Exercise: Put-call parity with continuous dividends

A stock pays a continuous dividend yield $q = 2\%$. $S_0 = 50$, $r = 4\%$, $T = 0.5$ years, $K = 50$.

Tasks

1. State the put-call parity relation with continuous dividend yield $q$.

2. If the market quotes $C = 2.30$, what is the arbitrage-free put price?

3. Modify the replication argument to account for dividends reinvested in the stock. (The "long stock" leg of portfolio B now holds $e^{-qT}$ shares initially, reinvesting the dividend stream so that at $T$ we hold 1 share.)

4. Suppose the same stock is quoted with $C = 2.30$, $P = 2.10$. Is there arbitrage? If so, compute the per-unit profit.

Hint

The adjusted forward price is $F = S_0 e^{(r - q)T}$, and put-call parity becomes $C - P = S_0 e^{-qT} - K e^{-rT}$.