How to Derive Put-Call Parity and Price a Call Option

Put-Call Parity is a fundamental concept in options pricing that establishes a relationship between the prices of European call and put options with the same strike price and expiration date. Understanding this relationship ensures that no arbitrage opportunities exist in the market.

If you’re under pressure from graduate deadlines, it helps to focus on the assumptions that drive the method and the exact steps your course expects. Online tutoring can help you build a clear solution path and avoid common pitfalls.

This post explains the topic as it is typically covered in graduate programs, emphasizing definitions, assumptions, and step-by-step reasoning. The aim is to clarify how the method works and how to apply it correctly across common problem types. The discussion is written to be accurate, self-contained, and coursework-ready.

---

Step 1: Constructing Two Portfolios

Consider two portfolios with the same payoff at expiration T:

1. Portfolio A: Long one European call option (C) and short one European put option (P), both with strike price K.
2. Portfolio B: Long one share of the underlying stock (S) and borrow an amount equal to the present value of the strike price PV(K) at the risk-free rate r.

At expiration, the payoffs for both portfolios are identical:

• If ST > K: Portfolio A pays ST - K, Portfolio B pays ST - K.
• If ST < K: Portfolio A pays ST - K, Portfolio B pays ST - K.

Since the payoffs are the same, the principle of no-arbitrage implies that their costs today must be equal:

C - P = S - PV(K)

Rearranging to solve for the call price:

C = P + S - PV(K)

Here:

• C = price of the European call option
• P = price of the European put option
• S = current price of the underlying asset
• PV(K) = present value of the strike price discounted at the risk-free rate

---

Step 2: Example – Calculating the Price of a Call Option

Suppose we know the following market data:

• Current stock price: S = $100
• Strike price: K = $105
• Time to expiration: 1 year
• Risk-free interest rate: r = 5% annually (continuous compounding)
• Price of a European put option: P = $7

Step 2a: Calculate Present Value of the Strike Price

Discount the strike price to present value:

PV(K) = K × e-rT = 105 × e-0.05 × 1 ≈ 105 × 0.9512 ≈ $99.88

Step 2b: Apply Put-Call Parity

Using C = P + S - PV(K):

C = 7 + 100 - 99.88 ≈ $7.12

Conclusion: The fair price of the European call option is approximately $7.12.

---

This subject often arises in graduate homework, exams, or research-focused coursework. Careful walkthroughs can help clarify both assumptions and results. Online tutoring support is available for graduate-level courses.