zulfiqar hasan 1. 2 lecture contents options; call options; put options; selling options;...
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Zulfiqar Hasan 1
Zulfiqar Hasan 2
Lecture Contents
Options; Call Options; Put Options; Selling Options; Combinations of Options; Valuing Options; An Option‑Pricing Formula
Stocks and Bonds as Options; Capital-Structure Policy and Options; Mergers and Options; Investment in Real Projects and Options; Summary and Conclusions
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What is an Option?
1. An option is a contract which gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time.
2. An option gives the holder the right, but not the obligation, to buy or sell a given quantity of an asset on (or perhaps before) a given date, at prices agreed upon today.
So options allows an investor to “lock in” the followings:
1. a specified number of Shares
2. at a fixed price per share, called strike or exercise price
3. for a limited length of time
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1. Call option: Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset at some time in the future, at prices agreed upon today. When exercising a call option, you “call in” the asset.
2. Put option: Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset at some time in the future, at prices agreed upon today. When exercising a put, you “put” the asset to someone.
3. Exercising the Option: The act of buying or selling the underlying asset through the option contract.
4. Exercise (or strike) price: The price stated in the option contract at which the security can be bought or sold.
5. Option price: The market price of the option contract.
Option Terminology
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Option Terminology6. Expiration date: The date the option matures.7. Exercise value: The value of a call option if it were exercised
today = Current stock price - Strike price. Note: The exercise value is zero if the stock price is less than the strike price.
8. In-the-money call: A call whose exercise price is less than the current price of the underlying stock.
9. At-the-Money: The exercise price is equal to the spot price of the underlying asset.
10. Out-of-the-money call: A call option whose exercise price exceeds the current stock price.
Strike/Exercise Price (Call) Current Price Condition
$50> $50 In-the-Money
= $50 At-the-Money
< $50 Out-of-the Money
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11. Covered option: A call option written against stock held in an investor’s portfolio.
12. Naked (uncovered) option: An option sold without the stock to back it up.
13. LEAPS: Long-term Equity Anticipation Securities that are similar to conventional options except that they are long-term options with maturities of up to 2 1/2 years.
14. European option: An option that can only be exercised at expiration.
15. American option: An option that can be exercised at any time before and including at expiration.
Option Terminology
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Intrinsic value
Intrinsic value is the difference between the exercise price of the option and the spot price of the underlying asset. Intrinsic value can be defined as the amount by which the strike price of an option is in-the-money. It is actually the portion of an option's price that is not lost due to the passage of time. The following equations will allow you to calculate the intrinsic value of call and put options:
Call Options:Intrinsic value = Stock's Current Price - Call Strike Price Time Value = Call Premium - Intrinsic ValuePut Options:Intrinsic value = Put Strike Price - Stock's Current Price Time Value = Put Premium - Intrinsic Value
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Example 01: Call Option Intrinsic Value Explanation
If a call Option for 100 shares has a strike price of $35 and the stock is trading at $50 a share, what will be the intrinsic value of the call?
Intrinsic value (IV) = Stock's Current Price - Call Strike PriceIntrinsic value = $50 - $35So, Intrinsic value = $15 per shareTotal Intrinsic Value = IV x Number of Share
If the stock price is less than the strike price the call option has no intrinsic value
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Example 02: Put Option Intrinsic Value Explanation
If a put option for 100 shares has a strike price of $35 and the stock is trading at $20 a share. What will be the intrinsic value of the Put?
Intrinsic value = Put Strike Price - Stock's Current PriceIntrinsic value = $35 - $20So, Intrinsic value = $15 per share
If the stock price is greater than the strike price the put option has no intrinsic value.
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Example 03 : Understanding Options Quotes
Use the option quote information shown here to answer the questions that follow.
a. Are the call options in the money? What is the intrinsic value of an RWJ Corp. call option?
b. Are the put options in the money? What is the intrinsic value of an RWJ Corp. put option?
90909090
94949494
Answer: a. The calls are in the money. The intrinsic value of
the calls is $4.b. The puts are out of the money. The intrinsic
value of the puts is $0.
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Fundamentals of Option Valuation
C1 = 0 if S1 ≤ EOr C1 = 0 if S1 - E ≤0C1 = S1 - E if S1 >EOr EquivalentlyC1 = S1 -E, if S1 - E>0
The following notation will be useful:S1 = Stock price at expiration (in one period)
S0 = Stock price todayC1 =Value of the call option on the expiration date (in one period)C0 = Value of the call option todayE = Exercise price on the option
For example, suppose we have a call option with an exercise price of $10. The option is about to expire. If the stock is selling for $8, then we have the right to pay $10 for something worth only $8. Our option is thus worth exactly zero because the stock price is less than the exercise price on the option (S1 ≤ E). If the stock is selling for $12, then the option has value. Because we can buy the stock for $10, the option is worth S1- E= $12 – 10= $2.
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Example 04: Call Option Valuation: The Basic Approach
Rf 1
E C S 00
E = Exercise price on the optionRf = Risk Free Rate
T-bills currently yield 6.2 percent. Stock in Christina Manufacturing is currently selling for $55 per share. There is no possibility that the stock will be worth less than $50 per share in one year.
a. What is the value of a call option with a $45 exercise price? What is the intrinsic value?
b. What is the value of a call option with a $35 exercise price? What is the intrinsic value?
c. What is the value of a put option with a $45 exercise price? What is the intrinsic value?
Rf 1
E - S C 00
S0 = Stock price todayC0 = Value of the call option today
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Solution: Example 04
a.The value of the call is the stock price minus the present value of the exercise price, so:
C0 = S0 - E/(1 + Rf)
C0 = $55 – [$45/1.062] = $12.63Intrinsic value = $55 - $45 = $10b.The value of the call is the stock price minus the present value of the exercise price, so:
C0 = $55 – [$35/1.062] = $22.04The intrinsic value is the amount by which the stock price exceeds the exercise price of the call, so the intrinsic value is $20.
c.The value of the put option is $0 since there is no possibility that the put will finish in the money. The intrinsic value is also $0.
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Example 05: Valuing a Call Option
The market price per share is $45, with a strike price of $40. The call consists of 100 shares. Determine the value of the call.
Value of call = (Market Price of Stock - Exercise price of Call) x Number of Shares in callSo, Value of call = ($45 - $40) x 100 = $ 500
Example 06: Valuing a Call OptionA 2-month call option allows to buy 500 shares of ABC Company at $20 per share. What is the value of that option, if you exercise the option when the market price is $38 within that time period? What will happen if the market price decline from $20?
Value = $9000, Should not have exercised below $20 market price.
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Put-Call Parity (PCP)
The relationship between the prices of the underlying stock, a call option, a put option, and a risk less asset is called Put-Call parity condition.
where S = stock valueP = put value E = exercise price C = value of the call optionR = Risk Free Ratet = Duration
C -RteE P S
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Example 07: Put-Call Parity
$5.70 P
$40- 1$e45$ P
S- CeE P
CeE P S
)12
10.08(-
Rt-
-Rt
A share sells for $40. The continuously compounded risk-free rate is 8 percent per year. A call option with one month to expiration and a strike price of $45 sells for $1. What’s the value of a put option with the same expiration and strike?
C -RteE P S
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Practice at Home : Put-Call Parity
1. A stock is currently selling for $54 per share. A call option with an exercise price of $55 sells for $3.10 and expires in three months. If the risk-free rate of interest is 2.6 percent per year, compounded continuously, what is the price of a put option with the same exercise price?
2. A put option that expires in six months with an exercise price of $65 sells for $2.05. The stock is currently priced at $67, and the risk-free rate is 3.6 percent per year, compounded continuously. What is the price of a call option with the same exercise price?
3. A put option and a call option with an exercise price of $80 and five months to expiration sell for $2.05 and $4.80, respectively. If the riskfree rate is 4.8 percent per year, compounded continuously, what is the current stock price?
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Practice at Home: Put-Call Parity
4.A put option and call option with an exercise price of $65 expire in two months and sell for $2.50 and $0.90, respectively. If the stock is currently priced at $63.20, what is the annual continuously compounded rate of interest?
5.A put option with a maturity of five months sells for $6.33. A call with the same expiration sells for $9.30. If the exercise price is $75 and the stock is currently priced at $77.20, what is the annual continuously compounded interest rate?
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The Black-Scholes Model
t
trES
d
2
ln2
1tdd 12
Where; C0 = the value of a European option at time t = 0r = the risk-free interest rate.E = Exercise price on the optionN(d1) and N(d2) are probabilities that must be calculated.N(d) = Probability that a standardized, normally distributed, random
variable will be less than or equal to d.
The Black-Scholes Model allows us to value options in the real world just as we have done in the 2-state world.
)()( 210 dNeEdNSC rt
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Example 01: The Black-Scholes Model
S = $100; E =$90; Rf =4% per year, continuously compoundedd1 = .60; d2 = .30; t = 9 monthsBased on this information, what is the value of the call option, C?
As d1 = .60, So, N(d1). =0.7258, For d2, N(d2)= 0.6179. Using the Black-Scholes OPM,We calculate that the value of the call option is:
= $100 x .7258 - $90 x e.04(3/4) x .6179= $18.61
Notice that t, the time to expiration, is 9 months, which is 9/12 3/4 of one year.
)()( 210 dNeEdNSC rt
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Normal Distribution Table
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Example 02: BS-OPM
S= $70; E= $80; σ=60% per year; R= 4% per year, continuously compounded; t = 3 months. Calculate the value of options.
03.5$
2877.080$3974.070$
)()()4
1(04.0
210
e
dNeEdNSC rt
From table we get, N(d1)= .3974 and N(d2)= .2877
t
tRES
d
2
ln2
1
26.0
416.0
41
26.0
04.08070
ln2
1
d
56.04
16.026.0
tdd 12
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Practice 01: Call Option PricingSuppose you are given the following:S = $40; E = $36; R = 4% per year, continuously compounded; t = 3 months and . What’s the value of a call option on the stock?
yearper 70%σ
50.0
41
7.0
41
27.0
04.03640
ln2
1
d15.04
170.050.0
12
tdd
The values of N(d1) and N(d2) are .6915 and .5597, respectively. To get the second of these, we averaged the two numbers on each side, (.5557 + .5636)/2 = .5597.
71.7$5597.036$6915.040$
)()()4
1(04.0
210
e
dNeEdNSC rt
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Examples 04: BS-OPM
A share of stock sells for $40. The continuously compounded risk-free rate is 4 percent. The standard deviation of the return on the stock is 80 percent. What is the value of a put option with a strike of $45 and a three-month expiration?
values of N(d1) and N(d2) are .4721 and .3192, respectively. Notice that in both cases we average two values. Plugging all the numbers in:
Converting to a put
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The Black-Scholes Model
Find the value of a six-month call option on the Microsoft with an exercise price of $150
The current value of a share of Microsoft is $160
The interest rate available in the U.S. is r = 5%.
The option maturity is 6 months (half of a year).
The volatility of the underlying asset is 30% per annum.
Before we start, note that the intrinsic value of the option is $10—our answer must be at least that amount.
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The Black-Scholes Model
Let’s try our hand at using the model. If you have a calculator handy, follow along.
Then,
First calculate d1 and d2
T
TσrESd
)5.()/ln( 2
1
5282.05.30.0
5).)30.0(5.05(.)150/160ln( 2
1 d
31602.05.30.052815.012 Tdd
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The Black-Scholes Model
N(d1) = N(0.52815) = 0.7013
N(d2) = N(0.31602) = 0.62401
)N()N( 210 dEedSC rT
5282.01 d
31602.02 d
92.20$
62401.01507013.0160$
0
5.05.0
C
eC
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Practice at Home: Black-Scholes
06.What are the prices of a call option and a put option with the following characteristics?
Stock price = $32Exercise price = $30Risk-free rate = 5% per year, compounded continuouslyMaturity = 3 monthsStandard deviation = 54% per year
07. What are the prices of a call option and a put option with the following characteristics?
Stock price $98Exercise price $105Risk-free rate 4% per year, compounded continuouslyMaturity 9 monthsStandard deviation 62% per year
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Practice at Home: Black-Scholes
8. A call option matures in six months. The underlying stock price is $85, and the stock’s return has a standard deviation of 20 percent per year. The risk-free rate is 4 percent per year, compounded continuously. If the exercise price is $0, what is the price of the call option?
9. A call option has an exercise price of $75 and matures in six months. The current stock price is $80, and the risk-free rate is 5 percent per year, compounded continuously. What is the price of the call if the standard deviation of the stock is 0 percent per year?
10.Astock is currently priced at $35. Acall option with an expiration of one year has an exercise price of $50. The risk-free rate is 12 percent per year, compounded continuously, and the standard deviation of the stock’s return is infinitely large. What is the price of the call option?
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Summary and Conclusions
• The most familiar options are puts and calls.– Put options give the holder the right to sell stock
at a set price for a given amount of time.– Call options give the holder the right to buy stock
at a set price for a given amount of time.
• Put-Call parity
c0– (1+ r)T
E= S0 + p0
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Summary and Conclusions
• The value of a stock option depends on six factors:1. Current price of underlying stock.2. Dividend yield of the underlying stock.3. Strike price specified in the option contract.4. Risk-free interest rate over the life of the contract.5. Time remaining until the option contract expires.6. Price volatility of the underlying stock.
• Much of corporate financial theory can be presented in terms of options.1. Common stock in a levered firm can be viewed as a call option on the
assets of the firm.2. Real projects often have hidden option that enhance value.
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Market Value, Time Value and Intrinsic Valuefor an American Call
The value of a call option C0 must fall within max (S0 – E, 0) < C0 < S0.
25
Op
tion
pay
offs
($) Call
ST
loss
E
Profit
ST
Time value
Intrinsic value
Market Value
In-the-moneyOut-of-the-money
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Binomial Option Pricing Model
Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. S0= $25 today and in one year S1is either $28.75 or $21.25. The risk-free rate is 5%. What is the value of an at-the-money call option?
$25
$21.25 = $25×(1 –.15)
$28.75 = $25×(1.15)S1S0
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Binomial Option Pricing Model
1. A call option on this stock with exercise price of $25 will have the following payoffs.
2. We can replicate the payoffs of the call option. With a levered position in the stock.
$25
$21.25
$28.75S1S0 C1
$3.75
$0
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Binomial Option Pricing Model1. Borrow the present value of $21.25 today and buy 1 share. 2. The net payoff for this levered equity portfolio in one
period is either $7.50 or $0. 3. The levered equity portfolio has twice the option’s payoff
so the portfolio is worth twice the call option value.
$25
$21.25
$28.75S1S0 debt
– $21.25portfolio$7.50
$0
( – ) ==
=
C1
$3.75
$0– $21.25
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Binomial Option Pricing Model
The value today of the levered equity portfolio is today’s value of one share less the present value of a $21.25 debt:
)1(
25.21$25$
fr
$25
$21.25
$28.75S1S0 debt
– $21.25portfolio$7.50
$0
( – ) ==
=
C1
$3.75
$0– $21.25
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Binomial Option Pricing Model
We can value the call option today as half of the value of the levered equity portfolio:
)1(
25.21$25$
2
10
frC
$25
$21.25
$28.75S1S0 debt
– $21.25portfolio$7.50
$0
( – ) ==
=
C1
$3.75
$0– $21.25
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If the interest rate is 5%, the call is worth:
The Binomial Option Pricing Model
38.2$24.2025$2
1
)05.1(
25.21$25$
2
10
C
$25
$21.25
$28.75S1S0 debt
– $21.25portfolio$7.50
$0
( – ) ==
=
C1
$3.75
$0– $21.25
$2.38
C0
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the replicating portfolio intuition.the replicating portfolio intuition.
Binomial Option Pricing Model
Many derivative securities can be valued by valuing portfolios of primitive securities when those portfolios have the same payoffs as the derivative securities.
The most important lesson (so far) from the binomial option pricing model is:
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Delta and the Hedge Ratio
• This practice of the construction of a riskless hedge is called delta hedging.
• The delta of a call option is positive.– Recall from the example:
The delta of a put option is negative.
2
1
5.7$
75.3$
25.21$75.28$
075.3$
Swing of callSwing of stock
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Delta
• Determining the Amount of Borrowing:
38.2$24.20$25$2
1
)05.1(
25.21$25$
2
10
C
Value of a call = Stock price × Delta – Amount borrowed
$2.38 = $25 × ½ – Amount borrowed
Amount borrowed = $10.12
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The Risk-Neutral Approach to Valuation
We could value V(0) as the value of the replicating portfolio. An equivalent method is risk-neutral valuation
S(0), V(0)
S(U), V(U)
S(D), V(D)
q
1- q
)1(
)()1()()0(
fr
DVqUVqV
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The Risk-Neutral Approach to Valuation
S(0) is the value of the underlying asset today.
S(0), V(0)
S(U), V(U)
S(D), V(D)
S(U) and S(D) are the values of the asset in the next period following an up move and a down move, respectively.
q
1- q
V(U) and V(D) are the values of the asset in the next period following an up move and a down move, respectively.
q is the risk-neutral probability of an “up” move.
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The Risk-Neutral Approach to Valuation
• The key to finding q is to note that it is already impounded into an observable security price: the value of S(0):
S(0), V(0)
S(U), V(U)
S(D), V(D)
q
1- q
A minor bit of algebra yields:)()(
)()0()1(
DSUS
DSSrq f
)1()()1()(
)0(fr
DSqUSqS
)1(
)()1()()0(
fr
DVqUVqV
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Example of the Risk-Neutral Valuation of a Call:
Suppose a stock is worth $25 today and in one period will either be worth 15% more or 15% less. The risk-free rate is 5%. What is the value of an at-the-money call option?
The binomial tree would look like this:
$21.25,C(D)
q
1- q
$25,C(0)
$28.75,C(D)
)15.1(25$75.28$
)15.1(25$25.21$
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Example of the Risk-Neutral Valuation of a Call:
$21.25,C(D)
2/3
1/3
The next step would be to compute the risk neutral probabilities
$25,C(0)
$28.75,C(D)
)()(
)()0()1(
DSUS
DSSrq
f
3250.7$
5$
25.21$75.28$
25.21$25$)05.1(
q
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Example of the Risk-Neutral Valuation of a Call:
$21.25, $0
2/3
1/3
After that, find the value of the call in the up state and down state.
$25,C(0)
$28.75, $3.75
]0,75.28$25max[$)( DC
25$75.28$)( UC
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Example of the Risk-Neutral Valuation of a Call:
Finally, find the value of the call at time 0:
$21.25, $0
2/3
1/3
$25,C(0)
$28.75,$3.75
$25,$2.38
)1(
)()1()()0(
fr
DCqUCqC
)05.1(
0$)31(75.3$32)0(
C
38.2$)05.1(
50.2$)0( C
Zulfiqar Hasan 51
This risk-neutral result is consistent with valuing the call using a replicating portfolio.
Risk-Neutral Valuationand the Replicating Portfolio
38.2$24.2025$2
1
)05.1(
25.21$25$
2
10
C
38.2$05.1
50.2$
)05.1(
0$)31(75.3$320
C
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Call Option Payoffs
–20
12020 40 60 80 100
–40
20
40
60
Stock price ($)
Op
tion
pay
offs
($) Buy
a ca
ll
Exercise price = $50
50
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Call Option Payoffs
–20
12020 40 60 80 100
–40
20
40
60
Stock price ($)
Op
tion
pay
offs
($)
Sell a call
Exercise price = $50
50
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Call Option Profits
Exercise price = $50; option premium = $10
Sell a call
Buy a call
–20
12020 40 60 80 100
–40
20
40
60
Stock price ($)
Op
tion
pay
offs
($)
50–10
10
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Put Option Payoffs
–20
0 20 40 60 80 100
–40
20
0
40
60
Stock price ($)
Op
tion
pay
offs
($)
Buy a put
Exercise price = $50
50
50
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Put Option Payoffs
–20
0 20 40 60 80 100
–40
20
0
40
–50
Stock price ($)
Op
tion
pay
offs
($)
Sell a put
Exercise price = $50
50
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Put Option Profits
–20
20 40 60 80 100
–40
20
40
60
Stock price ($)
Op
tion
pay
offs
($)
Buy a put
Exercise price = $50; option premium = $10
–10
10Sell a put
50
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Selling Options
Exercise price = $50; option premium = $10 Sell a call
Buy a call
50 6040 100
–40
40
Stock price ($)
Op
tion
pay
offs
($)
Buy a put
Sell a put
The seller (or writer) of an option has an obligation.
The purchaser of an option has an option.
–10
10
Buy a call
Sell a
put
Buy a put
Sell a call