Hull 182.3 vs Hull 181.5

[Tipo]

182.3. Assume a one-year American call option (C) and a one-year American put option (P) both have a strike price (K) of $51.00 when the price of a non-dividend-paying stock (S) is $50.00. The riskless rate is 5.0%. What are the lower bounds, respectively, of the American call and American put?

a. C >= 0, P >= 0
b. C >= 0, P >= $1.00
c. C >= $1.49, P >= 0
d. C >= $1.49, P >= $1.00<--answer

Im abit confused with the below formulas

C>=S(0) - K * exp(-rT)
P >= K - S(0)

and

S(o)-K=<C-P=<S(0)-K*exp(-rT)

When is one prefered over the other?
If i do the above question using the C-P, i end up with
-1=<C-P=<1.5
However the ans is d, which fits all
What is wrong here?C=0 and p=0 definitely fits and is the lowest

The C-P formula was used in 181.5 to calculate the lower bounds

181.5. The price of an American call on a non-dividend-paying stock is $3.00. The stock price is $40.00, the strike price is $42.00, and the expiration date is in six (6) months. The risk-free interest rate is 4.0%. What are the lower and upper bounds for the price of an American put on the same stock with the same strike price and expiration date?
a. $1.17 and $2.00
b. $2.17 and $3.33
c. $3.06 and $5.17
d. $4.17 and $5.00<--ans

 

[David Harper CFA FRM]

Hi @Tipo

Interesting observation! The key difference between those two questions, both of which ask about the lower bound of an American put, is that one question provides you the additional information of the corresponding American call option.

• Imagine if 181.5 did not give us this assumption: The price of an American call on a non-dividend-paying stock is $3.00. Then the lower bound of the put would be P >= K - S(0) --> P >= 42-40 = $2; i.e., if we don't have the corresponding call, we don't really have access to put-call parity and the lower bound is simply the intrinsic value (time value is at least zero).
• But as the question does give us the price of the American call, we can access put-call parity to inform P >= K*exp(-rt) - S + C --> P >= $4.17, which maintains the truth of P>=2, but is superior. So, basically, if we have the corresponding option price, we are better to try and utilize put-call parity. I hope that helps!