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.

Finance Train Subscription

Unlock full access to Finance Train and see the entire library of member-only content and resources.