Put-Call Parity and Synthetic Arbitrage Bounds

Scenario

An equity derivatives market maker observes the following quotes for a non-dividend-paying stock and its European options:

• Spot price: S_0 = \100.00$
• Continuously compounded risk-free interest rate: $r = 5\%$ per annum
• Time to maturity: $T = 1.0$ year
• Strike price: K = \100.00$
• European call option market price: C = \12.00$
• European put option market price: P = \7.00$

Your Task

1. State the fundamental European Put-Call Parity relationship.
2. Determine whether an arbitrage opportunity exists in these market quotes.
3. Construct the exact replicating cash-and-carry or reverse cash-and-carry portfolio to exploit the mispricing, and compute the risk-free profit realized at terminal maturity $T$.

Constraints

• European exercise style (no early exercise).
• Frictionless borrowing and lending at constant rate $r$.
• No short-selling restrictions or transaction costs.

Examples Given $S_0 = 100, K = 100, r = 0.05, T = 1.0$, the theoretical difference $C - P$ must equal $S_0 - K e^{-r T} \approx 100 - 95.1229 = 4.8771$. With $C - P = 5.00$, synthetic forward is overpriced by \0.1229$ in present value terms.