P&L Dynamics

A Synthetic Option is a proprietary trading algorithm that aims to replicate the P&L performance of a traditional option (the ‘Theoretical Option’) by continuously adjusting an underlying portfolio of tokens and perpetual derivatives (the ’Replicating Portfolio’).

The P&L of a synthetic or traditional option can be expressed as:

Option P&L = Delta P&L - Theta Decay

1. Delta P&L: Driven by moves in the underlying token price

2. Theta Decay: Driven volatility in the underlying token price, and time

As explained later, at option expiry this converges to:

For Calls:

Option P&L = Max(0, Token Price - Strike) * Notional - Theta Decay

For Puts:

Option P&L = Max(0, Strike - Token Price) * Notional - Theta Decay

Delta P&L in detail

Delta P&L is driven by moves in the underlying token price and operates differently for Calls and Puts.

• For Calls, an increase in the underlying token price will generate positive P&L, and a decrease in the underlying token price will generate negative P&L.

• For Puts, a decrease in the underlying token price will generate a positive P&L, and an increase in the underlying token price will generate a negative P&L.

For synthetic options, Delta P&L is driven by a change in the valuation of the replication portfolio.

• For Calls, the replication portfolio is mostly long positions in synthetics and tokens, so an increase in the underlying token price will increase the value of the replicating portfolio.

• For Puts, the replication portfolio is mostly short positions in synthetics and tokens, so an increase in the underlying token price will decrease the value of the replicating portfolio.

At expiry, Delta P&L converges to the following formulae: -For a Call:

Delta P&L = Max(0, Token Price - Strike) * Notional

-For a Put:

Delta P&L = Max(0, Strike - Token Price) * Notional

Call Value as a Function of Token Price

Put Value as a Function of Token Price

Theta Decay in detail

Theta Decay operates in the same way for both Puts and Calls and is almost always negative for a long position.

With a traditional option, theta decay is expressed as a gradual decline in the value of the option. -If the option does not expire in-the-money, theta decay will drive its value to zero at expiry. -With a traditional option, we can say you pay your theta decay ‘up front’ in a lump sum as the option premium.

With synthetic options, theta decay is caused by 2 factors:

1. Funding P&L: -Perpetual derivatives held in the replicating portfolio must pay or receive a funding rate periodically. -For Puts, this is usually (though not always) beneficial, and reduces theta decay. -For Calls, this is usually (though not always) detrimental, and increases theta decay.

2. Trading Costs: -As the price of the underlying token moves around, the synthetic option algorithm needs to trade to adjust the replicating portfolio. -This trading is costly due to commissions and trading spreads, but also because the algorithm has ‘negative gamma’, meaning it must ‘buy high and sell low’ when it adjusts positions.

Theta Decay Uncertainty

With traditional options, the entire theta decay is paid up front in a lump sum: this is the option premium. As such, theta decay to expiry is known in advance with certainty.

With synthetic options, theta decay is paid over time, and gradually reduces the collateral associated with the synthetic option. In this case, theta decay is not known with certainty in advance and depends on how volatile the underlying token price is over the life of the synthetic option.

We can’t say with certainty in advance whether theta decay will be lower for a synthetic option versus a traditional option in a specific instance. However, our historical analysis shows that, on average, synthetic options have a slightly lower theta decay than traditional options.

Theta Decay will be Higher for Volatile Markets

The more the price of the underlying token jumps back-and-forth, the higher Trading Costs will be, and so the higher Theta Decay will be.

Theta Decay will be Higher for Strikes Close to Current Token Price

Theta Decay is higher for options with a strike close to the current token price. For example, if the current price of BTC is 60,000, a Call with a strike of 61,000 will have much higher theta decay (i.e. it will be much more expensive to run) than a Call with a strike of 65,000.