# Synthetic Options and Rationale

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

c0 + X/(1+rF)T = p0 + S0

• c0 = Current price of the European call
• p0 = Current price of the European put
• X = Strike price of the put and the call
• T = Time to expiration
• rF = Risk free rate
• S0 = Current spot price of the underlying asset

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.

c0 = p0 + S0 - X/(1+rF)T

• 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.

S0 = c0 - p0 + X/(1+rF)T

• 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.