Stock Option Properties and Trading Strategies
Read Chapters 10 and 11
Outline
 How Variables Influence an Option's Price
 Arbitrage
 PutCall Parity
 Early Exercise of American Options
 Bounds for European and American Options
 Principal Protected Note
 Spreads
 Combinations
How Variables Influence an Option's Price
 Define the notation
 European call option price, c
 European put option price, p
 American call option, C
 American put option, P
 Stock price today,S_{0}
 Stock price at maturity, S_{T}
 Strike price, K
 Life of option, T
 Volatility of stock price, σ
 Present value of dividends during option's life, D
 Riskfree rate with continuous compounding, r
 How changes in a variable influences an option's price
Variable 
c 
p 
C 
P 
S_{0} 
+ 
 
+ 
 
K 
 
+ 
 
+ 
T 
? 
? 
+ 
+ 
σ 
+ 
+ 
+ 
+ 
r 
+ 
 
+ 
 
D 
 
+ 
 
+ 
 How to read chart
 Option Pricing Call option – when exercised – buy at strike price and sell at spot price, or S_{T}  K
 If spot price rises, call option is more likely to be exercised.
 Thus, option's cost must be greater, c (+)
 If the strike price rises, call option is less likely to be exercised.
 Thus, option's cost must decrease, c ()
 Put option – when exercised – buy at spot and sell at strike price, or K  S_{T}
 If the time increases, the spot price can change more.
 The holder has more time to exercise the options.
 A crisis or event can make markets more volatile
 American calls and puts will cost more (+)
 The ? assumes the European options have a fixed time and cannot increase
 If volatility increases, both the put's and call's option price increases
 Means the spot price has more variation and more likely to be exercised.
 Interest rates – complicated
 If dividends increase, the holder does not receive dividends
 After corporation pays dividends, the spot price decreases.
 Holder receives less money if he exercises a call option and receives more money if he exercises a put.
 Call option price decreases while put option price increases
 American vs European Options
 An American option equals or exceeds European option
 Holder can exercise American option any time.
 American options are more likely to be exercised
 Insurance – more likely a person uses insurance, the more he or she has to pay.
Arbitrage
 Arbitrage Opportunity for calls
 Suppose that
 c = 3
 S_{0} = 20
 T = 1
 r = 10%
 K = 18
 D = 0
 Does an arbitrage opportunity exist?
 Lower bound for european call option prices
 c ≥ max(S_{0} – K e^{–rT}, 0)
 If option is not in the money, then it equals zero
 If the option is in the money, the value equals S_{0} – K e^{–rT}
 Does 3≥ max(3.7129, 0)
 No, so you should buy low and sell high
 Buy the call option for 3 because your theoretical value is 3.7129.
 Short the stock for $20
 Buy the call for $3
 Invest the proceeds, $20  $3 = $17 at 10% for one year
 At maturity
 If stock price > 18
 Exercise call and buy stock at $18
 Use money from bank
 Close the short
 profit = 18.7879  18 = 0.7879
 If stock price ≤ 18
 Buy stock from spot market using money at bank
 Close the short
 profit = 18.7879  S_{T}
 Call option is undervalued
 Arbitrage opportunity for a put
 Supposed that
 p = 1
 S_{0} = 37
 T = 0.5
 r =5%
 K = 40
 D = 0
 Does an arbitrage opportunity exist?
 Lower bound for European put prices
 p ≥ max(K e^{–rT} – S_{0}, 0)
 If a put is out of the money, it has a value of zero.
 If a put is in the money, then it has a value, K e^{–rT} – S_{0}
 Does 1 ≥ max(2.0124, 0)?
 No, so buy low and sell high.
 Borrow $38 at 5% for six months
 Buy the stock for $37
 Buy the put for a $1
 At maturity
 If stock price > 40
 Do not exercise the put
 Sell stock on the spot market
 Repay loan with interest of 38.9620
 profit = S_{T}  38.9620
 If stock price ≤ 40
 Exercise put and buy at stock at $40
 Repay bank loan of 38.9620
 profit = $40  38.9620 = 1.0380
PutCall Parity
 Proof
 Construct the portfolios:
 Portfolio A: Buy call option and bond
 European call has K strike price
 Zerocoupon bond that pays K at time T
 Value of portfolio at Time 0: c + K e^{rT}
 Portfolio C: Buy put option and stock
 European put has strike price K
 The stock has value S_{0}
 Value of portfolio at Time 0: p + S_{0}
Value of Portfolio at T 
S_{T} > K 
S_{T} < K 
Portfolio A 
Call option 
S_{T} − K 
0 

Zerocoupon bond 
K 
K 

Total 
S_{T} 
K 
Portfolio C 
Put Option 
0 
K− S_{T} 

Share 
S_{T} 
S_{T} 

Total 
S_{T} 
K 
 The trick to reading the table
 Evaluate the value of Portfolio A and Portfolio C if S_{T} > K
 Then evaluate portfolio values for S_{T} < K
 Both portfolios are worth max(S_{T} , K ) at the maturity
 Therefore, the options must have the same value today
 This means, c + K e^{rT} = p + S_{0}
 Arbitrage
 Suppose that
 c = 3
 S_{0} = 31
 T = 0.25
 r = 10%
 K =30
 D = 0
 What are the arbitrage possibilities if p = 1 ?
 Substitute info into putcall parity
 c + K e^{rT} = p + S_{0}
 3 + 30 e^{0.1x0.25} = p + 31
 32.2593 = p + 31
 if p = 1
 Then 32.2593 > p + 31
 Buy low and sell high
 Short call on left side of equation, +$3
 Buy stock on right side of equation, $31
 Buy put on right side of equation, $1
 Borrow +$3$31$1 = $29 from bank
 At maturity
 If S_{T} > 30
 Holder exercise the call option
 That means you must sell your stock to him for 30
 You do not exercise the put
 Repay the bank 29 e^{0.25x0.1} = 29.7341
 profit = 30  29.7341 = 0.2659
 If S_{T} < 30
 Holder does not exercise the call
 You exercise the put and sell your stock for 30
 Repay the bank 29.7341
 profit = 30  29.7341 = 0.2659
 What are the arbitrage possibilities if p = 22.5?
Early Exercise of American Options
 A trader may exercise an American option early
 A trader should never exercise an American call early that pays no dividend
 Based on a famous proof
 American and European call options are the same, C = c
 For example, an American call option:
 S_{0} = 100
 T = 0.25
 K = 60
 D = 0
 Should you exercise immediately?
 payoff = 100  60 = 40
 What should you do if you want to hold the stock for the next 3 months?
 You do not feel that the stock is worth holding for the next 3 months?
 Reasons for not exercising a call early
 Only applies to calls that pay no dividends
 No income is sacrificed
 You delay paying the strike price
 Holding the call provides insurance against stock price falling below strike price
 Are there any advantages to exercising an American put early
 For example
 S_{0} = 60
 T = 0.25
 r=10%
 K = 100
 D = 0
 Investor would exercise early
 Payoff = 100  60 = 40
Bounds for European and American Options
 Bounds for European call with dividends
 c ≥ max(S_{0} – D – K e^{–rT}, 0)
 Or as c ≥ max( [S_{0} – D} – K e^{–rT}, 0)
 When corporation pays a dividend, the stock price, S_{0}, falls by the amount of dividend
 Bounds for European put with dividends
 p ≥ max(D + K e^{–rT} – S_{0}, 0)
 p ≥ max(K e^{–rT} – [S_{0} – D], 0)
 Extension of PutCall Parity
 European options; D > 0
 c + D + K e^{rT} = p + S_{0}
 Or rewrite as c + K e^{rT} = p + [S_{0} – D]
 American inequality, D = 0
 S_{0}  K < C  P < S_{0}  K e^{rT}
 Note: S_{0} – K is the payoff if you exercise an American call option today
 American inequality, D > 0
 S_{0}  D  K < C  P < S_{0}  K e^{rT}
Principal Protected Note
 Principal Protected Note
 Allows investor to take a risky position without risking any principal
 Example:
 $1000 instrument consisting of
 Principal: 3year zerocoupon bond with principal of $1000
 Profit: 3year atthemoney call option on a stock portfolio currently worth $1000
 On maturity:
 Investor gets the 1,000 face value of the discount bond.
 If the stock value rises in value, the call option is exercised.
 Buy at strike and sell at spot
 Investor receives profit from rise in value
 If the stock value drops in value, the call option is not exercised
 Investor earns nothing from the call option
 Viability depends on
 Level of dividends
 Level of interest rates
 Volatility of the portfolio
Spreads
 Spread  trading strategy that uses two or more options of the same type
 Bull market  majority of investors believe stock market will rise
 Bear market  most investors believe stock market will fall
 Bull spread  investors believe market will increase
 Use European options
 Buy a call option on a stock for a low strike price, k_{1}
 Sell a call option on a stock for a high strike price, k_{2}
 k_{2} > k_{1}
 Both calls have the same maturity
 Case 1, S_{T} ≤ k_{1}
 Long (bought) call  you do not exercise
 Short (sold) call  holder does not exercise
 No one will buy at k_{1} and sell for S_{T}
 profit = c_{2} – c_{1}
 Case 2, k_{1} < S_{T} < k_{2}
 Long (bought) call  you exercise because you can buy at k_{1} and sell at S_{T}
 Short (sold) call  holder does not exercise
 profit = c_{2} – c_{1} + S_{T} – k_{1}
 Case 3, S_{T} ≥ k_{2}
 Long (bought) call  you exercise and buy at k_{1} and sell at S_{T}
 Short (sold) call)  holder exercises
 You must sell at k_{2} and buy at S_{T}
 payoff = k_{2} – S_{T}
 profit = c_{2} – c_{1} +S_{T} – k_{1} + k_{2} – S_{T} = c_{2} – c_{1} + k_{2} – k_{1}
 Graph shows a bull spread
 Investor makes money during bull market
 Bull spread using puts
 Use European options
 Buy a put option on a stock for a low strike price, k_{1}
 Sell a put option on a stock for a high strike price, k_{2}
 k_{2} > k_{1}
 Both calls have the same maturity
 Case 1, S_{T} ≤ k_{1}
 Long (bought) put  you exercise
 You buy at S_{T} and sell at k_{1}
 payoff = k_{1} – S_{T}
 Short (sold)put  holder exercises
 You must buy at k_{2} and sell at S_{T}
 payoff = S_{T} – k_{2}
 profit = p_{2} – p_{1} + k_{1} – S_{T} + S_{T} – k_{2} = p_{2} – p_{1} + k_{1} – k_{2}
 Case 2, k_{1} < S_{T} < k_{2}
 Long (bought) put  you do not exercise
 Short (sold) put  holder exercises
 You must buy at k_{2} and sell at S_{T}
 payoff = S_{T} – k_{2}
 profit = p_{2} – p_{1} + S_{T} – k_{2}
 Case 3, S_{T} ≥ k_{2}
 Long (bought) put  you do not exercise
 Short (sold) put  holder does not exercise
 profit = p_{2} – p_{1} > 0
 Graph shows a bull spread
 Investor makes money during bull market
 Bear spread using puts
 Use European options
 Sell a put option on a stock for a low strike price, k_{1}
 Buy a put option on a stock for a high strike price, k_{2}
 k_{2} > k_{1}
 Both calls have the same maturity
 Case 1, S_{T} ≤ k_{1}
 Long (bought) put  you exercise
 You buy at S_{T} and sell at k_{2}
 payoff = k_{2} – S_{T}
 Short (sold)put  holder exercises
 You must buy at k_{1} and sell at S_{T}
 payoff = S_{T} – k_{1}
 profit = p_{1} – p_{2} + S_{T} – k_{1} + k_{2} – S_{T} = p_{1} – p_{2} – k_{1} + k_{2}
 Case 2, k_{1} < S_{T} < k_{2}
 Long (bought) put  you exercise and buy at S_{T} and sell at k_{2}
 Short (sold) put  holder does not exercise
 profit = p_{1} – p_{2} + k_{2} – S_{T}
 Case 3, S_{T} ≥ k_{2}
 Long (bought) put  you do not exercise
 Short (sold) put  holder does not exercise
 profit = p_{1} – p_{2} < 0
 Graph shows a bear spread
 Investor makes money during bear market
 Box Spread
 Combine a call bull spread and a put bear spread
 Strategy
 Buy a call option for a low strike price k_{1}
 Sell a put for a low strike price k_{1}
 Sell a call option for a high strike price k_{2}
 Buy a put for a high strike price k_{2}
 k_{2} > k_{1}
 premium = p_{1} + c_{2} – p_{2} – c_{1} < 0
 Because p_{1} – p_{2} < 0
 And c_{2} – c_{1} < 0
 Case 1 S_{T} ≤ k_{1}
 payoff = 0 for calls
 payoff = k_{2} – k_{1} for puts
 Total payoff = k_{2} – k_{1}
 Case 2 k_{1} < S_{T} < k_{2}
 payoff = 0 + S_{T} – k_{1} = S_{T} – k_{1} for calls
 payoff = 0 + k_{2} – S_{T} = k_{2} – S_{T} for puts
 total = S_{T} – k_{1} + k_{2} – S_{T} = k_{2} – k_{1}
 Case 3 S_{T} ≥ k_{2}
 payoff = –k_{1} + k_{2} for calls
 payoff = 0 for puts
 total payoff = k_{2} – k_{1}
 Graph of box spread is shown below
 The graph shows the payoff exceeds the premium cost
 Butterfly Spread
 Writer believes the market will stay the same
 Strategy
 Buy a European call for a low price, k_{1}
 Buy a European call for a high price, k_{3}
 Sell two calls for midprice, k_{2}
 k_{1} < k_{2} < k_{3}
 premium = 2 c_{2} – c_{1} – c_{3}
 Case 1 S_{T} ≤ k_{1}
 Buy (bought) calls: You do not exercise
 You would never buy at k_{1} to sell at S_{T}
 Sell (sold) calls: Holder does not exercise
 profit = 2 c_{2} – c_{1} – c_{3}
 Case 2 k_{1} < S_{T} < k_{2}
 Buy (bought) calls: You exercise call (k_{1}) because you buy at k_{1} and sell at S_{T}
 You do not exercise call (k_{3})
 payoff = S_{T} – k_{1}
 Sell (sold) calls: Holder does not exercise
 profit = 2 c_{2} – c_{1} – c_{3} + S_{T} – k_{1}
 Case 3 k_{2} ≤ S_{T} < k_{3}
 Buy (bought) calls: You exercise the call (k_{1}) but not call (k_{2})
 Sell (sold) calls: Holder exercises
 He buys at k_{2} and sells at S_{T}
 You must sell at k_{2} and buy at S_{T}
 payoff = 2(k_{2}  S_{T})
 Total profit = 2c_{2} – c_{1} – c_{3} + S_{T} – k_{1} + 2k_{2} – 2S_{T} = 2c_{2} – c_{1} – c_{3} + 2k_{2} – k_{1} – S_{T}
 Case 4 S_{T} ≥ K_{3}
 Buy (bought) calls: You exercise both calls
 payoff = S_{T} – k_{1} + S_{T} – k_{3}
 Sell (sold) calls: Holder exercise both calls
 payoff = –2(S_{T} – k_{2})
 Total profit = 2c_{2} – c_{1} – c_{3} + 2S_{T} – k_{1} – k_{3} – 2S_{T} + 2k_{2} = 2c_{2} – c_{1} – c_{3} + 2k_{2} – k_{1} – k_{3}
 If k_{2} is the average of k_{1} and k_{3}, then
 total profit = 2c_{2} – c_{1} – c_{3} + 2k_{2} – k_{2} – k_{2} = 2c_{2} – c_{1} – c_{3}
 Graph is below
 Calendar spread  the same strike price but different maturities
 Strategy
 Sell a call for a strike price k and maturity T_{1}
 Buy a call at strike price k and maturity T_{2}
 T_{2} > T_{1}
 premium = c_{1} – c_{2} < 0
 because longer maturities cost more
 At time T_{1}, close out and sell the longer maturity call
 If S_{T} ≤ k
 Sell (sold) call: Holder does not exercise
 Buy (bought) call: Holder does not excercise
 No one buys at k to sell at S_{T}
 profit = c_{1} – c_{2} < 0
 If k > S_{T}
 Sell (sold) call: Holder exercises. You must buy at S_{T} and sell at k
 Buy (bought) call: You cannot exercise. You can sell your option because it is in the money.
 profit = c_{1} – c_{2} + k – S_{T} + gain
 Graph is below
Combination
 Combination  mix calls and puts
 Long Straddle  investor believes market will change big but not sure which direction
 Buy both a call and a put at same strike price, k
 If S_{T} ≤ k
 You do not exercise the call but you exercise the put
 You buy at S_{T} and sell at k
 payoff = k – S_{T}
 profit = k – S_{T} – c – p
 If S_{T} > k
 You exercise the call but not the put
 You buy at k and sell at S_{T}
 payoff = S_{T} – k
 profit = S_{T} – k – c – p
 Graph below shows a long straddle

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