-free rate:

• Example: In our example, X = $50, PU = $55, PD = $45, h = 2 and suppose the risk-free rate is 1%.

32

r

PDS

hC

1

1

72.2$01.1

4550

2

1

1

1

r

PDS

hC



• Unlike direct approaches to valuation, where we estimate expected cash flow on the call and then use a risk-adjusted discount rate, here we use a hedge to generate a risk-free payoff discounted at the risk-free rate

• This is an indirect approach to valuation.• We can also observe that the call option will be worth more if:

– The price of the underlying asset increases– The risk-free rate increases– The risk of the asset increases– The strike price increases– The up-price increases

• These observations are consistent with the impact of these variables discussed earlier, but time to expiration cannot be included because the binomial model used here is a single period model



• Risk neutral is the state of ignoring the risk involved in determining expected rates of return.

• Risk-neutral probabilities are derived and ensure that the asset price goes up with the risk-free rate. These are not the true or actual probabilities of the asset price increasing or decreasing, but rather the probabilities that would exist if the investor were risk neutral instead of risk averse.

• Since the call option is valued as if the investor were risk-neutral, we can use risk-neutral probabilities to determine the expected payoff on the call and value it directly.



Example: As before, there is a 60% the price increases from $50 to $55 and a 40% chance it falls to $45. The risk-free rate is 1%.• We can calculate the probabilities that will generate a 1% return on

the asset given values of either $55 or $45 next period.• 50(1.01) = 55P + 45(1 – P), or P = (50.5 – 45) / 10 = 0.55• So, there is a 55% chance of increasing to $55 and a 45% chance of

getting zero payoff from the option• The expected payoff is: 0.45 ($0) + 0.55 ($5) = $2.75• We can now value the option directly by discounting the expected

payoff next period of $2.75 by 1%:– Value of call option = $2.75 / 1.01 = $2.72



© John Wiley & Sons Canada, Ltd. 36

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Laurence Booth Sean Cleary. LEARNING OBJECTIVES Options 12 12.1 Describe the basic nature of call options and the factors that influence their value.

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