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, 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
- Risk-free 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
- 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 price
-
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 price
-
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
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 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 |
|
Zero-coupon 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 put-call 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 possibility 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 Put-Call Parity
- European options; D > 0
-
c + D + K e -rT = p + S 0
- Or re-write 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:
- $1,000 instrument consisting of
- Principal: 3-year zero-coupon bond with principal of $1,000
- Profit: 3-year at-the-money call option on a stock portfolio currently worth $1,000
- On maturity:
- Investor gets the 1,000 face value of the discount bond
- If the stock value rises in value, the holder exercises the call option
- Buy at strike and sell at spot
- Investor receives profit from rise in value
- If the stock value drops in value, the holder does not exercise the call option
- 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 mid-price, 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
-
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
- 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
|