Interpolating SABR Implied Volatility Between Expiries

# SABR volatility interpolation


# SABR volatility interpolation












I am wondering how in practice the SABR model is used to calculate implied volatility in-between listed expiries? Imagine we have list of fixed expiries $T_1<\ldots<T_N$ and SABR parameters $\{\alpha_j, \beta_j, \rho_j, \nu_j\}$, calibrated to expiry $T_j$.

Do in practice we calculate the volatiltiy at $t\in[T_j, T_{j+1}]$ by interpolating total variance linearly for the fixed strike: $$ \frac{\sigma(K,t)^2 t - \sigma(K,T_j)^2 T_j}{t-T_j}=\frac{\sigma(K,T_{j+1})^2 T_{j+1} - \sigma(K,T_j)^2 T_j}{T_{j+1}-T_j} $$ or we calculate by interpolating in space of parameters of SABR? For example by using $\{\alpha_{j+1}, \beta_{j+1}, \rho_{j+1}, \nu_{j+1}\}$ directly or taking some linear combination of $\{\alpha_{j}, \beta_{j}, \rho_{j}, \nu_{j}\}$ and $\{\alpha_{j+1}, \beta_{j+1}, \rho_{j+1}, \nu_{j+1}\}$ to calculate $\sigma(K,t)$?

## Answer by Jesper Tidblom (score 2)

https://quant.stackexchange.com/a/83695

It depend on what you mean by "in practice". If you are not into pricing, but rather rough estimates, interpolating in total variance, as you suggest, is often recommended as the most natural way of doing it. However, specifying SABR parameters at each expiry is not automatically part of some full volatility surface model for which there is a model specified "correct" way to obtain the volatilities in between the expiries.

Specifying the four SABR parameters globally, instead of one setup for each expiry, will in itself give you an implied volatility surface for all expiries and strikes (where you get a good approximation of the volatility by the standard Hagan/Obloy formulas). The resulting surface will of course be very rough and inaccurate, since you only have four model parameters to fit to all vanilla call/puts on the market.

You could generalize the SABR model to the piecewise constant case where your parameters are constant for each period between the expiries. Notice that this is not the same as fitting a standard constant parameter model for each expiry separately. This will lead to a full surface model with well defined volatilities between the expiries. However, in that case you cannot use the Hagan formulas directly though, you then need to generalize the formulas to the piecewise constant case.

If you need some model where you match all the given volatilities on the market accurately, you should avoid using the SABR model at all. If you just need some rough way of getting some decent volatilities, then just go ahead with the standard linear interpolation in variance, as you suggested.

## Answer by Micio Geremia (score 1)

https://quant.stackexchange.com/a/83970

I suggest to check Patrick S. Hagan, Andrew S. Lesniewski, Diana E. Woodward, "Managing Vol Surfaces", Wilmott, January 2018, Pages 24-43 (https://doi.org/10.1002/wilm.10643).

In this paper the authors propose an advanced version of the original SABR model, said "Dynamical SABR model", with time-dependent coefficients. For any given option's expiry date $T$ they derive the effective SABR parameters $\theta_{eff}=\{\alpha_{eff}, \rho_{eff}, \nu_{eff}\}$ such that the original SABR model with these constant parameters gives the same terminal probability density as the dynamic SABR model. As a consequence, for expiry $T$, all European option prices under the dynamic SABR model are identical to their prices under the original SABR model with these effective parameters, and the implied volatility smile $\sigma(T,K,\theta_{eff})$ is given by the original SABR formula. This result is not exact, but has the same accuracy as the original SABR analysis. They also show how this model can be calibrated to volatility surfaces using piecewise constant instantaneous forward volatility $\sigma(T)$, volatility of volatility $\nu(T)$, and forward-volatility correlation $\rho(T)$.

I personally did not test this version of the SABR model, but in my opinion it seems promising.