Notes 5
Finance 5108
Learning Objectives
1. Understand the bounds for a stock option
2. Learn and be able to apply Put/call parity
3. The effects of dividends on the bounds of options and Put/call parity
4. Understand and be able to calculate the binomial model for the pricing of call
and put options using the one step and twostep model for the pricing of options
5. The relationship of volatility to u and d
6. The application of the model with a continuous dividend or for foreign currency
option evaluation
As we learned earlier the factors that affect the value of an option are the current
price of the underlying asset, the strike price of the option, and the dividends
expected during the life of the option. These affect the intrinsic value of the
option. The volatility of the underlying asset, the time to expiration, and the risk
free rate affect the time premium of the option.
Call price relationships. The price of a call must exceed or be equal to the current
market price minus the present value of the strike price. This can be shown to be
true if you have two portfolios. Portfolio A consists of the call option with the
strike price X and purchase a bond that matures at the expiration that has a
terminal value of X. The cost of this portfolio would be C
0
plus the present value
of the bond X e
rt
. Portfolio B would consist of owning the stock at a price of S
0
. If
the price of the stock at expiration, S
T,
is greater than strike price, Portfolio A
would have a value of S
T
 X (the intrinsic value of the call plus the bond value X
(maturity value of the bond). The total is S
T
which is the value of the stock. On
the other hand if the stock price at expiration is less than the strike price, a
rational investor would let the call expire and the value of portfolio A would be X
or the strike price. Since the other component was a bond that matures at X.
Portfolio B would have the value of the stock. Since by definition in this state of
the world the X > S
T.
Now the question is which of the portfolio would you pay more for today? It is
obvious that the Portfolio A has a greater payoff when the market price of the
stock is less than the strike price of the call at expiration option and in any other
states the portfolios have equal value. Therefore the cost of Portfolio A should be
greater than Portfolio B or
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View Full DocumentSolving we can see the price of the call must be greater than the price of the
stock minus the present value of the strike price.
The same logic can be applied to a put option and the results would be
rT
0
0
P = Xe
 S
These establish the lower bound of a Call or Put option.
The upper bound for a
call option would be the stock price.
The upper bound for an put option would be
the present value of the strike price of the option.
Put/Call Parity
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 Fall '09
 Rader
 Finance, Derivatives, Options

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