Put-Call Parity

It is a general phenomenon that long position on a call option combined with short position on the put option with the same strike and maturity is equivalent to long position on futures contract (with the same parameters). Mathematically it can be formulated as follows:

Long Call + Short Put = Long Futures

or equivalently

Call - Put = Futures.

The above parity gives us a chance of arbitrage - whenever there is a market inefficiency such that the above parity no longer holds, we are in a good position to earn without risk.

Note that this is only true for Quedex-style inverse options (i.e. options with fixed USD value). Options with 1 BTC notional, which are traded elsewhere, cannot be easily hedged with inverse futures.

Example

Assume on the market one observes the following:

  • Mark Price = Spot Price = $11000,
  • Futures Price = $10000,
  • Call Option Premium = 0.00000155 BTC,
  • Put Option Premium = 0.00001085 BTC, where both options are on Strike = $11000 and with 1-week maturity.

We could open a short position on a futures contract on the same strike and maturity (and Quantity = 10000) and our initial Unsettled P/L would equal

Unsettled P/L = Side * Quantity * (1 / Entry Price - 1 / Mark Price) = -1 * 10000 * (1 / 10000 - 1 / 11000) ~ -0.091 BTC.

We could add to that a short position on 10000 call options and long position on the same quantity of put options. Whence our Unsettled P/L on option positions equals

Unsettled P/L = Quantity * ( Put Option Premium - Call Option Premium ) = = 10000 * (0.00001085 - 0.00000155) = 0.093 BTC.

and thus our Unsettled P/L is positive and equals 0.093 - 0.091 = 0.002 BTC. But our portfolio consists of long put, short call and short futures, that is our instruments compensate - Put - Call - Futures = 0. This means our P/L won’t change over time and we have gained risk free 0.002 BTC.