Tuesday, July 29, 2014

Another SVI Initial Guess

The SVI formula is:
$$w(k) = a + b ( \rho (k-m) + \sqrt{(k-m)^2+ \sigma^2}$$
where k is the log-moneyness, w(k) the implied variance at a given moneyness and a,b,rho,m,sigma the 5 SVI parameters.

A. Vogt described a particularly simple way to find an initial guess to fit SVI to an implied volatility slice a while ago. The idea to compute rho and sigma from the left and right asymptotic slopes. a,m are recovered from the crossing point of the asymptotes and sigma using the minimum variance.

Later, Zeliade has shown a very nice reduction of the problem to 2 variables, while the remaining 3 can be deduced explicitly. The practical side is that constraints are automatically included, the less practical side is the choice of minimizer for the two variables (Nelder-Mead) and of initial guess (a few random points).

Instead, a simple alternative is the following: given b and rho from the asymptotic slopes, one could also just fit a parabola at-the-money, in a similar spirit as the explicit SABR calibration, and recover explicitly a, m and sigma.

To illustrate I take the data from Zeliade, where the input is already some SVI fit to market data.
3M expiry - Zeliade data

4Y expiry, Zeliade data
We clearly see that ATM the fit is better for the parabolic initial guess than for Vogt, but as one goes further away from ATM, Vogt guess seems better.

Compared to SABR, the parabola itself fits decently only very close to ATM. If one computes the higher order Taylor expansion of SVI around k=0, powers of (k/sigma) appear, while sigma is often relatively small especially for short expiries: the fourth derivative will quickly make a difference.

On implied volatilities stemming from a SABR fit of the SP500, here is how the various methods behave:
1M expiry on SABR data
4Y expiry on SABR data
As expected, because SABR (and thus the input implied vol) is much closer to a parabola, the parabolic initial guess is much better than Vogt. The initial guess of Vogt is particularly bad on long expiries, although it will still converge quite quickly to the true minimum with Levenberg-Marquardt.

In practice, I have found the method of Zeliade to be very robust, even if a bit slower than Vogt, while Vogt can sometimes (rarely) be too sensitive to the estimate of the asymptotes.

The parabolic guess method could also be applied to always fit exactly ATM vol, slope and curvature, and calibrate rho, b to gives the best overall fit. It might be an idea for the next blog post.

1 comment :

  1. I'm trying to implement the quasi-explicit calibration of the Gatheral's SVI inspired model as laid out in the Zeliade wp. This is the algorithm I follow:

    1. Select some initial parameters values for (a, b, rho, m, sigma)
    2. For the initial m and s, do a constrained optimization - P(m, sigma)
    3. Finally, using (a*, b*, rho*) from step (2), use simplex (Nelder-Mead) to estimate m* and sigma* by minimizing P.

    Is this enough to calibrate the model? More specifically, do I need to run multiple iterations of the algorithm or the 3 steps sufficient? I'm a bit confused because in step (2), the solution (a*, b*, rho*) depend on the initially chosen (m*, sigma*) and so different choices of initial values of these should yield different solutions for (a*, b*, rho*).