# SABR volatility model

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In mathematical finance, the SABR model is a stochastic volatility model, which attempts to capture the volatility smile in derivatives markets. The name stands for "stochastic alpha, beta, rho", referring to the parameters of the model. The SABR model is widely used by practitioners in the financial industry, especially in the interest rate derivative markets. It was developed by Patrick Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward.

## Dynamics

The SABR model describes a single forward $F$, such as a LIBOR forward rate, a forward swap rate, or a forward stock price. The volatility of the forward $F$ is described by a parameter $\sigma$. SABR is a dynamic model in which both $F$ and $\sigma$ are represented by stochastic state variables whose time evolution is given by the following system of stochastic differential equations:

$dF_t=\sigma_t F^\beta_t\, dW_t,$
$d\sigma_t=\alpha\sigma^{}_t\, dZ_t,$

with the prescribed time zero (currently observed) values $F_0$ and $\sigma_0$. Here, $W_t$ and $Z_t$ are two correlated Wiener processes with correlation coefficient $-1<\rho<1$. The constant parameters $\beta,\;\alpha$ satisfy the conditions $0\leq\beta\leq 1,\;\alpha\geq 0$.

The above dynamics is a stochastic version of the CEV model with the skewness parameter $\beta$: in fact, it reduces to the CEV model if $\alpha=0$ The parameter $\alpha$ is often referred to as the volvol, and its meaning is that of the lognormal volatility of the volatility parameter $\sigma$.

## Asymptotic solution

We consider a European option (say, a call) on the forward $F$ struck at $K$, which expires $T$ years from now. The value of this option is equal to the suitably discounted expected value of the payoff $\max\left(F_T-K,\;0\right)$ under the probability distribution of the process $F_t$.

Except for the special cases of $\beta=0$ and $\beta=1$, no closed form expression for this probability distribution is known. The general case can be solved approximately by means of an asymptotic expansion in the parameter $\varepsilon=T\alpha^2$. Under typical market conditions, this parameter is small and the approximate solution is actually quite accurate. Also significantly, this solution has a rather simple functional form, is very easy to implement in computer code, and lends itself well to risk management of large portfolios of options in real time.

It is convenient to express the solution in terms of the implied volatility of the option. Namely, we force the SABR model price of the option into the form of the Black model valuation formula. Then the implied volatility, which is the value of the lognormal volatility parameter in Black's model that forces it to match the SABR price, is approximately given by:

$\sigma_{\text{impl}}=\alpha\; \frac{\log\left(F_0/K\right)}{D\left(\zeta\right)}\; \left\{1+\left[\frac{2\gamma_2-\gamma_1^2+1/F_{\text{mid}}^2}{24}\;\left(\frac{\sigma_0 C\left(F_{\text{mid}}\right)}{\alpha}\right)^2+\frac{\rho\gamma_1}{4}\;\frac{\sigma_0 C\left(F_{\text{mid}}\right)}{\alpha}+\frac{2-3\rho^2}{24} \right]\varepsilon\right\},$

where, for clarity, we have set $C\left(F\right)=F^\beta$. The value $F_{\text{mid}}$ denotes a conveniently chosen midpoint between $F_0$ and $K$ (such as the geometric average $\sqrt{F_0 K}$ or the arithmetic average $\left(F_0+K\right)/2$). We have also set

$\zeta=\frac{\alpha}{\sigma_0}\;\int_K^{F_0}\frac{dx}{C\left(x\right)} =\frac{\alpha}{\sigma_0\left(1-\beta\right)}\;\left(F_0^{1-\beta}-K^{1-\beta}\right),$

and

$\gamma_1=\frac{C'\left(F_{\text{mid}}\right)}{C\left(F_{\text{mid}}\right)} =\frac{\beta}{F_{\text{mid}}}\;,$
$\gamma_2=\frac{C''\left(F_{\text{mid}}\right)}{C\left(F_{\text{mid}}\right)} =-\frac{\beta\left(1-\beta\right)}{F_{\text{mid}}^2}\;.$

The function $D\left(\zeta\right)$ entering the formula above is given by

$D\left(\zeta\right)=\log\left(\frac{\sqrt{1-2\rho\zeta+\zeta^2}+\zeta-\rho}{1-\rho}\right).$

Alternatively, one can express the SABR price in terms of the normal Black's model. Then the implied normal volatility can be asymptotically computed by means of the following expression:

$\sigma_{\text{impl}}^{\text{n}}=\alpha\; \frac{F_0-K}{D\left(\zeta\right)}\; \left\{1+\left[\frac{2\gamma_2-\gamma_1^2}{24}\;\left(\frac{\sigma_0 C\left(F_{\text{mid}}\right)}{\alpha}\right)^2+\frac{\rho\gamma_1}{4}\;\frac{\sigma_0 C\left(F_{\text{mid}}\right)}{\alpha}+\frac{2-3\rho^2}{24} \right]\varepsilon\right\}.$

It is worth noting that the normal SABR implied volatility is generally somewhat more accurate than the lognormal implied volatility.