# Put-Call Parity

Put-Call Parity is a commonly referenced term. It is primarily used to identify pricing discrepancies between European put and call options with identical underliers, expirations, and strike prices.

According to Put-Call Parity, simultaneously owning a short European put and long European call on the same underlier with the same strike will result in the same return as holding one forward contract on the same underlier.

Put Call Parity demonstrates the relationship between European put and call options with the same underlier, expiration, and strike. The concept applies specifically to European-style options, as American-style options allow for execution at any time prior to expiration.

The put-call parity also relies on a few other assumptions, such as:

• The interest rate is constant and does not vary over time

• There are no transfer restrictions because the underlying stock is liquid.

You can determine put-call parity by using the formula:

$\text{ Call option (C) + Present Value of strike price PV(x) = Put option (P) + Underlying Asset price (S) }$ If we move the equation around to:

$\text{Call Option (C) = Put Option (P) + Underlying asset price (S) - Present value of strike price PV(x) }$

We can see that the call option should be equal to a portfolio with a long position on the put option, a long position on the underlying asset price, and a short position in the present value of the strike price.

#### Arbitrage Opportunities

Put-call parity is often used by traders to identify arbitrage opportunities. An arbitrage opportunity arises when the actual market prices violate the put-call parity equation. In this case, a trader could theoretically lock in a risk-free profit by taking advantage of the price discrepancy.

For instance, if the price of a call option and the present value of the strike price is less than the price of the put option plus the price of the underlying asset, the trader could buy the call, short the put, sell the underlying asset, and lend the strike price. The transactions would result in a net inflow of cash and, at expiration, they will have to pay out the strike price, which they have lent, thus realizing a risk-free profit.

#### Implications

Beyond identifying arbitrage opportunities, the put-call parity is an important tool in financial management. For instance, it plays a key role in the pricing and valuation of corporate securities, especially in convertible bonds and warrants. Furthermore, the put-call parity also serves as a benchmark to assess whether options are fairly priced in the market.

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