Question

An American call option on a non-dividend-paying stock is currently trading in the CBOT market. The price of the underlying stock is $36, the option strike price is $30, and the option expiration date is in three months. The risk free interest rate is 8%

a. Calculate the upper and lower bounds for the price of this American call option --> Calculated as max (So - Ke^ -rT, 0)

b. Explain the arbitrage opportunities presented to an arbitrageur when this American call option is trading at $37 and when it is trading at $4. Calculate the risk-free profits that can be earned by an arbitrageur in both cases. (Note: calculate the risk-free profit also for both scenarios at maturity i.e. when ST is greater than $30 and when ST is lower than $30).

Answer #1

a.

S_{0:} the current price of the underlying asset=
$36

K: the exercised (strike) price =$30

T: the time to expiration of option =3 months

r: the risk-free interest rate= 8%

Upper bounds for this American call option, C
<=S_{0}

<=36

Lower bounds for this American call option, : C >=
S_{0} - Ke ^{-rT}

>= 36-30*e
^{-0.08*(3/12)}

>=6.59

b.

When S_{T}> $30,

When C=$37

C+ Ke ^{-rT} =37+30*e ^{-0.08*(3/12)} =66.4

When C=$4

C+ Ke ^{-rT} =4+30*e ^{-0.08*(3/12)} =33.40

Arbitrage opportunity exists with a risk-free profit of $33

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