The Put-Call Symmetry (PCS) simply explained

Put-Call Symmetry (PCS) is a financial principle that provides a relationship between the prices of European put and call options with different strike prices but connected through the forward price of the underlying asset.

In order for put-call symmetry (PCS) to hold, the following assumptions are necessary:

• Frictionless Markets: There are no transaction costs, taxes, or restrictions on short selling, and securities can be divided into any fraction.
• No Arbitrage: There are no opportunities to make a risk-free profit from discrepancies in the pricing of securities.
• Zero Drift: The expected return on the underlying asset is equal to the risk-free rate.
• Symmetry Condition: The returns of the underlying asset are symmetrically distributed, meaning that the price of the underlying asset has no skewness or bias in either direction.
  

Under these conditions, PCS asserts that the adjusted prices of European calls and puts should be equal when their strikes are such that the geometric mean of the call strike (K) and the put strike(H) equals the forward price (F) of the underlying asset.

The forward price F is calculated by taking the current spot price S₀ and adjusting it for the risk-free rate r over time T:

F = S₀ * e^(rT)

We set the strike prices for the call (K) and the put (H) so that their geometric mean equals the forward price:

√(K * H) = F

This relationship makes the two options behave in a balanced way around the forward price. If the asset's future price is exactly at the forward price, both the call and the put have the same"distance" to the money, making them mirror images of each other in terms of their payout structure.

Which means:

K * H = F²

We define the European symmetry relationship, where the price of the call option divided by the square root of its strike price equals the price of the put option divided by the square root of itsstrike price: 

C(K) / √K = P(H) / √H 

Put-Call Symmetry (PCS) has practical applications in the pricing and hedging of exotic options. PCS can be particularly useful in the construction of static hedges for certain types of exoticoptions.

In a practical scenario, a trader might use PCS to price a down-and-in put option, which is active only if the underlying asset hits a certain barrier. By constructing a portfolio of European putsand calls with strikes set according to PCS, the trader can create a position whose value at expiration matches that of the exotic option under various scenarios.

The real power of PCS in hedging exotic options lies in its ability to reduce the complexity of managing such products. While PCS-based hedges might not be perfect due to real-world market frictionsand other factors, they can offer a close approximation that is sometimes easier and more cost-effective than dynamic hedging strategies.