Practical Applications from the Experts - February 2010

categories: Product Documentation, general

SABR Model

SABR Model is a stochastic volatility model, which has been widely used by practitioners in the interest rate derivatives markets. The name stands for "Stochastic Alpha, Beta, Rho", referring to the parameters of the model. It has been released to the Swaption template and will be extended to Cap and Floor templates early this year. The dynamics of the volatility skew/smile as interest rates change are well captured by calibrating SABR model to a richer implied volatility data set.

Key assumptions of SABR Model

The underlying forward rate evolves according to a constant elasticity of variance process, while volatility evolves according to geometric Brownian motion.

Example

Let’s take a look at 3 pairs of swaptions below. All 3 pairs have the same set of inputs except for the strike rate and the volatility model. The inputs are as follows:

• Each has a notional of $1,000,000
• Time to swaption expiry is 3 months
• 1 year swap
• Semiannual coupon with the reference curve of USD Swap
• European option, yield-based model (SABR is not applicable for price-based model)
• The 3 pairs shown are in the money, at the money and out of the money scenarios respectively. Within each pair, one is modeled using the traditional ATM-vol-only model while the other is modeled using the SABR model

The sample report below illustrates the comparison of risk numbers between the two models across the 3 different moneyness scenarios.

As can be seen in the report, there is no variation of the volatility number (82.96) under the traditional ATM vol model, which is not a perfect reflection of reality. Meanwhile, SABR model generates different volatilities for different moneyness scenarios, which is able to capture the volatility skew/smile. It hence generates more accurate PV and Vega results.

The above comparison will be more straightforward by looking at the volatility skew graph below drawn from this example.

Stress Test

In RiskManager, users also have the ability to run stress tests on the SABR model to give different shift amounts along the volatility skew. For instance, assume one feels the volatility skew in the previous example should be flatter, one can define a SABR model shift where one shifts the in-the-money part of the skew up by 10%, shift the out-of-the-money part down by 10% and keeps the at-the-money volatility unshifted. The results in the table below are as expected--the PV of the in-the-money swaption goes up while the PV of the out-of-the-money swaption goes down. The uneven changes prove that the shifts are carried out in a non-parallel fashion.

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