SABR volatility model

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 S. 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_{t}^{\beta }\,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 $-1<\rho <1$ :

dW_{t}dZ_{t}=\rho dt

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.

SABR for the negative rates

A SABR model extension for Negative interest rates that has gained popularity in the recent years is the shifted SABR model, where shifted forward rate is assumed to follow a SABR process

dF_{t}=\sigma _{t}(F_{t}+s)^{\beta }\,dW_{t},

d\sigma _{t}=\alpha \sigma _{t}\,dZ_{t},

for some positive shift $s$ . Since shifts are included in a market quotes, and there is an intuitive soft boundary for how negative rates can become, shifted SABR has become market best practice to accommodate negative rates.

The SABR model can also be modified to cover Negative interest rates by:

dF_{t}=\sigma _{t}|F_{t}|^{\beta }\,dW_{t},

d\sigma _{t}=\alpha \sigma _{t}\,dZ_{t},

for $0\leq \beta \leq 1/2$ and a free boundary condition for $F=0$ . Its exact solution for the zero correlation as well as an efficient approximation for a general case are available.^[1]

An obvious drawback of this approach is the a priori assumption of potential highly negative interest rates via the free boundary.

Arbitrage problem in the implied volatility formula

Although the asymptotic solution is very easy to implement, the density implied by the approximation is not always arbitrage-free, especially not for very low strikes (it becomes negative or the density does not integrate to one).

One possibility to "fix" the formula is use the stochastic collocation method and to project the corresponding implied, ill-posed, model on a polynomial of an arbitrage-free variables, e.g. normal. This will guarantee equality in probability at the collocation points while the generated density is arbitrage-free .^[2] Using the projection method analytic European option prices are available and the implied volatilities stay very close to those initially obtained by the asymptotic formula.

Another possibility is to rely on a fast and robust PDE solver on an equivalent expansion of the forward PDE, that preserves numerically the zero-th and first moment, thus guaranteeing the absence of arbitrage.^[3]

Extensions

The SABR model can be extended by assuming its parameters to be time-dependent. This however complicates the calibration procedure. An advanced calibration method of the time-dependent SABR model is based on so-called ``effective parameters".^[4]

References

↑ Antonov, Alexandre and Konikov, Michael and Spector, Michael, The Free Boundary SABR: Natural Extension to Negative Rates (January 28, 2015). Available at SSRN: http://ssrn.com/abstract=2557046
↑ Grzelak, Lech and Oosterlee Kees, "From arbitrage to arbitrage-free implied volatilities". Journal of Computational Finance, Forthcoming. Available at SSRN: http://ssrn.com/abstract=2529684
↑ Le Floc'h, Fabien and Kennedy, Gary, "Finite difference techniques for arbitrage-free SABR". Journal of Computational Finance, August 2016. Available at http://www.risk.net/journal-of-computational-finance/technical-paper/2465429/finite-difference-techniques-for-arbitrage-free-sabr
↑ Van der Stoep, A.W.; Grzelak, L.A.; Oosterlee, C.W. (2015). "The Time-Dependent FX-SABR Model: Efficient Calibration based on Effective Parameters". International Journal of Theoretical and Applied Finance. p. 1550042.

Managing Smile Risk, P. Hagan et al. – The original paper introducing the SABR model.
Probability Distribution in the SABR Model of Stochastic Volatility, P. Hagan et al. - Introduced the normal SABR model, heat kernel expansion, and asymptotic probability distribution.
Hedging under SABR Model, B. Bartlett – Refined risk management under the SABR model.
LIBOR market model with SABR style stochastic volatility, P. Hagan and A. Lesniewski – LMM extension of SABR for term structure modeling.
Arbitrage Free SABR, P. Hagan et al. - Refined treatment of near zero forwards.
Fine Tune Your Smile – Correction to Hagan et al.
A summary of the approaches to the SABR model for equity derivatives smile
Unifying the BGM and SABR models: a short ride in hyperbolic geometry
Asymptotic Approximations to CEV and SABR Models
Test SABR (with calibration) online
SABR calibration
Advanced Analytics for the SABR Model - Includes exact formula for zero correlation case
Small-Strike Implied Volatility Expansion in the SABR Model - Arbitrage-free asymptotic formula for small strikes and for long-dated options
The Free Boundary SABR: Natural Extension to Negative Rates - SABR for the negative rates

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