# The relationship between valuation models for call...

## The relationship between valuation models for call and put options (parity)

All classical models for estimating the fair price of an option are developed for a call option. This is not accidental, since there is a strict dependence between the call price and the put price, called "put/call parity" (put! call parity ).

Put/call parity - the existence of a rigid relationship between the behavior of fair prices for put and call options on the same underlying asset when the lifetime is equal and the same external environment.

To understand this dependence, consider the hedged strategy of an investor buying options and put and call to reduce the risk of investing in underlying assets. The investor's goal is to minimize the risk as much as possible.

The investor can form the following portfolio: 1) sell the call option to the stock and buy the put option with the strike price X, 2) buy the stock at the current price 5. Such a hedged strategy does not have any risk for the investor and provides a gain in any situation with respect to the dynamics of the share price in the amount of X.

Make sure that this is so. At the current time, the cash flow of the investor (outflows) is equal to (S - C + P), where 5 - the current share price; С - call price; P is the price of the put; C and P are the required variables in the model (fair prices for options).

In Table. 31.3 shows the net payments that an investor has with a hedged portfolio at a stock price of S * on the settlement date.

Table 31.3

Option value at maturity

 Cash flows from the investor as of the date of execution of the action to form a portfolio at the time t = 0 If S * & gt; X if in * & lt; X Sell option call money flow (S * -X) 0 Buying a put option The costs are P 0 X-S * Purchase of shares The costs are S (the price of the stock at time t = 0) S * S * Finalized cash flow for the selected strategy X X

Thus, in the absence of risk, an investor can earn money X through such a hedging strategy. It is obvious that the current estimate of X must coincide with the current costs of implementing this strategy.

The current valuation of the hedge strategy win = PV (X) = The current cash flow estimate for the cost of the strategy = S-C-P.

Thus, knowing the current price of the stock 5, the strike price, the risk-free bet (as the discount rate to bring the future flow X to the current time) and estimating the call price by the model, you can calculate the put price.

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