# Black-Scholes Model

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.

<figure><img src="/files/fDxxcdaw7s7ux2oRo3qC" alt=""><figcaption><p>Pre-populated price using modified Black-Scholes model</p></figcaption></figure>

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:&#x20;

$$
C(S,t)=SN(d\_1 )-Ke^{-rT} N(d\_2 )
$$

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*


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