I am trying to understand the SABR model. Specifically, I am having difficulty to understand how to calibrate the model parameters, that is,
Small example on the above would be useful. Thanks in advance
The SABR model of Hagan is described by the following Stochastic differential equations: $$\begin{align} & d{{f}_{t}}={{\alpha }_{t}}f_{t}^{\beta }d{{W}_{t}}^{1} \\ & d{{\alpha }_{t}}=v\,{{\alpha }_{t}}d{{W}_{t}}^{2} \\ & {{E}^{Q}}[d{{W}_{t}}^{1},d{{W}_{t}}^{2}]=\rho dt \\ \end{align}$$ In these equations, $f_t$ is the forward rate, $\alpha$ is the initial variance, $\beta$ is the exponent for the forward rate and $v$ is the volatility of variance.
It is well-known the prices of European call options in the SABR model are given by Black's model. For a current forward rate $f$, strike $K$, and implied volatility $\sigma_{B}$ the price of a European call option with maturity $T$ is $$C(f,K,{{\sigma }_{\beta }},T)={{e}^{-rT}}(f\,N({{d}_{1}})-K\,N({{d}_{2}}))$$ where \begin{align} & {{d}_{1}}=\frac{\ln \left( \frac{f}{K} \right)+\frac{1}{2}\sigma _{B }^{2}T}{{{\sigma }_{B }}\sqrt{T}} \\ & {{d}_{2}}=\frac{\ln \left( \frac{f}{K} \right)-\frac{1}{2}\sigma _{B}^{2}T}{{{\sigma }_{B }}\sqrt{T}} \\ \end{align} and
Estimating $\alpha$, $\rho$ and v: This can be accomplished by minimizing the errors between the model and market volatilities {$\sigma_{i}^{market}$}(from interest rate derivatives, for example) with identical maturity T. Hence, for example, we can use SSE, which produces $$(\widehat{\alpha },\widehat{\rho },\widehat{v})=\underset{\alpha ,\rho ,v}{\mathop{\arg \min }}\,{{\sum\limits_{i}{\left( \sigma _{i}^{market}-{{\sigma }_{B }}({{f}_{i}},{{K}_{i}};\alpha ,\rho ,v) \right)}}^{2}}$$
Estimating $\beta$:
The at-the-money volatility $\sigma_{ATM}$ is obtained by setting $f = K$ in equation $\sigma (K,\beta)$, which produces $${{\sigma }_{ATM}}={{\sigma }_{\beta }}(f,f)=\frac{\alpha \left( 1+\left[ \frac{{{(1-\beta )}^{2}}}{24}\times \frac{{{\alpha }^{2}}}{{{f}^{2-2\beta }}}+\frac{1}{4}\frac{\rho \beta v\alpha }{{{f}^{1-\beta }}}+\frac{2-3{{\rho }^{2}}}{24}{{v}^{2}} \right]T \right)}{{{f}^{1-\beta }}}$$ Taking logs produces $$\ln {{\sigma }_{ATM}}\approx \ln \alpha -(1-\beta )\ln f$$ Edit for Gordon
In practice, the choice of $\beta$ has little effect on the resulting shape of the volatility curve produced by the SABR model, so the choice of is not crucial. The choice of $\beta$, however, can affect the Greeks. Barlett provides more accurate Greeks and shows that they are less sensitive to the choice of $\beta$.Indeed The case $\beta=0$ produces the stochastic normal model, $\beta=1$ produces the stochastic log-normal model, $\beta=\frac{1}{2}$ produces the stochastic CIR model.
one of my friend recently wrote about SABR model and calibration. I highly recommend you to read it to get your answers http://janroman.dhis.org/stud/EXJOBB/SABR.pdf
