# Volatility Curve Surface Construction

OpusARM implied volatility term surface calibration keeps OpusAMM option prices in line with  the market and overall trading conditions.&#x20;

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:&#x20;

For a day d, we let&#x20;

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

Then daily variance between days d(t) and d(t+1) is defined as:&#x20;

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

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:&#x20;

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

Now the interpolated A(d) is:&#x20;

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

Yielding:&#x20;

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