Volatility Curve Surface Construction

OpusARM implied volatility term surface calibration keeps OpusAMM option prices in line with the market and overall trading conditions.

Volatility interpolation methods use accumulated weighted daily variances between curve tenors, with weights reflecting each individual day’s value according to business days, non-business days, and important events. Holidays and weekends are substituted with a constant weight and inwards interpolation towards the one day tenor and outwards interpolation is used:

The weighted daily variance:

For a day d, we let

$$A(d) = \text{vol}(d)*\text{vol}(d)*d$$

Then daily variance between days d(t) and d(t+1) is defined as:

$$D(v) = \frac{A(d(t+1))-A(d(t))}{d(t+1)-d(t)}$$

For each day $d_i$ we then define $W_i$ to be the weight associated to that day, and the total weight

$$W(d) = ∑_{d_i \text{ days}}W_i$$

Then weighted daily variance between days $d(t)$ and $d(t+1)$ is defined as:

$$dVw = \frac{A(d(t+1))-A(d(t))}{W(d(t+1))-W(d(t))}$$

Now the interpolated A(d) is:

$$A(d) = A(d(t)) + dVw * [W(d)-W(d(t))]$$

Yielding:

$$\text{Vol}(d) = \frac{\sqrt{A(d)}}{d}$$