SABR Model

The SABR (Stochastic Alpha, Beta, Rho model) model is a stochastic volatility evaluating model that attempts to depict volatility smile patterns in financial markets. It does so by modeling stochastic volatility dynamics with single forward rates. Volatility smiles are patterns formed as a result of financial option pricing.

Market-implied volatility may vary with both the strike value and the time to maturity. The model attempts to capture the volatility smile structure by modeling a single forward rate with stochastic volatility dynamics. It is a substitute model that covers the Black-Scholes model's shortcomings (such as why volatility smiles and skew occur).

The SABR model (Stochastic Alpha, Beta, Rho model) in mathematical finance is a method that attempts to study forward prices that are often used in modeling interest rate derivatives. The parameters used in the process are Alpha, Beta and Rho. Alpha in the SABR model parameter helps understand the magnitude or range of variations in the underlying asset's price. The beta in the SABR model helps in understanding the sensitivity of the price movements portrayed by forwards toward the spot price. The rho parameter helps in understanding the relationship between forward price and underlying asset price's volatility movements.

It is used as an alternative that covers certain shortcomings in the original Black-Scholes model, in which volatility is assumed constant. However, options possess different volatilities according to the market prices. The implied volatility in equity markets, when plotted on a graph against the strike price, will result in one that is downward sloping. This is known as "volatility skews," and those curved like a valley or smile are known as "volatility smiles." The SABR (Stochastic Alpha, Beta, Rho) model predicts the accurate dynamics that cause these better when compared to other models and prevents unstable hedges.

It is preferred to other models for various reasons. Under this stochastic volatility model, the volatility and asset price are related. The model has a simpler form that accommodates the inclusion of market prices and risks such as Volga and Vanna risks. Additionally, it assumes that the forward price volatility is a stochastic variable. Hence, the model is often used to price options, manage risks and hedge portfolios.

Analytical assessment of the probability distribution from the SABR model can be complicated and challenging. The use of Mthe onte Carlo simulation technique can be helpful in deciphering the pmodel's properties and behavior The model assumes that tsset prices behave in accordance with the geometric Brownian motion, and tolatility's movement is based on the Ornstein-Uhlenbeck process. 

These are parameterized by alpha, beta and rho. Alpha in the SABR model deals with initial volatility levels, beta in the SABR model deals with variance elasticity and rho deals with price volatility correlation. It also involves volatility's volatility as another parameter. The model involves closed-form approximation usage for implied volatility. The Hagan formula used in the approximation helps in fast and accurate strike price implied volatility computations. Finite difference methods shall also be used to find option pricing, but they are time-consuming. Artificial neural networks that use the universal approximation theorem shall be used to calibrate the SABR parameters.

Let us look into some examples to understand the topic better

Example #1

Imagine Dan is an investor who wants to price a forward option. He goes with the popular Black-Scholes model. However, he remembers that the model assumes volatility as constant, so instead, he opts for the SABR model to cbetter apture the dynamics of the market volatility smile graphs This would give him a bmoreaccurate implied volatility reading. Additionally, the model as simpler and easier to use comparatively.

Example #2

The study presents extensions to the SABR (Stochastic Alpha, Beta, Rho) models to be applied in a swaption market that requires negative rates. It includes shifted SABR model, mixture and free boundary SABR models. Swaptions are interest rate options that give the holder the right to enter interest rate swap contracts at a specified time with a predetermined swap rate in the future. The SABR models were used as an interpolation technique for the swaptions cubes. The study aimed to extend the model to negative rates. It concluded that each extension method had its benefits and drawbacks. The Mixture SABR model had been found to be the state solution for parameterization issues and proved to be efficient and best calibrated (as compared to shifted sabr model and free boundary models). However, it also had drawbacks, such as not being able to accommodate market prices with extreme negative strikes.

Given below are some of the disadvantages of the SABR model:

• One of the major drawbacks of the model is that the forward rate's probability density function becomes negative at extremely low strike prices. This shall be critical in cases of a negative rate environment where the derivatives in the market are traded at low or negative strikes. 
• It assumes log-normal distribution for an asset, which may not hold in markets that possess high volatility.
• It triggers risk management actions due to jumps in the model's parameters that are spread across valuation dates or contract expiries.
• Although the model is simple, the analysis of probabilistic distributions can be challenging. 

The differences between the two concepts are given as follows :

Concept 

• The SABR (stochastic alpha, beta, rho) Model is a stochastic model that captures volatility smiles.
• Heston Model is also a stochastic model that evaluates volatility.  

Calculation 

• The SABR (stochastic alpha, beta, rho) model accommodates closed form for an asset's implied volatility, allowing faster calculations.
• Comparatively, the Heston model provides semi-analytical formulas for prices of call options and requires additional time for computation. 

Nature 

• The SABR (stochastic alpha, beta, rho) model uses a single or one-factor model to find the volatility of underlying assets.
• The Heston model is a two-factor model to find the volatility of underlying assets.