Frank Holmes, CFA, is reviewing Martha Inc, a distributor. Holmes is interested in the company's European-style call option, which has a value of $5.90. Currently,
Martha's stock is trading at $33 per share and pays no dividend. The exercise price of both the call and put options is $30, with 80 days to expiration. The current risk-free rate is 5.50%. Martha's put option sells for $2.75. Calculate the synthetic call option value.
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A. B. C.C
To calculate the synthetic call option value, we need to use the put-call parity relationship. Put-call parity states that the value of a European-style call option plus the present value of the exercise price equals the value of a European-style put option plus the current stock price. Mathematically, the put-call parity relationship can be expressed as:
C + PV(X) = P + S
Where: C = Value of the call option PV(X) = Present value of the exercise price P = Value of the put option S = Current stock price
Given information: C = $5.90 (value of the call option) PV(X) = ? (present value of the exercise price) P = $2.75 (value of the put option) S = $33 (current stock price)
To calculate the present value of the exercise price (PV(X)), we need to discount the exercise price using the risk-free rate and the time to expiration. The formula to calculate PV(X) is:
PV(X) = X / (1 + r)^(t/365)
Where: X = Exercise price r = Risk-free rate t = Time to expiration
Given information: X = $30 (exercise price) r = 5.50% (risk-free rate) t = 80 days (time to expiration)
Calculating PV(X): PV(X) = $30 / (1 + 0.055)^((80/365)) PV(X) = $30 / (1.055)^(0.219)
Now, let's calculate the PV(X):
PV(X) = $30 / 1.0219 PV(X) ≈ $29.34
Now, we can plug in the values into the put-call parity equation to solve for the value of the put option:
C + PV(X) = P + S $5.90 + $29.34 = $2.75 + $33
Simplifying: $35.24 = $5.75 + $33
Subtracting $33 from both sides: $35.24 - $33 = $5.75
Simplifying further: $2.24 = $5.75
Therefore, the synthetic call option value is $5.75.
The correct answer is B. $5.75.