Options Analytics Guide
This page documents the Eris Options Analytics that are produced by the Eris Price Engine™. These analytics includes option price, implied volatility, and a range of additional risk measures (collectively the option “greeks”), that are published on the Eris website and in various Eris data files. Below are the symbols, conventions, and formulaic expressions being utilized for the data produced.
Symbols
The following symbols are used in the tables below:
| Symbol | Description | Symbol | Description |
|---|---|---|---|
| \(F\) | Futures price of the underlying delivery futures contract.* | \(\bar{\sigma}_{LN}\) | Implied lognormal rate volatility. |
| \(K\) | Options strike, in the unit of the futures price. | \(\bar{\sigma}_N\) | Implied normal rate volatility (bpVol). |
| \(T\) | Time to maturity in years, measured in trading time, divided by 252.** | \(N(\cdot)\) | Standard normal cumulative distribution function. |
| \(DV01_F\) | DV01 of the underlying delivery futures contract. | \(n(\cdot)\) | Standard normal probability density function. |
| \(\sigma_{LN}\) | Implied lognormal futures price volatility. | \(c\) | Fixed coupon of the underlying delivery futures contract. |
| \(\sigma_N\) | Implied normal futures price volatility. | \(PAI\) | Eris Price Alignment Interest® on the underlying delivery futures contract, equal to the daily accumulated SOFR interest on the previous day’s SwapNPV, mirroring the daily interest on variation margin that takes place with cleared swaps. |
| \(s\) | Par swap rate. | \(PV01\) | $ price change for a 1 bp (0.01%) change in “c”, the fixed coupon of the underlying delivery futures contract. |
| SwapNPV | Present value of the swap embedded in the underlying futures contract. |
* Futures Price = 100 + SwapNPV + Past Coupons - PAI, but for the underlying delivery futures contract, Past Coupons will always be 0.
** Fractional trading days commence at 2 am CT (3 am ET) for a total of 12 hours, such that time to expiry in days as of 2 am CT (3 am ET) on any day is the remaining whole trading days plus the fraction of time remaining in the day prior to 2 pm CT (3 pm ET) that day. After 2 pm CT (3 pm ET), the next trading day is treated as 1 whole trading day until 2 am CT (3 am ET), at which point time then declines fractionally utilizing a 12-hour denominator.
Model Introduction
Eris Option Analytics are built on the closed-form pricing frameworks of the Asay model, (Black-76 modified with a risk-free rate of zero, i.e., discount factor of 1), and the Bachelier model. In the tables below, we identify where each is used.
Asay Model
As Eris Options' underlying delivery instruments are contracts with a futures price, for which it may be assumed cannot be negative, and because the option is European-exercise with futures-style margining, we utilize the Black-76 model formula with the risk-free rate used for discounting the forward option value set to zero (or, discount factor omitted or set to 1).
For a call option:
For a put option:
Where:
Bachelier Model
In order to translate futures price-based Eris Options greeks to interest rate-based equivalents, for broader user application and comparison with rate-based models and markets, we utilize the Bachelier model formula. As Eris Options are European-exercise and utilize futures-style margining, the risk-free rate used for discounting forward option value is again set to zero (or discount factor omitted or set to 1). In which case, the Bachelier formula reads:
For a call option:
For a put option:
Where:
In rate space, the analogous expressions replace \(F\), \(K\), and \(\sigma\) with the par swap rate \(s\), strike rate \(k\), and the corresponding rate volatilities \(\bar{\sigma}_{LN}\) and \(\bar{\sigma}_N\).
Analytics
Eris generates two forms of datasets, Settlement Analytics and Advanced Analytics. Settlement Analytics are required to adhere to the minimum price increments of the contract specifications. Advanced Analytics are the outputs from the Eris Price Engine, without any rounding to meet the minimum price increments of the contract specifications. These model-based Advanced Analytics are prefixed with “M”.
Settlement Analytics
Subsets of these analytics are available on the following pages: Volume & Open Interest, Trade Log and our Public File Server.
| Output | Description | Model | Expression | Example |
|---|---|---|---|---|
| Volatility | Implied lognormal futures price volatility. | Asay | \(\sigma_{LN}\) | 0.0525 5.25% annualized |
| Rate Vol LN | Implied lognormal rate volatility. | Asay | \(\bar{\sigma}_{LN}\) | 0.1077 10.77% annualized |
| Rate Vol Nor | Implied normal rate volatility (bpVol). | Bachelier | \(\bar{\sigma}_N\) | 0.821 82.1 bp annualized |
| Swaption Equivalent Strike | Implied par swap rate of the option strike. | N/A | \(k = c - \dfrac{K - 100 + PAI}{PV01}\) | 3.75% |
| BP OTM | Measure of basis points (BP) that the par swap rate implied by the option strike is out-of-the-money (OTM) relative to the par swap implied by the futures price. | N/A | \(\dfrac{K-F}{PV01_F} \times 1000\) | 70.0 70 bp OTM |
| Delta | Change in quoted option price with respect to the change in underlying futures price. | Asay | Call: \(N(d_1)\) Put: \(-N(-d_1)\) | 0.253 25.3% |
| DV01 | Dollar change in option price with respect to a 1bp parallel shift in the rate curve. | Asay | Call: \(N(d_1)DV01_F\) Put: \(-N(-d_1)DV01_F\) | 21.44 $21.44 |
| Gamma | Change in Delta of the option with respect to a change in futures price. | Asay | \(\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}}\) | 0.2231 22.31% |
| Gamma01 | Change in DV01 of the option for a 1bp parallel shift in the rate curve, aka GM01 or Dollar Gamma. | Asay | \(\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}} \times \dfrac{DV01_F^2}{1000}\) | 0.43 $0.43 |
| Theta | Change in option price for 1 trading day of time decay. | Asay | \(-\frac{1}{2}\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}}F^2\sigma_{LN}^2\dfrac{1}{252}\) | -0.00452 |
| Theta($) | Dollar change in option price for 1 trading day of time decay, aka Theta1d. | Asay | \(-\frac{1}{2}\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}}F^2\sigma_{LN}^2\dfrac{1}{252} \times 1000\) | -4.52 -$4.52 |
| Vega | Change in option price for 1% change in Volatility (implied lognormal futures price volatility). | Asay | \(\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}}\sigma_{LN}TF^2 \times 0.01\) | 0.01002 |
| Vega($) | Dollar equivalent of Vega. | Asay | \(\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}}\sigma_{LN}TF^2 \times 0.01 \times 1000\) | 10.02 $10.02 |
| Vega (Rate Vol LN) | Change in option price for 1% change in Rate Vol LN (implied lognormal rate volatility). | Asay | \(\dfrac{n(d_1)}{s\sigma_{LN}\sqrt{T}}\sigma_{LN}Ts^2 \times 0.01\) | 0.00051 |
| Vega($) (Rate Vol LN) | Dollar equivalent of Vega for lognormal rate volatility. | Asay | \(\dfrac{n(d_1)}{s\sigma_{LN}\sqrt{T}}\sigma_{LN}Ts^2 \times 0.01 \times 1000\) | 0.51 $0.51 |
| Vega (Rate Vol Nor) | Change in option price for 1 bp change in Rate Vol Nor (implied normal rate volatility). | Bachelier | \(n(d)\sqrt{T} \times 1bp\) | 0.00001 |
| Vega($) (Rate Vol Nor) | Dollar equivalent of Vega for Rate Vol Nor (implied normal rate volatility). | Bachelier | \(n(d)\sqrt{T} \times 1bp \times 1000\) | 0.01 $0.01 |
| Vanna | Change of Delta for a change in Volatility (implied lognormal futures price volatility), OR, change of Vega (futures price vega) for a change in futures price. | Asay | \(-\dfrac{n(d_1)d_2}{\sigma_{LN}} \times 0.01\) | 0.0525 5.25% |
| DvegaDvol | Change in Vega for a change in implied volatility, also known as Volga or VegaV. | Asay | \(F\sqrt{T}n(d_1)\dfrac{d_1d_2}{\sigma_{LN}} \times 0.01^2\) | 0.0501 5.01% |
Advanced Analytics
These analytics are currently not provided publicly by Eris Innovations. We plan to make them available over the next few months after the June 2026 launch of Eris Options.
| Output | Description | Model | Expression | Example |
|---|---|---|---|---|
| MPriceRaw | Model price raw generated from model vol raw. | Asay | \[ \begin{aligned} C(F,K,T,\sigma_{LN}) &= F N(d_1) - K N(d_2) \\ P(F,K,T,\sigma_{LN}) &= K N(-d_2) - F N(-d_1) \end{aligned} \] | 0.02438 |
| MVolatility | Implied lognormal futures price volatility. | Asay | \(\sigma_{LN}\) | 0.0525 5.25% annualized |
| MRate Vol LN | Implied lognormal rate volatility. | Asay | \(\bar{\sigma}_{LN}\) | 0.1077 10.77% annualized |
| MRate Vol Nor | Implied normal rate volatility (bpVol). | Bachelier | \(\bar{\sigma}_N\) | 0.821 82.1 bp annualized |
| MDelta | Change in quoted option price with respect to the change in underlying futures price. | Asay | Call: \(N(d_1)\) Put: \(-N(-d_1)\) | 0.253 25.3% |
| MDV01 | Dollar change in option price with respect to a 1 bp parallel shift in the rate curve. | Asay | Call: \(N(d_1)DV01_F\) Put: \(-N(-d_1)DV01_F\) | 21.44 $21.44 |
| MGamma | Change in Delta of the option with respect to a change in futures price. | Asay | \(\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}}\) | 0.2231 22.31% |
| MGamma01 | Change in DV01 of the option for a 1 bp parallel shift in the rate curve, aka GM01 or Dollar Gamma. | Asay | \(\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}} \times \dfrac{DV01_F^2}{1000}\) | 0.43 $0.43 |
| MTheta | Change in option price for 1 trading day of time decay. | Asay | \(-\frac{1}{2}\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}}F^2\sigma_{LN}^2\dfrac{1}{252}\) | -0.00452 |
| MTheta($) | Dollar change in option price for 1 trading day of time decay, aka Theta1d. | Asay | \(-\frac{1}{2}\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}}F^2\sigma_{LN}^2\dfrac{1}{252} \times 1000\) | -4.52 -$4.52 |
| MVega | Change in option price for 1% change in Volatility (implied lognormal futures price volatility). | Asay | \(\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}}\sigma_{LN}TF^2 \times 0.01\) | 0.01002 |
| MVega($) | Dollar equivalent of Vega. | Asay | \(\dfrac{n(d_1)}{F\sigma_{LN}\sqrt{T}}\sigma_{LN}TF^2 \times 0.01 \times 1000\) | 10.02 $10.02 |
| MVega (Rate Vol LN) | Change in option price for 1% change in Rate Vol LN (implied lognormal rate volatility). | Asay | \(\dfrac{n(d_1)}{s\sigma_{LN}\sqrt{T}}\sigma_{LN}Ts^2 \times 0.01\) | 0.00051 |
| MVega($) (Rate Vol LN) | Dollar equivalent of Vega for lognormal rate volatility. | Asay | \(\dfrac{n(d_1)}{s\sigma_{LN}\sqrt{T}}\sigma_{LN}Ts^2 \times 0.01 \times 1000\) | 0.51 $0.51 |
| MVega (Rate Vol Nor) | Change in option price for 1 bp change in Rate Vol Nor (implied normal rate volatility). | Bachelier | \(n(d)\sqrt{T} \times 1bp\) | 0.00001 |
| MVega($) (Rate Vol Nor) | Dollar equivalent of Vega for Rate Vol Nor (implied normal rate volatility). | Bachelier | \(n(d)\sqrt{T} \times 1bp \times 1000\) | 0.01 $0.01 |
| MVanna | Change of Delta for a change in Volatility (implied lognormal futures price volatility), OR, change of Vega (futures price vega) for a change in futures price. | Asay | \(-\dfrac{n(d_1)d_2}{\sigma_{LN}} \times 0.01\) | 0.0525 5.25% |
| MDvegaDvol | Change in Vega for a change in implied volatility, also known as Volga or VegaV. | Asay | \(F\sqrt{T}n(d_1)\dfrac{d_1d_2}{\sigma_{LN}} \times 0.01^2\) | 0.0501 5.01% |
