How Return–Volatility Correlation Shapes Implied Volatility

# How inaccurate is the BSIV as compared to the Heston IV due to unaccounted correlation?


# How inaccurate is the BSIV as compared to the Heston IV due to unaccounted correlation?












To provide more color to the question, after reading Christoffersen et al. (2013), I found out recently that the Heston model provides a more accurate future realized volatility estimate in the form of ATM implied volatility (as compared to the Black-Scholes implied volatility) in the case of correlation between the innovations in returns and volatility existing i.e. $dz_1$ and $dz_2$. The correlation $\rho$ I am talking about is:

$$dS=rSdt+\sqrt{V}Sdz_1$$

$$dV=\kappa(\theta-V)dt+\sigma_V\sqrt{V}dz_2$$

$$\rho dt=dz_1 dz_2$$

Thus, how much of a correlation exists in reality? Is it negative or positive? Is the magnitude sufficiently large that the BSIV is not that reliable of an estimate of future realized volatility? (as compared to the Heston IV).

General opinions or papers references are always welcome.

## Answer by Mats Lind (score 3, accepted)

https://quant.stackexchange.com/a/81433

The correlation between vola innovations and returns is negative for indices as for instance this study shows. Stocks with options that do not display a volatility skew, which should be expected for low-beta stocks, have a correlation near zero. OP:s 2013 reference has it that correlation is sufficiently large so yes, Heston should accordingly be better than BS to give an implied volatility estimate more in line with future realized volatility for the underlying studied there. Remember though that IV for indices ususally is higher than RV reflecting a vola-premium.

## Answer by Frido (score 3)

https://quant.stackexchange.com/a/81435

There are a few things in your question which confuse me, but perhaps by pointing out my confusion your question is answered.

First of all the title: comparing "the Heston IV" to "the BS IV" does not make sense to me.

There is no such thing as the Heston IV; the Heston option price takes as input a single instantaneous volatility. This instantaneous vol is never called IV.

Next, there is no single BS IV. The BS IVs are the IVs you observe in the market. As there is a skew/smile/smirk, there is not one unique IV but one IV for each strike.

It is possible that the BS IVs are generated by the Heston model, but other processes, not just Heston, can generate the observed skew

Perhaps what you mean is the BS volatility parameter as it was originally intended to be used in the BS model. Unlike the stochastic spot vol in the Heston model the BS volatility was originally a deterministic volatility which at most can generate a term structure (not a skew).

If your question is then how inaccurate is the BS model constant / deterministic spot vol compared to the Heston stochastic spot vol, then the answer is very inaccurate for two reasons: 1. the BS model does not take stochastic spot vol into account and 2. it does not take correlation into account. (Among others.) These 2 (stoch spot vol and correlation) lead to the skew (neg/pos slope) and curvature of market IVs.

In equities correlation is generally negative, in volatility it is usually positive, in FX it can be both.