The SABR (Stochastic Alpha, Beta, Rho) model is a widely used stochastic volatility model for pricing interest rate derivatives. It was introduced by Hagan, Kumar, Lesniewski, and Woodward in 2002. The main advantage of the SABR model is its ability to capture the volatility smile observed in the market prices of options, which is not possible using the standard Black-Scholes model.
The SABR model describes the joint dynamics of two correlated stochastic processes: the forward rate F(t) and its volatility σ(t). The model consists of the following two stochastic differential equations (SDEs):
$$\begin{aligned}
dF(t) &= \sigma(t) F(t)^{\beta} dW_{1}(t), \\
d\sigma(t) &= \alpha \sigma(t) dW_{2}(t),\end{aligned}$$
where W1(t) and W2(t) are two correlated standard Brownian motions with correlation coefficient ρ:
dW1(t)dW2(t) = ρdt.
In the SABR model, there are four parameters:
1. α: This parameter represents the initial (at t = 0) value of the volatility process. It is called the "vol-of-vol" because it determines how volatile the volatility process σ(t) is. Higher values of α lead to larger changes in the instantaneous volatility.
2. β: This parameter determines the forward rate process’s elasticity. Common choices for β are 0, 0.5, and 1, corresponding to normal, lognormal, and CIR models, respectively.
3. ρ: This parameter represents the correlation between the two Brownian motions driving the forward rate and volatility processes, and thus the correlation between the forward rate and its volatility. A positive correlation implies that when the forward rate increases, its volatility is also more likely to increase, while a negative correlation implies the opposite.
4. ν: This is the vol-of-vol parameter, which measures the rate of mean reversion of the volatility process.
To use the SABR model for pricing interest rate derivatives, we need to determine the implied volatility for different strikes and maturities. The implied volatility is defined as the market-observed volatility that, when plugged into the Black-Scholes formula, would produce the market-observed option price.
Hagan et al. derived an approximate closed-form expression for the implied volatility under the SABR model:
$$\begin{aligned}
\sigma(K, T) \approx \frac{\alpha S(x, \rho, \nu)}{(F(0)K)^{(1 - \beta)/2} \left[ 1 + \frac{(1 - \beta)^2}{24}\log^2\left(\frac{F(0)}{K}\right) + \frac{(1 - \beta)^4}{1920}\log^4\left(\frac{F(0)}{K}\right) \right]} ,\end{aligned}$$
where
K is the option’s strike price,
T is the option’s time to maturity,
F(0) is the forward rate at the current time,
S(x, ρ, ν) is a scaling function involving $x = \log\left(\frac{F(0)}{K}\right)$, ρ, and ν. This function adjusts for the skew/shape of the volatility smile.
Once we have the SABR-implied volatility for different strike prices and maturities, we can use the Black-Scholes formula or similar models to price various interest rate derivatives, including European options, swaptions, and caps/floors.
In summary, the SABR model is a versatile and powerful tool for capturing the volatility smile in interest rate derivatives pricing. It involves modeling the joint dynamics of the forward rate and its volatility using correlated stochastic processes and provides an approximate closed-form formula for the implied volatility.