A stock price is currently $40. It is known that at the end of three months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of a three-month European put option on the stock with an exercise price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.
A stock price is currently $40. It is known that at the end of three months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of a three-month European put option on the stock with an exercise price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.
The price of the European put option will be $$5 when the stock price is $35, otherwise $0, when the stock price is $45 at the end of three months.The portfolio consists of the followin:
−
Δ
:
s
h
a
r
e
s
+
1
:
o
p
t
i
o
n
The delta of the put option is taken as negative. To get a positive endowment, one need to take the portfolio to +1 option and
−
Δ
shares rather than -1 option, and
+
Δ
shares.The value of the portfolio will be
−
35
Δ
+
5
otherwise
−
45
Δ
If:
−
35
Δ
+
5
=
−
45
Δ
which gives:
Δ
=
−
0.5
22.5 will be the price of the portfolio. In this value of delta, the portfolio will be riskless. The current value of the portfolio will be:
−
40
Δ
+
f
f is the option price. To earn the risk-free rate of interest, portfolio should be as follows:
(
40
×
0.5
+
f
)
×
1.02
=
22.5
It equals to:
f
=
2.06
Therefore, the price of the option will be $2.06.
b. Verifying whether the risk-neutral valuation type gives the same answer.P is the probability of an increase in the stock value of a risk-neutral situation, which is shown below:
45
p
+
35
(
1
−
p
)
=
40
×
1.02
Equals to:
10
p
=
5.8
Or:
p
=
0.58
The expected value of the option will be:
0
×
0.58
+
5
×
0.42
=
2.10
Present value will be:
2.10
1.02
=
2.06
Hence, the answers are same.
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