Black-Scholes Model

Modified Black-Scholes model for digital assets

Portfolio modelling is centered around the Black-Scholes model with key modifications for digital assets. Modifications including the new Opportunity Cost (O) of collateral term have been introduced to reflect the characteristics of the 0xdx platform more accurately.

When you click on an option type within the orderbook, the order price field pre-populates with the value calculated by our modified Black-Scholes model.

Pre-populated price using modified Black-Scholes model

Below is the modified Black-Scholes model used:

$$d_1 = (ln(S/K) +(r+O+σ^2/2))/(σ√T)$$

$$d_2=d_1-σ√T$$

Premium price for calls:

$$C(S,t)=SN(d_1 )-Ke^{-rT} N(d_2 )$$

Premium price for puts:

$$P(K,t)=Ke^{-rT} N(-d_2 )-SN(-d_1 )$$

Where:

C(S,t) = Call Option (Premium) Price

P(K,t) = Put Option (Premium) Price

N( ) = Cumulative Distribution (Density) Function

T = Time Left to Maturity (in years)

S = Underlying Asset Price

K = Strike Price

r = Risk Free Rate

O = Opportunity Cost of Collateral

σ = Volatility