# Spot and Vol Correlation in Idealised Regimes of the Volatility Surface

From http://www.globalvolatilitysummit.com/wp-content/uploads/2015/10/Santander-Volatility-Trading-Primer-Part-II.pdf it states that there are the four idealised regimes of volatility surface.

1) Sticky delta (or sticky moneyness). Sticky delta assumes a constant volatility for options of the same strike as a percentage of spot. For example, ATM or 100% strike volatility has constant volatility. As this model implies there is a positive correlation between volatility and spot, the opposite of what is usually seen in the market, it is not a particularly realistic model (except over a very long time horizon).

(2) Sticky strike. A sticky strike volatility surface has a constant volatility for options with the same fixed currency strike. Sticky strike is usually thought of as a stable (or unmoving) volatility surface as real-life options (with a fixed currency strike) do not change their implied volatility.

(3) Sticky local volatility. Local volatility is the instantaneous volatility of stock at a certain stock price. When local volatility is static, implied volatility rises when markets fall (ie, there is a negative correlation between stock prices and volatility). Of all the four volatility regimes, it is arguably the most realistic and fairly prices skew.

(4) Jumpy volatility. We define a jumpy volatility regime as one in which there is an excessive jump in implied volatility for a given movement in spot. There is a very high negative correlation between spot and volatility. This regime usually occurs over a very short time horizon in panicked markets (or a crash).

The chapter then goes on to explain the four regimes in more detail, however, I am unclear as to how this can be used in a practical sense? If not, why did the author introduce these regimes? Moreover, it states that skew and the effects of remarking the volatility surface is very different in each regime (see pg 221). Therefore I would have thought it would be very hard to trade skew as you may have to change between the regimes?

In black-scholes world, correlation between volatility and spot is zero. From the above details you can estimate how the implied volatility for a given option (note options have FIXED strikes) might change for a given move in spot. If when stock goes up, the option's implied vol goes down, this would be a violation of the black-scholes model (which scenario 1/2/3/4 would the correlation be negative?). In particular, your black scholes delta will be biased. Given the option's vega, can you tell me in what way the delta would be biased?

specifically what is dBS(S, sigma(S))/dS ? where dsigma(s)/ds < 0

This might give you some intuition for how this information would be used in a practical sense.

• I am not sure how, given the options vega, would the delta be biased and I dont understand what the second from last line is about – Permian Mar 4 '17 at 19:55