# Put-call parity theory

Put-call parity theory links put and call option values via ‘no arbitrage’ assumptions and the related:

- underlying asset price
- option strike price
- time to maturity and
- risk-free rate of return.

So for example if the put option value, underlying asset price, strike price, time to maturity, and risk-free rate of return are known, then the call option value can be calculated using the put-call parity relationship:

Underlying asset price + Put value *less* Call value = Present Value of option strike price

Call value = Underlying asset price + Put value *less* Present Value of option strike price

In the special case where the strike price of the options is equal to the forward price of the underlying asset, the Put value and the Call value are exactly equal.

## Theoretically risk-free portfolios

The no-arbitrage pricing relationship is based on the theory that combinations of market assets and liabilities with the same terminal cash flows, must also have the same present values (i.e. the same theoretical current market prices).

For example both the left side and the right side of the put-call parity formula represent portfolios with the same terminal value:

Underlying asset + Put *less* Call = Present Value of option strike price

The left side portfolio is built by buying the underlying asset, buying a put option, and selling a call option with the same strike price.

The theoretically risk free terminal value of this portfolio is the equal strike price of the two options.

The present value of this left side portfolio is the present value of the strike price.

The right side of the portfolio is a deposit of cash.

This also produces a theoretically risk free terminal value, equal to the strike price of the options.

The pricing of these two portfolios must in theory be exactly the same.

If this relationship did not hold, there would be an arbitrage opportunity to buy the cheaper portfolio and sell the more expensive one, to earn an immediate risk free profit.

Therefore market supply and demand pressures will act to quickly re-establish the no arbitrage pricing relationship, following any temporary mis-alignments.