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Chapter 15: Options Intro
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Options: American vs European
Option: gives the holder the right, but not the obligation, to buy or sell the underlying asset at the exercise price by the exercise date
Call: right, but not the obligation to buy…
Put: right, but not the obligation to sell…
European option: option can be exercised only on exercise date T
American option: option can be exercised on or before exercise date T
Options Notation
Time subscript if necessary: C0 or Ct
Time subscript if necessary: P0, Pt
X or K: exercise price (or strike price)
S, S0, St: stock (underlying security) price
r: interest rate
s: standard deviation of stock (underlying asset) return
Call option cash flows
Put option cash flows
Option buyers and writers
Options can be bought (long) or sold i.e. written (short)
Like long or short positions in stock or other assets
Long call option:
Pays option premium up front, receive payoff/money if call ends in the money, i.e., if ST – X > 0
Short call option:
Receives option premium up front, pays (or loses) money if call ends in the money, i.e., if ST – X > 0
Why buy a call on a stock?
Call: right, but not the obligation, to buy the stock in future
Example: PG price on 17 April 2014: $78.53
Call = $3.00; Exercise price = $80; Time to expiration: 17 Jan 2015
Why is this an attractive strategy?
For $3 you get to bet(!) that the stock will go over $80 by 17 Jan 2015
Low initial cash requirement
Option increases in value only if the stock goes above $83 ($80 + $3 option premium paid)
Instead of buying the stock today, you get to wait 9 months to buy the stock? Delayed purchase
What’s worth more?
Right to buy PG in next 18 days for $80?
Right to buy PG in next 9 months for $80?
Calls with longer expiration times are worth MORE
PG January 2015 call with exercise = $80
Costs $3 today
Between April 2014 and January 2015: Only has non-negative cash flow
PG X=$80 call, T = expiration = 17 Jan 2015, C=cost = $3
If on 17 Jan 2015, PG stock is worth $100?
Exercise the call option i.e. buy the stock for $80
Immediate profit = $20 = $100 – $80
Taking into account option premium paid, net profit = $17 = $20 – $3
If on 17 Jan 2015, PG stock is worth $150:
Net profit = $67 = $150 – $80 – $3 = max[ST – X,0] – C
If on 17 Jan 2015, PG stock is worth $60?
Will you use your right to buy the stock for $80? No!
Immediate profit = max[ST-X,0] = 0. Net profit: max[ST-X,0] – C = -$3
Option expires worthless
Put option on PG
Put is the right, but not the obligation, to SELL a share of PG stock on or before 17 Jan 2015
A PG X=$80 Jan 2015 Put, on 17 April 2014 costs $6.65
If on 17 Jan 2015, PG stock price = $100, will you want to exercise the put?
i.e. do you want to sell a share of PG for $80, if the stock price is $100? No!
Immediate profit in January from the Put = $0 = max(X – ST, 0)
Net profit = -$6.65 = cost of put
If on 17 Jan 2015, PG stock price = $50, will you exercise the put?
Immediate profit = $30 = $80 – $50 = max(X – ST, 0)
Net profit = $30 – $6.65 = $23.35
Payoff & Profit for calls and puts
Long Call option:
Payoff on a call = max[ST – X, 0]
Profit on a call = max[ST – X, 0] – C
Long Put option:
Payoff on a put = max[X – ST, 0]
Profit on a put = max[X – ST, 0] – P
Call vs Put option price
January X = 80 Call has price in April 2014 of $3
January X = 80 Put has price in April 2014 of $6.65
Call is a bet(!) that PG will go above $80 by January 2015
Put is a bet(!) that PG will fall below $80 by January 2015
Why is the put more valuable than the call?
Market thinks that chance of PG going above $80 less than the chance of going below $80
Call options with different Exercise prices
PG X = 35 Jan 2015 call price > PG X = 50 Jan 2015 call price
X = 35 call gives you the right,, but not the obligation, to buy the stock for $35
X=50 call gives you the right, but not the obligation, to buy stock for $50
More attractive to be able to buy the stock for $35 than $50
Conclusion: as X ↑, call price ↓
PG X = 35 Jan 2015 put price < PG X = 50 Jan 2015 put price
More attractive to be able to buy the stock for $35 than $50
Conclusion: as X ↑, put price ↑
How to think about options - summary
Leveraged bet(!) on direction of the price of the underlying security
Long Call: anticipating stock price will go up in the future
Delay the purchase of stock
Unlimited gain if stock price goes up
Lower immediate liquidity requirement of outright stock purchase
Possibility of option expiring worthless and losing the option premium paid
Long Put: anticipating stock price will go down in the future
Delay the sale of stock
Unlimited (i.e., until stock price hits 0) gain if stock price goes down
Can protect the value of the stock if currently owned (portfolio insurance or protective put)
Possibility of option expiring worthless and losing the option premium paid
Options are one-sided bets
General properties of calls & puts
↑ exercise price, ↓ price of the Call
Higher the exercise price, higher the security price must rise for call option to be valuable
↓ exercise price, ↑ price of the Call
Lower the exercise price, lower the security price must rise for call option to be valuable
↑ exercise price, ↑ price of the Put
Higher the exercise price, the higher the profit from the price of the security declining
↓ exercise price, ↓ price of the Put
Lower the exercise price, the lower the profit from the price of the security declining
General properties of calls & puts
Longer the maturity of option, ↑ price of the Call or Put
Longer the maturity i.e., 9 months vs 3 months, more time for the security price to increase or decrease
More risky the underlying asset, i.e., ↑ s (standard deviation), ↑ price of the Call or Put
More risky security, expected change in price is higher, either up or down
Option combination profit patterns
Popular sport! Graph payoffs for call option, put option, stock at exercise date T as function of stock price ST
Graph payoffs of combinations
Stock + put (“protective put”)
Two calls with different exercise prices (“spread”)
Three calls or puts with different exercise prices (“butterfly)
See slides in Appendix for details
Option arbitrage propositions
Facts about option prices
Derived without much/any assumptions about stochastic process of stock price
Derived only from definitions
Arbitrage position 0
Consider an American call costing C0, with exercise price X, where the stock price is S0. Then C0 must be > Max(S0 – X, 0)
Proof by example: Suppose C0 = 5, S0 = 50, and X = 40
Make immediate profit = $5
Buy call: -5
Exercise immediately: -40
Sell stock immediately: +50
Arbitrage proposition 1
Consider a European call costing C0, with exercise price X, where the stock price is S0. Then C0 must be > Max(S0- PV(X), 0)
Proposition 1 is deep (proof in book)
Arbitrage proposition 2
It is never optimal to early-exercise an American call written on a stock which doesn’t pay dividends before the option maturity T
Another interpretation: If you’re thinking about early-exercising a call:
SELL THE CALL, don’t exercise it
Preserve the time-value in the price of the option
Example: you own a call T = 0.5 years, X = 50, S = 80, r = 6%
Immediate early exercise: Payoff = S – X = 30
By Prop. 1, call price is at least Max(S – PV(X), 0) = 80 – (exp-0.5*6%)50
= 80 – 0.97 * 50 = 31.45
Better off selling the call than exercising
Conclusion from Prop. 2
The American feature of calls is often worthless
In many cases: American call and European call have same value
Not true for puts: European put worth less than American put
Proposition 3: Put-call parity
Consider a European put and call on the same stock. Put and call have same exercise price X. Stock pays no dividends before option exercise date T
Then: P0 + S0 = C0 + PV(X)
Price of put + stock
Price of call + PV of X
Proposition 5: Call price convexity
Consider three calls on same stock with same maturity T
Assume that Call1 has exercise price X1, Call2 has X2, Call3 has X3
Assume X1 < X2 < X3 and equally spaced: X2 = (X1+X3) / 2
When Proposition 5 condition is violated, there is an arbitrage opportunity
No arbitrage: Sometimes you win, sometimes you lose.
Proposition 5: No arbitrage
ARBITRAGE:
You NEVER lose!
Proposition 5: Arbitrage
App 15 - 30
Slides following have been included for additional detail and a more advanced exploration of option strategies
Please reach out to the instructor or TA if further clarification would be helpful
App 15 - 31
Sample information on Yahoo! Finance
App 15 - 32
Sample information on Yahoo! Finance
Importing data from Web into Excel (subscription)
App 15 - 33
Data|From Web
Opens Microsoft Explorer: Got to URL
Mark arrows of desired tables
App 15 - 34
App 15 - 35
Make money on bought stock if price rises
Make money on shorted stock if price falls
Stock profit patterns
App 15 - 36
Call option profit pattern
App 15 - 37
Put option profit pattern
Protective put
Buy stock today at price S0 and Buy a put for P0 with exercise price X
Cost today: S0 + P0 ; Payoff at time T: ST + Max(X - ST, 0)
App 15 - 38
Payoff pattern looks like that of a call.
Is Stock + Put = Call?
Call spread
Buy call with exercise price, Xhigh; Write call with exercise price, Xlow. Both with same time T to maturity
Profit at T: -C(Xhigh) + C(Xlow) + Max(ST – Xhigh, 0) – Max(ST – Xlow, 0)
App 15 - 39
Combination of three puts or calls with different exercise prices (some long, some short)
Total number of positions (short + long) adds to zero
App 15- 40
App 15 - 41
Purchase call option,
cash flow < 0
Terminal call payoff,
Cash flows
of call buyer
Between times 0 and T:
Cash flow = 0 for European option
0 for American option
Write (I.e., issue) call option,
cash flow > 0
Pay terminal call payoff
Cash flows
of call writer
Between times 0 and T:
Cash flow = 0 for European option
0 for American option
Call Option Payoff Patterns
Purchase put option,
cash flow < 0
Terminal put payoff,
Cash flows
of put buyer
Between times 0 and T:
Cash flow = 0 for European option
0 for American option
Write (I.e., issue) call option,
cash flow > 0
Pay terminal call payoff
= – Max[X – S
Cash flows
of put writer
Between times 0 and T:
Cash flow = 0 for European option
0 for American option
Put Option Payoff Patterns
PG JAN2015 CALL, X = $80, CALL PRICE TODAY = 3
ST60<-- stock price in January
Call price, C 3
X, exercise80
Net profit
max(ST-X,0)-C
-3<-- =MAX(B3-B5,0)-B4
Data table: profit as function of ST
ST -3<-- =B6, call option profit in January
-1001020304050607080020406080100120140160Call Profit as Function of ST
PG JAN2015 PUT, X = $80, PUT PRICE TODAY = 6.65
ST50<-- stock price in January
Put price, P6.65
X, exercise80
Net profit
max(X-ST,0)-P
23.35<-- =MAX(B5-B3,0)-B4
Data table: profit as function of ST
ST23.35<-- =B6, call option profit in January
-20-1001020304050607080020406080100120140160Put Profit as Function of ST
expiration
stock priceClosing
Short description of underlying
1XLF17 Call17-Nov-1216.110.060.01433,9982,358,344Tracks index of financial stocks
2SPY143 Put17-Nov-12143.411.94-0.2389,6171,803,974Tracks SP500
3QQQ65 Put17-Nov-1266.020.81-0.2552,780759,554Tracks Nasdaq 100
4IWM78 Put17-Nov-1281.830.54-0.0943,168448,734Tracks Russell 2000
5MSFT29 Call17-Nov-12280.23-0.2036,371206,064Microsoft
6HPQ17 Put18-May-1314.713.23-0.1826,4256,984Hewlett-Packard
7SLV34 Call17-Nov-1231.390.150.0126,250121,974Tracks silver price
8UTX72.5 Put17-Nov-1277.830.38-0.0124,51522,674United Technologies
9FB21 Call17-Nov-1219.320.780.1024,390183,944Facebook
10INTC22 Call17-Nov-1221.460.200.0123,339190,434Intel
11ECA26 Call17-Nov-1223.020.18-0.2320,252219,634Encana Corp.
12GE22 Call17-Nov-1221.70.31-0.1519,603227,984General Electric
13BTU16 Call19-Jan-1329.9513.003.0019,091218,994
14NLY8 Put17-Jan-1515.940.660.0516,771177,934Annaly Capital Management (a REIT)
15CSCO19 Call17-Nov-1218.190.360.0016,654569,894Cisco
16AET45 Call17-Nov-1244.21.260.3516,48622,004Aetna
17EEM41 Put22-Dec-1241.91.14-0.2215,454444,464Tracks MSCI emerging markets index
18MS19 Call17-Nov-1217.450.14-0.0514,654289,574
19NXY20 Put22-Dec-1224.140.770.2714,64571,564Nexen (energy)
20FXI37 Call17-Nov-1237.671.160.2513,9481,127,324Tracks China 25 index
MOST ACTIVE OPTIONS, 22 OCTOBER 2012
Option price
Call1Call2Call3
Exercise price 203040
Call price 1064
Number of calls purchased 1-21
PROPOSITION 5 CONDITION HOLDS
Call1Call2Call3
Exercise price 203040
Call price 1084
Number of calls purchased 1-21
PROPOSITION 5 CONDITION VIOLATED
Max(50,0)15
CallbuyersprofitSXC
Max(0,50)10
PutbuyersprofitXSP
Call1Call2Call3
Exercise price 203040
Call price 1064
Number of calls purchased -12-1
, terminal stock price
Profit, Call1 -20<-- =B5*(MAX($B$8-B3,0)-B4)
Profit, Call1 28<-- =C5*(MAX($B$8-C3,0)-C4)
Profit, Call1 -6<-- =D5*(MAX($B$8-D3,0)-D4)
Total 2<-- =SUM(B9:B11)
BUTTERFLY WITH THREE CALLS
Payoff example
Call1Call2Call3
Exercise price 203040
Call price 1064
Number of calls purchased 1-21
, terminal stock price
Profit, Call1 20<-- =B5*(MAX(B8-B3,0)-B4)
Profit, Call1 -28<-- =C5*(MAX(B8-C3,0)-C4)
Profit, Call1 6<-- =D5*(MAX(B8-D3,0)-D4)
Total -2<-- =SUM(B9:B11)
SWITCH THE POSITIONS
Payoff example
1 call bought with exercise price 20
2 call written with exercise price 30
1 call bought with exercise price 40
-4-20246810010203040506070Total profitTerminal stock price S
Butterfly Payoff Pattern
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