The prices of put and call options have an identity relationship through the concept of put-call parity.

- c
_{0}= Current price of the European call - p
_{0}= Current price of the European put - X = Strike price of the put and the call
- T = Time to expiration
- r
_{F}= Risk free rate - S
_{0}= Current spot price of the underlying asset

c

_{0}+ X/(1+r_{F})^{T}= p_{0}+ S_{0}The formula translation is: the price of a call with strike X plus the present value of strike price X equals the price of the put with strike X plus the current spot price.

**Synthetic Call Option:**If an investor believes that a call option is over-priced, then he/she can sell the call on the market and replicate a synthetic call.- Borrow the present value of the strike price at the risk free rate and purchase the underlying stock and a put.
**Synthetic Put Option:**Similar to the synthetic call option. A synthetic put can be created by re-arranging the put-call parity relationship, if the trader believes the put is overvalued.**Synthetic Stock:**A synthetic stock can also be created by rearranging the put-call parity identity. In this case, the investor will buy the call, sell the put, and lend the present value of the strike at the current risk free rate.- If the stock rises in value, then the long call will provide the upside; if the stock falls, then the short put will replicate the downside.
- Rationale for a “Synthetic”
- Rational investors would not arbitrarily enter into “synthetic” position; it is done to exploit a perceived mispricing. For example, if the investor believes that put call parity is showing that a stock’s call is overvalued, then he/she may execute a synthetic call strategy.
- The key to a synthetic strategy is to buy the undervalued asset, sell the overvalued asset and invest or borrow the difference at the risk free rate.

c

_{0}= p_{0}+ S_{0}– X/(1+r_{F})^{T}S

_{0}= c_{0}– p_{0}+ X/(1+r_{F})^{T}
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