The Arrow Markets pricing engine fits SVI curves for implied volatility
For our benchmark prices, we use SVI models (so-called stochastic volatility-inspired models), which provide tractable volatility curve parameterizations. Options market makers and traders commonly use these models. Here are some key references on the SVI approach: URL.
The SVI model specifies the implied volatility curve as essentially a second-order expansion around log moneyness
IVSVI(k)=a+b(km)+c(km)2+d2\begin{align*} IV_{SVI}(k) = a + b \,(k-m) + c \sqrt{(k-m)^2 + d^2} \end{align*}
where we enforce the relation
b=cρ where ρ<1.b = c \rho \text{ where } |\rho| < 1.
The parameters
{a,ρ,c,d,m}\{a, \rho, c, d, m\}
are obtained by fitting this model to market-implied volatility quotes. In practice, this curve is refit at some frequency, and data cleaning needs to occur. Arrow's up-to-date fit frequency and data cleaning steps are maintained here: URL.
A separate set of parameters is fitted for each expiration. To get the midpoint price for an option with strike
and expiration
, we calculate the log moneyness
k=logK/Stk = \log{K/S_{t}}
and consult the fitted curve associated with expiration
to get the implied volatility
σT,k=IVSVI,T(k).\sigma_{T,k} = IV_{SVI,T}(k).
The midpoint price is then the BSM price of the option given