Interpreting Rho in the Surface SVI Implied-Volatility Model

Summary

The document asks how to interpret the rho parameter in the Surface SVI formula when the model is used to fit implied volatility. It notes two descriptions found in papers: rho is proportional to the implied-volatility smile's slope near at-the-money, and rho is related to correlation between the underlying price and its instantaneous volatility.

The author is unsure how a correlation with instantaneous volatility applies when Surface SVI models an implied-volatility surface rather than a stochastic-volatility process. The text does not include an answer, derivation, or empirical evidence resolving that distinction. It raises a useful conceptual question about the parameter's interpretation, but does not establish when either description is appropriate or how to estimate the quantity.

Key ideas

• The Surface SVI formula includes a rho parameter constrained to the interval from minus one to one.
• The document reports that rho is described as proportional to the at-the-money smile slope.
• It also reports an interpretation of rho as correlation between the underlying price and instantaneous volatility.
• The document does not resolve how that correlation interpretation applies when fitting an implied-volatility surface.

Tags

# what does correlation $\rho$ means in surface SVI?


# what does correlation $\rho$ means in surface SVI?












Why does everyone say $\rho$ is correlation in Surface SVI?

$w = \frac{\theta_t}{2}(1+\rho\psi(\theta_t)k + \sqrt{(\psi(\theta_t)k+\rho)^2+1-\rho^2})$, with $\rho \in [-1,1]$

This paper says it is proportional to the slope of smile at the ATM point.

This paper says it is the correlation between the stock price and its instantaneous volatility. What is the instantaneous volatility? I am confused because I am using surface SVI to model the implied volatility instead of stochastic volatility.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.