The Put-Call Parity Theorem is a principle that defines the relationship between the price of European put and call options of the same class. The same class means that the options have the same underlying asset, strike price, and expiration date. The theorem says that the price of a call option implies a certain fair price for the corresponding put option, and vice versa.
The theorem also shows that a portfolio of a long call and a short put is equivalent to a forward contract on the same underlying asset, with the same strike price and expiration date. A forward contract is an agreement to buy or sell an asset at a fixed price in the future. This means that if you hold a long call and a short put, you will have the same return as if you hold a forward contract, regardless of the price of the asset at expiration.
The theorem is based on some assumptions, such as the existence of a risk-free interest rate, no transaction costs, no dividends, and the ability to borrow and lend at the same rate. If these assumptions are violated, the theorem may not hold exactly, and there may be arbitrage opportunities. Arbitrage is the possibility of making a risk-free profit by exploiting price differences in different markets.
Understanding the Put-Call Parity Theorem
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. The theorem ensures that in an efficient market, there is an equilibrium among these option prices. The formula for Put-Call Parity is as follows:
C – P = S – PV(X)
Where:
- C is the price of the European call option.
- P is the price of the European put option.
- S is the current stock price.
- X is the option’s strike price.
- PV(X) is the present value of the strike price discounted at the risk-free interest rate.
Excel Formulas for Put-Call Parity
Let’s break down the formula into Excel-friendly components:
- Present Value (PV):
PV(X) = X / (1 + r)^tIn Excel, you can use the formula:
=X / (1 + r)^t
Where r is the risk-free interest rate and t is the time to expiration. - Put-Call Parity Formula:
C - P = S - PV(X)In Excel:
=C - P - S + PV(X)
Procedure for Implementing Put-Call Parity in Excel
- Gather the Necessary Data:
- Current stock price (S),
- Call option price (C),
- Put option price (P),
- Strike price (X),
- Risk-free interest rate (r),
- Time to expiration (t).
- Calculate Present Value (PV(X)):
PV(X) = X / (1 + r)^t - Implement Put-Call Parity Formula:
Put-Call Parity = C - P - S + PV(X)
Scenario: Real Numbers Example
Suppose:
- S = $50 (current stock price),
- C = $5 (call option price),
- P = $3 (put option price),
- X = $52 (strike price),
- r = 5% (risk-free interest rate),
- t = 1 (time to expiration in years).
Excel Table for Scenario
| A | B | |
|---|---|---|
| 1 | Data | |
| 2 | Current Stock Price | $50.00 |
| 3 | Call Option Price | $5.00 |
| 4 | Put Option Price | $3.00 |
| 5 | Strike Price | $52.00 |
| 6 | Risk-Free Interest Rate | 5% |
| 7 | Time to Expiration (Years) | 1 |
| 8 | Calculations | |
| 9 | Present Value (PV(X)) | =B5/(1+B6)^B7 |
| 10 | Put-Call Parity | =B3-B4-B2+B9 |
Calculation Results
PV(X) = $52 / (1 + 0.05)^1 ≈ $49.52
Put-Call Parity = $5 – $3 – $50 + $49.52 ≈ $1.52
In this scenario, the Put-Call Parity holds, indicating that the market is in equilibrium.
Other Approaches
- Verification Approach:Check if the equality
holds. If yes, the Put-Call Parity is satisfied. - Implied Put-Call Parity:Calculate the implied put price using the call option price and vice versa. Compare the calculated prices with the actual put and call prices.
- Visual Representation:Use Excel charts to visually represent the relationship between call and put option prices, stock price, and present value.
