Stock Option Properties and Trading Strategies

Read Chapters 10 and 11

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
  • 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,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 - + - +
  • How to read chart
    • 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

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, k1
        • Pay premium c1
      • Sell a call option on a stock for a high strike price, k2
        • Receive premium c2
      • 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

bull spread with call options

  • Bull spread using puts
    • Use European options
      • Buy a put option on a stock for a low strike price, k1
        • Pay premium p1
      • Sell a put option on a stock for a high strike price, k2
        • Receive premium p2
      • 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

bull spread with put options

  • Bear spread using puts
    • Use European options
      • Sell a put option on a stock for a low strike price, k1
        • Receive premium p1
      • Buy a put option on a stock for a high strike price, k2
        • Pay premium p2
      • 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

bear spread with put options

  • Box Spread
    • Combine a call bull spread and a put bear spread
    • 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

box spread with call and put options

  • Butterfly Spread
    • 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

butterfly spread with call options

  • 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

calendar spread with call options

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

combination long straddle

 

FOLLOW ME