(a) A stock is currently priced at $80. At the end of four months, the stock price is either $75 or $85 (the range gives information about volatility and ultimately, delta).
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By constructing a
portfolio, find the value of a four-month European put option with strike $80.
Assume continuous compounding and interest rates of 5%.
Answer
At the end of four months the value of the option will be
either $5 (if the stock price is $75) or $0 (if the stock price is $85).
Portfolio: 
(Note: The delta, 
of a put option
is negative. We have constructed the portfolio so that it is +1 option and 
shares rather
than 
option and 
shares so that
the initial investment is positive.)
The value of the portfolio is either 
or 
in four months.
If
Thus,
The value of the portfolio is certain to be 42.5. For this
value of the portfolio
is therefore riskless. The current value of the portfolio is:
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(Where 
is the value of
the option)
Since the portfolio is riskless
The value of the option is therefore $1.80.
Given that: 
, 
This can also be calculated directly from equation
And 
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