Black Volatility using SABR model

Question

As per the Wikipedia, the SABR model looks like below -

$dF_t = \sigma_t \left(F_t\right)^{\beta} dW_t$

$d \sigma_t = \alpha \sigma_t d Z_t$

I have 3 questions -

1. Let us assume that we know all the parameters. Then what does it mean when we are told that, estimate the Black volatility using SABR? Typically, when we estimate the Black volatility using BS formula, we take Option price with some maturity, and then estimate the $\sigma$ which is constant. In this case it itself stochastic. So what does the Black volatility mean in the SABR context? Is it the average Volatility of all $\sigma_t$ over the Option's life?

2. Now let say, we don't know the parameters, and we wish to estimate them. AFAIK, we generally look into market prices of Options for various strikes and maturities. So, do we have any closed form solution of the Option prices using SABR model?

3. As per wikipedia, the $F_t$ is Forward contract. Can we also use the SABR model for Futures assuming $F_t$ is future? So in turn, can we also estimate the Volatility of Spot, given that Volatility of Spot should be same as Futures as it is a linear derivative?

Answer 1

The SABR model represents the stochastic evolution of the price of some kind of assets under the measure for which it is a zero-drift martingale. For Forward contracts it's the so called "Forward measure", the one induced using the price of a zero-coupon bond that matures at the forward contract payment date as numeraire.

Now there is a difference between "estimating" and "calibrating" parameters: the first requires a statistics/econometric approach and a set of observed values of the random variable or random process. Parameters are estimated and confidence intervals are constructed in order to reject the null hypothesis formulated on what you are modeling.

When you want to calibrate your parameters, you are simply minimizing the difference between a predetermined function of those parameters and some observed quantities that such function should recover. The two approaches are completely different: calibrating a model is totally orthogonal to the true data-generating process nor you get any kind of way to assess the quality of your assumptions.

That being said, there's no such a thing as a SABR Option Formula at least in the same sense of Black and Scholes formula: there is an approximation of the Black-76 formula implied volatility as function of the SABR parameters. To be very precise there are many approximations (see references) for both the Black-76 formula implied volatility and the Bachelier (or Normal) implied volatility. To keep things simple I'll focus on Black-76 volatility.

Let $F_0$, $K$ and $T$ be respectively today's forward price, the option time to maturity and the strike price of the option. Then the implied volatility is a quantity

$$\sigma^{Black}_{Market}(K,F_0,T) : Black(K,F_0,T,\sigma^{Black}_{Market}(K,F_0,T)) = MarketPrice(K,T)$$

where $Black$ is the Black-76 option price (omitting call/put because theoretical volatility is identical). Now following the references you have some function

$$\sigma^{Black}(K,F_0,T) = \sigma^{Black}_{SABR}(\alpha_0(T), \beta(T), \nu(T), \rho(T), K; F_0, T) + error$$

such that $$Black(K,F_0,T,\sigma^{Black}_{Market}(K,F_0,T)) \simeq Black(K,F_0,T,\sigma^{Black}_{SABR}(K,F_0,T)) $$

So, in order to calibrate the parameters (I will skip the intricacies the actual calibration) you have to:

• Find a set of prices of European-style options for different strikes on a Forward contract with a givent maturity (notice indeed the parameters are correct only for that specific option maturity)
• Compute the implied volatility of such prices
• Find the parameters $\alpha_0(T), \beta(T), \nu(T), \rho(T)$ that minimize the difference between the SABR volatility approximation and the implied volatilities you extracted from market prices.
• Now you have calibrated the SABR model.

Notice how the SABR model is not actually a "model": it's a practical parametrization of the implied volatility surface, you will still price your options using the Black-76 formula.

The real utility of the SABR is to compute smile-corrected option sensitivities (the so-called Greeks) which you can see it improves your hedging variance. Notice indeed that's the name of the original paper by Hagan.

As per your last question: the volatility of a future (or forward whatever for what it matters) is not the same as the spot. It is only in case you admit completely deterministic interest rates, but this is nonsensical (especially in case of options on interest rates). In any other case the volatility of forward prices (or rates) is given by the combination of the variances of the underlying and the discounting rates (as well as any covariance).

By the way this is one of the reasons for which trading options on forward contracts is so common: the implied volatility already "contains" the aforementioned combination, therefore you don't need to estimate/calibrate separately the volatilities and the correlations.

References

Rebonato, Riccardo. Volatility and correlation: the perfect hedger and the fox. John Wiley & Sons, 2005.

Hagan, Patrick S., et al. "Managing smile risk." The Best of Wilmott 1 (2002): 249-296.

Oblój, Jan. "Fine-tune your smile: Correction to Hagan et al." arXiv preprint arXiv:0708.0998 (2007).

Answer 2

1. That is the Black IMPLIED volatility that we are talking about. The formula for Blacks Vol in SABR means that when you compute $\sigma$ using the formula then you can produce option prices using the closed from formula given by Black and the volatility parameter is now given.

2. Yes. You look at option market pricee and find parameter to match those prices. This is specially easy, as you mentioned, when you have a closed from solution. As mentioned above, we do have a closed from solution.