SαβρModel - crowlogic/arb4j GitHub Wiki

The SABR model (Stochastic Alpha, Beta, Rho) is a popular stochastic volatility model used in mathematical finance for the pricing and risk management of interest rate derivatives, foreign exchange options, and other financial instruments. It was developed by Hagan, Kumar, Lesniewski, and Woodward in the early 2000s.

The SABR model is characterized by four parameters: alpha (α), beta (β), rho (ρ), and nu (ν). It assumes that both the forward rate (F) and the volatility (σ) follow stochastic processes, which are correlated with each other. The model is described by the following system of stochastic differential equations (SDEs):

  1. $dF = \sigma F^{\beta} dW_1$
  2. $d\sigma = \nu \sigma dW_2$
  3. $dW_1 dW_2 = \rho dt$

where:

  • $F$ represents the forward rate,
  • $\sigma$ is the volatility,
  • $\beta$ is a parameter that controls the behavior of the forward rate ($0 \leq \beta \leq 1$),
  • $\alpha$ is a parameter determining the initial level of the volatility,
  • $\rho$ is the correlation between the two Brownian motions, $W_1$ and $W_2$, ($-1 \leq \rho \leq 1$),
  • $\nu$ is the volatility of volatility, which determines how the volatility changes over time.

The SABR model has several attractive features:

  1. It is able to capture the skew and smile effects observed in the implied volatility surface.
  2. It allows for closed-form approximations for option prices, making it computationally efficient and practical for calibration and risk management.
  3. It can accommodate a variety of forward rate dynamics through the $\beta$ parameter. For example, $\beta = 0$ corresponds to a lognormal process, while $\beta = 1$ corresponds to a normal process.

Overall, the SABR model is a flexible and widely-used tool for modeling the complex dynamics of financial instruments, particularly in the context of interest rate and foreign exchange options.