The famous option pricing Black-Scholes framework assumes that the asset price follows the stochastic process:
$dS_{t} = rd_t+{\sigma}dw_t$
where $w_t$ is a Brownian motion and $r$ is the risk free rate.
This simply means the dynamics of asset price is a random variable following a lognormal distribution with constant volatility of ${\sigma}$. This evaluates to the option pricing formula:
$C({\sigma}, r, S_t, K) = S_t{N(d_1)} - KN(d_2)$
where $d_1 =
\frac{(ln(\frac{S_t}{K})+\frac{1}{2}(r+\sigma^2)t}{\sigma\sqrt{t}}$
and $d_2 = d_1-\sigma\sqrt{t}$
However, in practice, the option price implied by the market indicates that there is different volatility for different strikes. Furthermore, stock prices has distribution with fat tail and has higher peak relative to Normal distribution. This implies that there is a mixture in distributions with different variances. This is the motivation for developing volatility models such as Local Volatility, Heston Model, SABR Model and etc. In this post, the SABR model will be introduced briefly.
The SABR model is a stochastic volatility model developed by Patrick Hagan et al. It has the following dynamics:
$dF_t = F_t^{\beta} {\sigma}_t dW_t$ , $0≤{\beta}≤1$
$d{\sigma}_t = v{\sigma}_t dB_t$ , $v>0$
$d{<}W,B{>}_t = {\rho}dt$ , ${\rho}ϵ[-1,1]$
${\sigma}_0 = {\alpha} >0$
$where$
${\beta}$ : Describes the forward rate model.
$v$ : Lognormal volatility of volatility
${\rho}$ : Correlation between Forward rate and Volatility
${\alpha}$ : ATM volatility
This dynamic is clearly different from the Black Scholes framework and the market is quoted in Black's volatility for instruments such as cap/floor, swaptions and etc. What Hagan et al. did was to use this dynamics to price a European option. In the process of this evaluation, it is found that with some estimation techniques, the pricing using SABR dynamics is close to the normal model. From the normal model, it is then further derived that the pricing using SABR will approximate to Black's lognormal model.
The formula for the implied Black's volatility is as follows:
${\sigma}_{implied} (K,f)=\frac{\alpha}{((fK)^(\frac{(1-{\beta})}{2}) (1+\frac{(1-{\beta})^2}{24} log^2(\frac{f}{K})+\frac{(1-{\beta})^4}{1920} log^4((f/K)+⋯ ) ) )} \frac{z}{X(z)})$
$(1+\frac{(1-{\beta})^2}{24} \frac{{\alpha}^2}{(fK)^(1-{\beta})} + \frac{1}{4} \frac{{\alpha}{\beta}v{\rho}}{(fK)^\frac{1-{\beta}}{2}} +\frac{2-3{\rho}^2 }{24} v^2 ) t_ex+⋯)$
$where$
$z=\frac{v}{{\alpha}} (fK)^(\frac{1-{\beta}}{2})log(\frac{f}{K})$
$X(z)=log(\frac{\sqrt{1-2{\rho}z+z^2 }+z-{\rho})}{(1-{\rho})})$
This series of proving shows that Black's volatility quoted in the market can be modeled with SABR model. All that needs to be done is to calibrate the SABR parameters such that the volatility matches market quotes.
