Calculate Call with Put Call Parity

Call-Put Parity is a fundamental principle in options trading that establishes a mathematical relationship between the price of a call option and the price of a put option on the same underlying asset with the same strike price and expiration date. This relationship holds true in an efficient market with no arbitrage opportunities.

What is Call-Put Parity?

Call-Put Parity is a theoretical relationship between the prices of European call and put options on the same underlying asset. It states that the price of a call option plus the present value of the strike price should equal the price of a put option plus the present value of the underlying asset's current price.

This relationship is derived from the fact that both options give the holder the right, but not the obligation, to buy or sell the underlying asset at a specified price. In an arbitrage-free market, the prices of these options should be related in a specific way.

Call-Put Parity Formula

The basic Call-Put Parity formula is:

Call Option Price + Strike Price × e^-rT = Put Option Price + Underlying Asset Price × e^-rT

Where:

C = Price of the call option
P = Price of the put option
S = Current price of the underlying asset
K = Strike price
r = Risk-free interest rate
T = Time to expiration (in years)
e^-rT = Discount factor

This formula shows that the difference between the call and put option prices should equal the difference between the strike price and the current price of the underlying asset, discounted to present value.

How to Use This Calculator

Our calculator allows you to verify the Call-Put Parity relationship for any given set of option parameters. Simply enter the current price of the underlying asset, the strike price, the risk-free interest rate, the time to expiration, and the prices of the call and put options. The calculator will then determine whether the relationship holds true.

The result will show you whether the calculated value matches the expected value based on the formula, helping you identify potential arbitrage opportunities or verify the market efficiency.

Example Calculation

Let's consider an example where:

Current price of the underlying asset (S) = $100
Strike price (K) = $105
Risk-free interest rate (r) = 5% or 0.05
Time to expiration (T) = 1 year
Price of the call option (C) = $8
Price of the put option (P) = $3

Using the formula:

8 + 105 × e^-0.05×1 ≈ 3 + 100 × e^-0.05×1

8 + 105 × 0.9512 ≈ 3 + 100 × 0.9512

8 + 99.867 ≈ 3 + 95.12

107.867 ≈ 98.12

In this case, the relationship does not hold, which might indicate an arbitrage opportunity or market inefficiency.

Interpretation of Results

The results from the calculator can be interpreted in several ways:

If the calculated value matches the expected value: The market is efficient, and there are no arbitrage opportunities.
If the calculated value differs from the expected value: There may be an arbitrage opportunity, or the market is inefficient.
If the difference is significant: It may indicate a trading opportunity or a need to adjust your trading strategy.

Always consider other factors such as transaction costs, bid-ask spreads, and market liquidity when interpreting the results.

Limitations

While Call-Put Parity is a useful concept, it has several limitations:

It assumes European-style options, which can be exercised only at expiration.
It ignores transaction costs, bid-ask spreads, and other market frictions.
It assumes a constant risk-free interest rate and no dividends.
It may not hold in practice due to market imperfections and trading costs.

These limitations mean that Call-Put Parity should be used as a guide rather than an absolute rule in options trading.

Frequently Asked Questions

What is the difference between Call-Put Parity and the Put-Call Parity?: There is no difference - "Call-Put Parity" and "Put-Call Parity" refer to the same principle. The terms are often used interchangeably.
Can Call-Put Parity be used for American options?: No, Call-Put Parity is specifically derived for European options. American options, which can be exercised early, do not satisfy the same relationship.
How does Call-Put Parity relate to arbitrage?: Call-Put Parity helps identify arbitrage opportunities. If the relationship does not hold, it suggests that there is an opportunity to profit from the discrepancy.
Is Call-Put Parity always true in practice?: No, Call-Put Parity is a theoretical concept. In practice, market frictions, transaction costs, and other factors can cause deviations from the ideal relationship.
Can Call-Put Parity be used for options on futures?: Yes, the principle can be extended to options on futures contracts, but the exact formula may need to account for the specific characteristics of futures options.