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SABR model calibration

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Don’t use this!

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. It was developed by Patrick Hagan, Deep Kumar, Andrew Lesniewski, and Diana Woodward.

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 σ. SABR is a dynamic model in which both F and σ 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
Constant parameters should satisfy the condition 0 \leq \beta \leq 1, \alpha \geq 0
Here, W_t and Z_t are two correlated Wiener processes with correlation coefficient -1\leq\rho\leq 1. For simplicity sake, we assume that \rho = 1, therefore, we put Z_t = W_t:
dF_t = \sigma_t F_t^{\beta} dW_t
d\sigma_t = \alpha\sigma_t dW_t

Calibration by simple linear regression

By naive approach, in finite differences the SABR model looks:
F_t - F_{t-1} = \sigma_t F_{t-1}^{\beta} dW_{t-1}
\sigma_t = \sigma_{t-1} + \alpha\sigma_{t-1} dW_{t-1}
Substitution and rearranging parts yields:
F_t - F_{t-1} = \left(\sigma_{t-1} + \alpha\sigma_{t-1} dW_{t-1} \right) F_{t-1}^{\beta} dW_{t-1}
F_t - F_{t-1} = \left[ \sigma_{t-1} dW_{t-1} + \alpha\sigma_{t-1} dW^2_{t-1} \right] F_{t-1}^{\beta}
\ln \left( F_t - F_{t-1}\right) = \beta \ln F_{t-1} + \ln \left[ \left(\sigma_{t-1} + \alpha\sigma_{t-1}dW_{t-1}\right) dW_{t-1} \right]
Therefore, we can get it as y=ax+\ln b where b is an error term:
y = \ln \left( F_t - F_{t-1}\right)
x = \ln F_{t-1}
a = \beta
\ln b = \ln \left( \sigma_{t-1} + \alpha\sigma_{t-1} dW_{t-1} \right) + \ln dW_{t-1}
b = \left( \sigma_{t-1} + \alpha\sigma_{t-1} dW_{t-1} \right) dW_{t-1}
By renormalization of dW_t = \sqrt{\Delta t} W_t and W_t = N(0,1):
b =  \left( \sigma_{t-1} + \alpha\sigma_{t-1} \sqrt{\Delta t} W_{t-1} \right) \sqrt{\Delta t} W_{t-1}
Calculation of mean and variance of b gives us the following
\mu_b = \alpha\sigma \Delta t
\sigma^2_b = 2\alpha^2\sigma^2\Delta t^2 + \sigma^2\Delta t = 2\mu_b^2 + \sigma^2\Delta t
Solving the equation system yields:
\beta  = a
\sigma = \frac{\sigma^2_b-2\mu_b^2}{\Delta t}
\alpha = \frac{\mu_b}{\sigma^2_b-2\mu_b^2}

Simple linear regression without intercept term gives the following estimation of \beta:
\beta = \frac{\sum_{i=1}^N {\ln F_{t-1} \ln \left( F_t - F_{t-1}\right) } }{ \sum_{i=1}^N {\left(\ln F_{t-1}\right)^2} }
Mean and variance of error term \left( \ln \left( F_t - F_{t-1}\right) - \beta \ln F_{t-1} \right) yields estimates of \alpha and \sigma.

Open questions

  1. Bias of estimator
  2. Total least squares method might be more suitable for regression on random variables
  3. Experimental calibration on real world data

Update: The major assumption is that the price is only growing. It is here: \ln \left( F_t - F_{t-1}\right)


Written by fxpaul

June 17, 2011 at 09:27

Posted in trading math

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