Objectives
• Options
– basic strategies
– introduction to some pricing restrictions
– introduction to binomial model
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Payoff Diagrams
Payoff
Long stock
Stock Price
Short stock
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European call
Payoffs
Long call
Underlying
price
X
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European Put
Options Payoff
Long put
Underlying Price
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2
Some strategies
•
•
•
•
Naked
Protective put
Covered call
Straddle
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Protective put
• Purchase underlying security
• Purchase put option, exercise price X
Value
Underlying
Put
Value of underlying
X
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3
Algebraically
Position
ST <X
Underlying
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ST >X
ST
ST
Put
X- ST
ST
Net
X
ST
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Why do this?
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4
Covered call
• Purchase underlying
• Write call option against it
Underlying
Position Value
X
Underlying
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Straddle
• Buy put and call, both at the same strike
Position Value
Put
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Call
Options
Underlying
10
5
Other spreads
• combination of 2+ calls or puts, same asset
with differing exercise prices or times to
expiration
– Vertical or money spread
• Same maturity and different exercise price
– Horizontal or time spread
• different maturities
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Main Points
• Lots of strategies possible
• Options allow you to customize cash flows
across states in the future...
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Put call parity
• Relationship between price of European call
and put
• Independent of assumptions about
randomness in underlying
– stocks
– indexes
– bonds, currencies, etc.
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Final Payoffs: Long call and
short put
Position Value
Underlying
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Arbitrage relationship
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Example
• Stock price = $10
• maturity 1 year, interest rate 10%
• exercise price = $9.90
– Put price = $1
– Call price ?
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Call price < stock price
• Suppose stock price = 10, call price = 11,
exercise price = 4
• Strategy:
– purchase stock
– short option
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Payoffs on strategy
Item
Long stock
Today Maturity
ST<4
ST ≥4
-$10
ST
ST
Short option +$11
0
4- ST
Net
ST
4
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+$1
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9
American calls
• Don’t exercise early without dividends
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Binomial Model
• Workhorse model for derivatives valuation
• Very flexible
• Basic Assumption about stock price moves
– binomial process for stock price: either goes up
or down next period
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1 period example
• S0=10, current stock price, r= 1.1=riskfree
rate
• probability stock increase = q=0.5
– u: multiplicative upward movement =2
– d: downward movement = 0.5
–u>r>d
• Why?
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Stock price movements
S1 =
%&uS
'dS
0
0
= 2(10) = 20,
= 0.5(10) = 5,
probability = 0.5
probability = 0.5
uS0=20, prob 0.5
S0=10
dS0=5, prob 0.5
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Call option,X=10, 1 period
maturity
Stock Payoffs
Call Payoffs
20
Max(20-10,0)
=10
?
10
5
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Max(5-10,0)
=0
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Valuation
• Basic idea: come up with strategy of stock
and bonds with same payoff as option
– arbitrage says: cost of strategy = cost of option
•
: number of shares, : number of bonds
– ‘u’: 20+ 1.05=10
– ‘d’: 5+ 1.05=0
– Solving: = (2/3),
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=-(2/3)(5/1.1)
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Call value
• Value of call=value of portfolio
– (10)+ =(2/3)(10)-(2/3)(5/1.1)=3.64
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Implications
• 2 states in future: needed 2 securities (stock
and bond) to hedge/price option
• Call = long stock and short bond: levered
position in portfolio
• What happened to probability of u and d?
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Recap
• Pricing via replication
• probabilities of up and down doesn’t
matter!
• Steps:
– check if #securities ≥ # states
– if so, match cash flows state by state
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Summary
• Options
– strategies
– basic pricing restrictions
– introduction to binomial model
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Next Time
• Extend binomial model to deal with
multiple periods
• Applying model to portfolio strategies
– Sharpe and Perold in readings packet
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