# SABR model for stochastic volatility modelling

The SABR model is a stochastic volatility model. It stands for Stochastic Alpha Beta Rho (SABR), which are the main variables of SABR equations. More practically it is a widely used model in the financial industry to model both the volatility smile and the underlying price.

## SABR equations

I bet you are in a hurry and don't necessarily have the time for a long read. I will be good to you: here are the differential equations used in the SABR model. Do feel free to come back to these later in the article as the whole article will comment these equations.

- The underlying : $dF_t = \Sigma_t . F_t^{\Beta} dW1_t$
- The volatility : $d\Sigma_t = \Alpha . \Sigma_t . dW2_t$
- The correlated Wiener processes: $ dW1_t . dW2_t= \Rho. dt$

## SABR principles

The intuition behind the SABR model is that the forward rate and the volatility on this rate are linked.

### The volatility of the volatility

The volatility in SABR follows a log-normal distribution. It means that each Brownian move of the volatility depends on the current level of the volatility and in addition it also depends on a flat parameter called Alpha here (also often called Nu).

Interestingly, Alpha here plays the role of a flat volatility parameter. For this reason Alpha is dubbed the volatility of the volatility or “VolVol”.

Then we are already quite familiar with the impact of a volatility parameter: if Volvol is high, the volatility is moving a lot, if it is small instead, the volatility becomes flatter.

### The correlation between the Weiner processes

As “usual”, the changes in the underlying depend on the level of volatility, but here the model adds that the volatility is a stochastic process (and not flat) and the stochastic moves on the underlying and the volatility are correlated. This correlation between the volatility and the underlying changes is a corner stone of the SABR model.

If the correlation between the Weiner processes is 0, the random moves in the volatility and the underlying are uncorrelated.

If instead this correlation is close to 1, it means that an increase in the forward rate will likely coincide with an increase in volatility, and a decrease would happen also at the same time for the forward rate and the volatility.

If this correlation is close to -1, then on the contrary any random move on the underlying will see the opposite move in the volatility.

### The underlying distribution

In the SABR world, we don’t exactly know if the forward rates are normally distributed or log-normally distributed. This might seem a hard to understand statement. It only means that we don’t know if the change in the forward at any point in time depends on the current value of the forward or is independent of the current level of the forward.

Take an equity price: there is usually some resistance for this price to go down below a certain level because even if the company goes bust, the assets of the company are not worthless and can repay some money to shareholders when liquidated (after debt has been repaid). Even mathematically, an equity price can’t be negative, but it can be very large without restriction. These observations don’t apply to interest rates that are traditionally considered to have a more normal distribution.

In practice, interest rates in SABR world can behave as a blend of normal and log-normal distribution, thanks to the Beta parameter. This parameter is usually calibrated once, between 0 and 1, and not modified so much afterwards.

If **Beta = 0**, the interest rates distribution will be assumed to be **normal** (changes in the rates level won’t be impacted by the current interest rate level).

If **Beta = 1**, the interest rates distribution will be assumed to be **log-normal** (changes in the rates level will be impacted by the current interest rate level).

## Impact of SABR parameters on the generated smile

In practice, the SABR model is often used only to model the volatility. The important output from this model is then often the volatility smile and not the forward rate. Let use analyze the impact of the main SABR parameters on the volatility smile.

### Alpha (volvol)

**Increasing Alpha increases the convexity:**

### Rho

The Rho defines the skew in the underlying price distribution. Mathematically it is the correlation between the random processes linking the volatility and the underlying price.

**Increasing** Rho will **decrease** the volatility on the lower stikes.

### Beta (distribution Lognormal versus normal blend)

**Increasing** beta makes the underlying final price distribution more lognormal. As a consequence the **overall level of the volatility decreases dramatically**, especially on lower strikes, which would be made harder to reach in the case of a lognormal distribution.

### Normal vs logNormal distribution

It is always good to remember what a logNormal distribution is, when we most often deal with normal distributions. They are often used in different contexts: LogNormality is often assumed for equity prices, but equity returns are often assumed to be normally distributed. This actually makes complete sense if we define the returns on some equity stock as $ln(P_1/P_0)$ where $P_1$ and $P_0$ are the stock prices respectively at ti,mes 1 and 0.

Remember that if Y = Ln(X) is a random variable that follows a Normal distribution, then X is a random variable that follows a LogNormal distribution.