Mark Price and Greeks

The mark price represents a fair value estimate of futures and options prices for risk management purposes. A reasonable mark price is essential for the margin calculation and is continuously updated to reflect market risk levels.

Future Mark Price

According to futures pricing theory, the future mark price with an expiration date $T$ is defined as:

$$F^T_t=I_t \cdot e^{ABR(T) \cdot (T-t)}$$

Where, $t$ is the current time, $ABR(T)$ is the annualized basis rate for futures contracts with expiration time $T$, and $I_t$ refers to the index price when more than 30 minutes remain until expiration. Within 30 minutes before expiration, the SettleTWAP is applied.

Annualized Basis Rate

We retrieve the latest price data for futures contracts with different expiration dates from Deribit, ${(T_i,F_t^{T_i})}_{i=1,2,...,n}$, and backsolve for $ABR(T_i)$:

$$ABR(T_i) = \frac{1}{T_i-t}\log(\frac{F_t^{T_i}}{S_t})$$

Where, $S_t$ is the Deribit index price of the corresponding underlying asset. By interpolating these values, we can derive the annualized basis rate $ABR$ for any given expiration date $T$.

Option Mark Price

Option Volatility and Volatility Surface

Volatility is a core component in options pricing. While the Black-Scholes model assumes constant volatility with geometric Brownian motion, real-world volatility varies with strike price and time to expiration. Therefore, fitting a volatility surface becomes essential to eliminate strict arbitrage, highlight pricing biases, and enable market makers to adjust quotes to reduce risk exposure.

We use the SVI model to fit the implied volatility surface from exchange-traded options. For a series of options with the same expiration $T$, given a parameter set $\chi_R=\left\{a,b,\rho,m,\sigma \right\}$, the total implied variance for strike price $K$ is expressed as:

$$\omega(k; \chi_R) = a + b \left[ \rho(k - m) + \sqrt{(k - m)^2 + \sigma^2} \right]$$

Denoted as $\omega(k)$, the implied volatility $IV(K,T)$ is:

$$IV(K,T)=\sqrt{\frac{\omega(k)}{T-t}}$$

Where, $t$ is the current time, $k=\log(K/F_t^T)$ is the log-moneyness of strike price $K$.

We retrieve implied volatility data ${(K_i,v_i)}_{i=1,2,...,n}$ from Deribit for out-of-the-money options. $F_t^T$ is the underlying futures price for the respective expirations. Then:

$$k_i=\log(\frac{K_i}{F_t^T})$$

$$\omega(k_i) = v_i^2 \cdot (T-t)$$

We can obtain the dataset ${(k_i,\omega_i)}_{i=1,2,...,n}$. We apply the Quasi-Explicit method to optimize the fitting of the original SVI formula to determine the optimal parameter set $\left\{a^*,b^*,\rho^*,m^*,\sigma^*\right\}$. Applying this calibrated parameter set enables to interpolate a smooth volatility surface.

Option Mark Price

The option mark price is calculated using the Black-76 model. $F^T_t$ is the mark price of the underlying futures, and $r$ is the risk-free rate:

$$V(F^T_t,\sigma,T,t,K,r,call) = e^{-r(T-t)}[F^T_tN(d_1) - KN(d_2)]$$

$$V(F^T_t,\sigma,T,t,K,r,put) = e^{-r(T-t)}[KN(-d_2) - F^T_tN(-d_1)]$$

Where:

$$d_1 = \frac{\ln(F^T_t/K) + (\sigma^2/2)(T - t)}{\sigma\sqrt{T - t}}$$

$$d_2 = d_1 - \sigma\sqrt{T - t}$$

Example

Emma holds a BTC-10JAN24-43000-C option. The underlying futures price is 42,563, with 0.0195 years remaining to expiration. Assuming an implied volatility of 0.353, the call option price is calculated as:

$$d_1 = \frac{\ln(42563/43000) + (0.353^2/2)(0.0195)}{0.353\sqrt{0.0195}} = -0.1827$$

$$d_2 = -0.1827 - 0.353\sqrt{0.0195} = -0.2319$$

$$V = 42563 \cdot 0.43 - 43000 \cdot 0.41 = 641$$

Smooth Mark Price

The smooth mark price for both futures and options is calculated using the IndexTWAP as input, where $\tilde{I}_t$ denotes the IndexTWAP at time $t$.

For any futures contract with expiration $T$:

$$\tilde{F}^T_t = \begin{cases} \tilde{I}_t \cdot e^{ABR(T) \cdot (T-t)}, & \text{before UTC 7:40 on the expiration date} \\[5pt] F^T_t, & \text{after UTC 7:40 on the expiration date} \end{cases}$$

For options with expiration $T$ and strike price $K$, the smooth mark price is calculated similarly to the mark price, but using the smooth mark price of the underlying futures.

Greeks

Given the futures price $F_t^T$, strike price $K$, implied volatility $\sigma$, expiration date $T$, current time $t$, and risk-free rate $r$, the Greeks are computed using central difference methods:

Delta measures the sensitivity of the option price to a $1 increase in the underlying futures price:

$$\Delta = \frac{V(F^T_t + 1,\sigma,T,t,K,r) - V(F^T_t - 1,\sigma,T,t,K,r)}{2}$$

Gamma measures the sensitivity of Delta to a $1 increase in the underlying futures price:

$$\Gamma = \frac{V(F^T_t + 1,\sigma,T,t,K,r) + V(F^T_t - 1,\sigma,T,t,K,r) - 2 \cdot V(F^T_t,\sigma,T,t,K,r)}{(1)^2}$$

Vega measures the sensitivity of the option price to a 1% change in implied volatility $\sigma$:

$$\nu = \frac{V(F^T_t,\sigma + 0.01,T,t,K,r) - V(F^T_t,\sigma - 0.01,T,t,K,r)}{2}$$

Theta measures the sensitivity of the option price to the passage of time (per day):

$$\Theta = \begin{cases} V(F^T_t,\sigma,0,t,K,r) - V(F^T_t,\sigma,T,t,K,r), & T \leq \frac{1}{365} \\[5pt] V(F^T_t,\sigma,T - \frac{1}{365},t,K,r) - V(F^T_t,\sigma,T,t,K,r),& T > \frac{1}{365} \end{cases}$$

Rho measures the sensitivity of the option price to a 1% change in the risk-free rate:

$$\rho = \frac{V(F^T_t,\sigma,T,t,K,r + 0.01) - V(F^T_t,\sigma,T,t,K,r - 0.01)}{2}$$