01-options-and-forward-pricing.md

Options and Forward Pricing

An option pays at expiry. A call pays $\max(S_T - K, 0)$; a put pays $\max(K - S_T, 0)$. Before expiry the option is worth more than intrinsic value because future moves still matter. That extra value is where volatility enters.

Why Black-76

The dashboard prices options against the forward/futures mark $F$, not spot $S$. Crypto has deep futures markets and no clean risk-free curve, so Black-76 is the natural model:

$$ \begin{aligned} C &= df \cdot [F\Phi(d_1) - K\Phi(d_2)] \\ P &= df \cdot [K\Phi(-d_2) - F\Phi(-d_1)] \\ d_1 &= \frac{\ln(F/K) + \tfrac{1}{2}\sigma^2T}{\sigma\sqrt{T}},\quad d_2 = d_1 - \sigma\sqrt{T}. \end{aligned} $$

For Deribit inverse options we use df = 1. The premium is quoted in coin, so a 0.05 BTC option premium becomes a USD-style model input by multiplying by the matching futures mark:

$$ \text{premium}_{USD} = \text{premium}_{coin} \cdot F $$

Practical Edges

• At expiry or zero volatility, price is intrinsic value.
• Time is ACT/365 years, not calendar days.
• Deribit's mark_iv is a percent value, so divide by 100 before comparing.
• Vectorized NumPy inputs let one pricing call cover a full option chain.