SABR Volatility Model

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 rates derivatives markets.

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^eta_t dW_t,

:dsigma_t=alphasigma^{}_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< ho<1. The constant parameters eta,;alpha satisfy the conditions 0leqetaleq 1,;alphageq 0.

The above dynamics is a stochastic version of the CEV model with the "skewness" parameter eta: 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 maxleft(F_T-K,;0 ight) under the probability distribution of the process F_t.

Except for the special cases of eta=0 and eta=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=Talpha^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 Black's 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_{ ext{impl=alpha;frac{logleft(F_0/K ight)}{Dleft(zeta ight)};Big{1+left [frac{2gamma_2-gamma_1^2+1/F_{ ext{mid^2}{24};left(frac{sigma_0}{alpha} ight)^2;Cleft(F_{ ext{mid ight)^2+frac{ hogamma_1}{4};frac{sigma_0}{alpha};Cleft(F_{ ext{mid ight)+frac{2-3 ho^2}{24} ight] varepsilonBig},

where, for clarity, we have set Cleft(F ight)=F^eta. The value F_{ ext{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 ight)/2). We have also set

:zeta=frac{alpha}{sigma_0};int_K^{F_0}frac{dx}{Cleft(x ight)}=frac{alpha}{sigma_0left(1-eta ight)};left(F_0^{1-eta}-K^{1-eta} ight),

and

:gamma_1=frac{C'left(F_{ ext{mid ight)}{Cleft(F_{ ext{mid ight)}=frac{eta}{F_{ ext{mid};,

:gamma_2=frac{C"left(F_{ ext{mid ight)}{Cleft(F_{ ext{mid ight)}=-frac{etaleft(1-eta ight)}{F_{ ext{mid^2};.

The function Dleft(zeta ight) entering the formula above is given by:

Dleft(zeta ight)=logleft(frac{sqrt{1-2 hozeta+zeta^2}+zeta- ho}{1- ho} ight).

ee also

*Volatility
*Stochastic Volatility
*Risk-neutral measure

External links

*PDF| [http://www.math.columbia.edu/~lrb/sabrAll.pdf Managing Smile Risk, Hagan et. al.] |539 KiB - The original paper introducing the SABR model.
* [http://arxiv.org/abs/0708.0998v3 Fine Tune Your Smile - Correction to Hagan et. al.]
* [http://www.riskworx.com/insights/sabr/sabr.html A SUMMARY OF THE APPROACHES TO THE SABR MODEL FOR EQUITY DERIVATIVE SMILES]
* [http://arxiv.org/pdf/physics/0602102v1 UNIFYING THE BGM AND SABR MODELS: A SHORT RIDE IN HYPERBOLIC GEOMETRY, PIERRE HENRY-LABORD`ERE]


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