# What does it mean that model can reflect the ”volatility smile”

I know that implied volatility is the value for which the Black Scholes model returns the correct option price. I also know that if we plot the volatility on the strike price chart, we will see "smile". This means that the Black Scholes model is inappropriate for valuation because it assumes constant volatility. But what does it mean that another model (eg Kou model) can reflect the volatility smile? This model also assumed constant volatility. Does it mean that in the Kou model the graph of implied volatility is flatter, i.e. that the volatility is more constant with respect to strike and maturity?

• That's precisely what it means. The Kou (2002) model has constant instantaneous volatility but allows for (asymmetric) jumps in the asset price which increase tails of the distributions. Thus, OTM options can be priced more accurately (still not perfect) and the model thus endogenously produces a volatility smile. Stochastic volatility model are better in reproducing smile behaviour than finite active jump models. – Alex Aug 18 '20 at 12:26

A model that reflects the volatility smile is one with dynamics that approximate pricing that yields an implied volatility smile. However, your question makes me suspect you are fuzzy on some of these pieces, so let's go through this in more detail.

Implied Volatilities $$\implies$$ Correct Price?

You mention that implied volatility in the Black-Scholes model gives the "correct" price. That is a bit bold since we do not know the correct price. We might assume the correct price is determined solely by market prices or by some model, if you believe in possible inefficiencies. (Note that by the Grossman-Stiglitz argument, you should believe in inefficiencies for short periods of time).

Implied volatilities are just the volatilities that equate market prices and Black-Scholes prices (i.e. implied by the Black-Scholes model).

Smile or Smirk?

You also mention the volatility smile although that shape is not universal. Port-1987 in most equity markets, the "smile" has been more of a smirk: asymmetrical with much higher volatility for lower strike prices. For commodities, the smirk is much more pronounced with implied volatilities being much higher as the strike price increases.

Is Black-Scholes Inappropriate?

Does assuming constant volatility mean the Black-Scholes model is inappropriate for valuation? No. Black-Scholes pricing systematically diverging from market prices means the model is wrong, but "all models are wrong" as George Box famously pointed out. However, the Black-Scholes model is still useful -- and thus appropriate.

Why Black-Scholes Diverges from Market Pricing

The Black-Scholes and Merton models presume a partial equilibrium (no interaction between buyer and seller in setting prices) and limits for log-returns that converge to normality. That makes the math easier -- even though it disagrees with what we observe.

There are three forces that disagree with the Black-Scholes assumptions:

• We know that volatility is not constant across time. This is usually not a major factor, but it helps explain why we sometimes look at volatility surfaces.
• More important: we believe asset returns exhibit fat tails; the likelihood of unusual log-returns is higher than normality would suggest. That means out-of-the-money options are more likely to expire in-the-money than Black-Scholes suggests -- and thus are worth more than the Black-Scholes price. This is true even if we guessed the underlier volatility correctly. The market understands this and so the market price is higher. That leads to implied volatilities being higher for strike prices away from the current underlier price.
• Also crucial: investors dislike losses more than they like gains. This leads to investors being willing to pay more for protection against downside than they would pay for upside: put options are more expensive than even fat tails would suggest.

Put these together and implied volatilities being higher away from the current underlier price is because of fat tails and investors' preference for avoiding losses. If we infer these implied volatilities from puts and calls and then plot them by the strike prices of those puts and calls, we get a curve that is, indeed, higher as we get farther from (ATM strike prices, i.e. the current underlier price).

What Keeps Black-Scholes Appropriate?

What keeps the Black-Scholes model appropriate is the regular behavior of that volatility curve. A good model can be adjusted to make it better -- and the Black-Scholes model allows us to do exactly that. We can use higher implied volatilities for strike prices away from ATM to correct for fat tails and investors disliking losses more than they like gains.

How Can A Model Reflect the Volatility Curve?

Once you understand all of that, it's easy to see how a model may better reflect the volatility curve: it can allow for non-constant variance, fatter tails, and investors preference to reduce downside risk.

Does the Kou model reflect the volatility curve? It reflects it better, because it incorporates jumps (which effectively yield fatter tails). The Heston volatility model also has fatter tails and thus better reflects the volatility curve.

Could one do better than these models? Yes: also incorporating investors greater dislike of downside returns would be smart. Exponential-GARCH models accommodate this, but you would need to modify the Kou or Heston model to do likewise.

• "investors dislike losses more than they like losses." – Is that a typo for "investors dislike losses more than they like gains"? – Tanner Swett Aug 18 '20 at 21:24
• Whoops! Yes; thanks! – kurtosis Aug 18 '20 at 22:16