Outline

• How Variables Influence an Option's Price
• Arbitrage
• Put-Call Parity
• Early Exercise of American Options
• Bounds for European and American Options
• Principal Protected Note
• 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,S0
• Stock price at maturity, ST
• Strike price, K
• Life of option, T
• Volatility of stock price, σ
• Present value of dividends during option's life, D
• Risk-free rate with continuous compounding, r
• How changes in a variable influences an option's price
• Refer to chart below
 Variable c p C P S0 + - + - K - + - + T ? ? + + σ + + + + r + - + - D - + - +
• Option Pricing Call option – when exercised – buy at strike price and sell at spot price, or ST - 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 - ST
• Everything is opposite.
• 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
• C ≥ c
• P ≥ p
• 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
• S0 = 20
• T = 1
• r = 10%
• K = 18
• D = 0
• Does an arbitrage opportunity exist?
• Lower bound for european call option prices
• c ≥ max(S0 – 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 S0 – 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 - ST
• Call option is undervalued
• Arbitrage opportunity for a put
• Supposed that
• p = 1
• S0 = 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 – S0, 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 – S0
• 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 = ST - 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

Put-Call Parity

• Proof
• Construct the portfolios:
• Portfolio A: Buy call option and bond
• European call has K strike price
• Zero-coupon 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 S0
• Value of portfolio at Time 0: p + S0
 Value of Portfolio at T ST > K ST < K Portfolio A Call option ST − K 0 Zero-coupon bond K K Total ST K Portfolio C Put Option 0 K− ST Share ST ST Total ST K
• The trick to reading the table
• Evaluate the value of Portfolio A and Portfolio C if ST > K
• Then evaluate portfolio values for ST < K
• Both portfolios are worth max(ST , K ) at the maturity
• Therefore, the options must have the same value today
• This means, c + K e-rT = p + S0
• Arbitrage
• Suppose that
• c = 3
• S0 = 31
• T = 0.25
• r = 10%
• K =30
• D = 0
• What are the arbitrage possibilities if p = 1 ?
• Substitute info into put-call parity
• c + K e-rT = p + S0
• 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 ST > 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 e0.25x0.1 = 29.7341
• profit = 30 - 29.7341 = 0.2659
• If ST < 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?
• Use the same technique

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:
• S0 = 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
• S0 = 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(S0 – D – K e–rT, 0)
• Or as c ≥ max( [S0 – D} – K e–rT, 0)
• When corporation pays a dividend, the stock price, S0, falls by the amount of dividend
• Bounds for European put with dividends
• p ≥ max(D + K e–rT – S0, 0)
• p ≥ max(K e–rT – [S0 – D], 0)
• Extension of Put-Call Parity
• European options; D > 0
• c + D + K e-rT = p + S0
• Or re-write as c + K e-rT = p + [S0 – D]
• American inequality, D = 0
• S0 - K < C - P < S0 - K e-rT
• Note: S0 – K is the payoff if you exercise an American call option today
• American inequality, D > 0
• S0 - D - K < C - P < S0 - 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: 3-year zero-coupon bond with principal of \$1000
• Profit: 3-year at-the-money 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

• 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, k1
• Sell a call option on a stock for a high strike price, k2
• k2 > k1
• Both calls have the same maturity
• Case 1, ST ≤ k1
• Long (bought) call - you do not exercise
• Short (sold) call - holder does not exercise
• No one will buy at k1 and sell for ST
• profit = c2 – c1
• Case 2, k1 < ST < k2
• Long (bought) call - you exercise because you can buy at k1 and sell at ST
• payoff = ST – k1
• Short (sold) call - holder does not exercise
• payoff = 0
• profit = c2 – c1 + ST – k1
• Case 3, ST ≥ k2
• Long (bought) call - you exercise and buy at k1 and sell at ST
• payoff = ST – k1
• Short (sold) call) - holder exercises
• You must sell at k2 and buy at ST
• payoff = k2 – ST
• profit = c2 – c1 +ST – k1 + k2 – ST = c2 – c1 + k2 – k1
• Graph shows a bull spread
• Investor makes money during bull market

• Use European options
• Buy a put option on a stock for a low strike price, k1
• Sell a put option on a stock for a high strike price, k2
• k2 > k1
• Both calls have the same maturity
• Case 1, ST ≤ k1
• Long (bought) put - you exercise
• You buy at ST and sell at k1
• payoff = k1 – ST
• Short (sold)put - holder exercises
• You must buy at k2 and sell at ST
• payoff = ST – k2
• profit = p2 – p1 + k1 – ST + ST – k2 = p2 – p1 + k1 – k2
• Case 2, k1 < ST < k2
• Long (bought) put - you do not exercise
• payoff = 0
• Short (sold) put - holder exercises
• You must buy at k2 and sell at ST
• payoff = ST – k2
• profit = p2 – p1 + ST – k2
• Case 3, ST ≥ k2
• Long (bought) put - you do not exercise
• payoff = 0
• Short (sold) put - holder does not exercise
• payoff = 0
• profit = p2 – p1 > 0
• Graph shows a bull spread
• Investor makes money during bull market

• Use European options
• Sell a put option on a stock for a low strike price, k1
• Buy a put option on a stock for a high strike price, k2
• k2 > k1
• Both calls have the same maturity
• Case 1, ST ≤ k1
• Long (bought) put - you exercise
• You buy at ST and sell at k2
• payoff = k2 – ST
• Short (sold)put - holder exercises
• You must buy at k1 and sell at ST
• payoff = ST – k1
• profit = p1 – p2 + ST – k1 + k2 – ST = p1 – p2 – k1 + k2
• Case 2, k1 < ST < k2
• Long (bought) put - you exercise and buy at ST and sell at k2
• payoff = k2 – ST
• Short (sold) put - holder does not exercise
• payoff = 0
• profit = p1 – p2 + k2 – ST
• Case 3, ST ≥ k2
• Long (bought) put - you do not exercise
• payoff = 0
• Short (sold) put - holder does not exercise
• payoff = 0
• profit = p1 – p2 < 0
• Graph shows a bear spread
• Investor makes money during bear market

• Strategy
• Buy a call option for a low strike price k1
• Sell a put for a low strike price k1
• Sell a call option for a high strike price k2
• Buy a put for a high strike price k2
• k2 > k1
• premium = p1 + c2 p2 c1 < 0
• Because p1 p2 < 0
• And c2 c1 < 0
• Case 1 ST ≤ k1
• payoff = 0 for calls
• payoff = k2 – k1 for puts
• Total payoff = k2 – k1
• Case 2 k1 < ST < k2
• payoff = 0 + ST – k1 = ST – k1 for calls
• payoff = 0 + k2 – ST = k2 – ST for puts
• total = ST – k1 + k2 – ST = k2 – k1
• Case 3 ST ≥ k2
• payoff = –k1 + k2 for calls
• payoff = 0 for puts
• total payoff = k2 – k1
• Graph of box spread is shown below
• The graph shows the payoff exceeds the premium cost

• Writer believes the market will stay the same
• Strategy
• Buy a European call for a low price, k1
• Buy a European call for a high price, k3
• Sell two calls for mid-price, k2
• k1 < k2 < k3
• premium = 2 c2 – c1 – c3
• Case 1 ST ≤ k1
• Buy (bought) calls: You do not exercise
• You would never buy at k1 to sell at ST
• Sell (sold) calls: Holder does not exercise
• profit = 2 c2 – c1 – c3
• Case 2 k1 < ST < k2
• Buy (bought) calls: You exercise call (k1) because you buy at k1 and sell at ST
• You do not exercise call (k3)
• payoff = ST – k1
• Sell (sold) calls: Holder does not exercise
• profit = 2 c2 – c1 – c3 + ST – k1
• Case 3 k2 ≤ ST < k3
• Buy (bought) calls: You exercise the call (k1) but not call (k2)
• payoff = ST – k1
• Sell (sold) calls: Holder exercises
• He buys at k2 and sells at ST
• You must sell at k2 and buy at ST
• payoff = 2(k2 - ST)
• Total profit = 2c2 – c1 – c3 + ST – k1 + 2k2 – 2ST = 2c2 – c1 – c3 + 2k2 – k1 – ST
• Case 4 ST ≥ K3
• Buy (bought) calls: You exercise both calls
• payoff = ST – k1 + ST – k3
• Sell (sold) calls: Holder exercise both calls
• payoff = –2(ST – k2)
• Total profit = 2c2 – c1 – c3 + 2ST – k1 – k3 – 2ST + 2k2 = 2c2 – c1 – c3 + 2k2 – k1 – k3
• If k2 is the average of k1 and k3, then
• total profit = 2c2 – c1 – c3 + 2k2 – k2 – k2 = 2c2 – c1 – c3
• Graph is below

• Calendar spread - the same strike price but different maturities
• Strategy
• Sell a call for a strike price k and maturity T1
• Buy a call at strike price k and maturity T2
• T2 > T1
• premium = c1 – c2 < 0
• because longer maturities cost more
• At time T1, close out and sell the longer maturity call
• If ST ≤ k
• Sell (sold) call: Holder does not exercise
• payoff = 0
• Buy (bought) call: Holder does not excercise
• payoff = 0
• No one buys at k to sell at ST
• profit = c1 – c2 < 0
• If k > ST
• Sell (sold) call: Holder exercises. You must buy at ST and sell at k
• payoff = k – ST
• Buy (bought) call: You cannot exercise. You can sell your option because it is in the money.
• profit = c1 – c2 + k – ST + 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
• premium = c + p
• If ST ≤ k
• You do not exercise the call but you exercise the put
• You buy at ST and sell at k
• payoff = k – ST
• profit = k – ST – c – p
• If ST > k
• You exercise the call but not the put
• You buy at k and sell at ST
• payoff = ST – k
• profit = ST – k – c – p
• Graph below shows a long straddle