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I am working on some Black-Scholes stuff and currently investigating implied volatility (IV). I understand that the typical volatility smile can be viewed as a criticism of the assumption about constant volatility. However, I want to also look at this phenomenon from the view of the Black-Scholes model.

So my questions is as following:

  1. How exactly can the case of $\sigma^{IV}>\sigma^{RV}$ where consequently the market price being higher than the real price, i.e. $C(\sigma^{IV})>C(\sigma^{RV})$, be seen as the market being risk averse or pricing in extreme events? I think this question is due to some lack of intuition...

  2. Why do we see that typically ATM options have lower IV than OTM/ATM - this means implicitly, that we value OTM/ITM options higher?

Best regards

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    $\begingroup$ Implied volatility exceeds realised volatility is due to the variance risk premium. The relationship between moneyness and option prices is known as volatility smile. $\endgroup$
    – Kevin
    Jun 3 at 19:40
  • $\begingroup$ Can you provide som intuition/explanation about why we value OTM/ITM options higher than ATM cf. the volatility smile? $\endgroup$
    – Sebastian
    Jun 3 at 20:11
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    $\begingroup$ Think of put OTM put options. Black and Scholes suggest that this option doesn't have much value. The market pays a lot more for these options. Thus, you need a high implied volatility (the number that converts option prices to volatility quotes). Why does the market believe OTM puts are more valuable? Fat tails, skewness, heteroscedasticity, rare crashes (jumps), ... these are all features that the Black Scholes formula ignores, but the market incorporates. These factors increase the market price of OTM puts and therefore translate to high implied volatilities $\endgroup$
    – Kevin
    Jun 3 at 21:41
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I do not think you can look at this without the BS model in any case. For example in equity markets where listed options are liquid and commonly traded, BS is essentially used in every vol surface construction. All methods like SVI, mixed lognormal etc rely on BS. This is because there is no continuum of option prices. The finite number of strikes and maturities available means you can only observe a set number of points. In order to get an actual vol surface you need to interpolate/extrapolate between these observed points.

In my experience, if you read (economics) papers on that subject, you get more or less obscure explanations and no consensus. Option-Implied Risk-Neutral Distributions and Risk Aversion is a decent summary.

Ignoring mathematical rigor and economically sound explanations, you can actually explain this very intuitively:

1 ) Why is RV the real price? If you have car insurance, do you expect them to charge exactly what expected loss they have or do you expect insurance companies to want to make a living from offering this service? Neither banks nor insurance companies lack pompous buildings, implying they actually earn (a lot of) money with what they do. The risk aversion argument is generally debatable as it should result in only one side of the vol surface being higher than ATM, which may arguably apply to some equity (indices). However, that will not really concern ATM but (deep) OTM puts (left hand side of vol surface). See point 2)

Side remark 1: Variance risk premium (as shown in the wikipedia link provided in the comment above) is NOT exactly the same as vol risk premia. Although, research papers frequently call the difference between current implied volatility, and (recent) historic realized price volatility the variance risk premium. One of the weird things just like in many areas of mathematics, log means the natural logarithm and the ln notation is seldom seen.

2 ) BS assumes that the underlying follows geometric Brownian motion; exhibits lognormal distribution of returns (meaning log returns are normally distributed). Now, they unfortunately are not normal in real life. FX offers the easiest way to explain what vol smiles actually do in my opinion. FX is primarily vol quoted (no liquid exchange trading and even the CME offers vol quoted options).

You can can read P.409 chapter 19 “OPTIONS, FUTURES, AND OTHER DERIVATIVES - John C. Hull: 8th edition” Side remark, I switched the notation to match FX (Hull uses Black Scholes for equity).

Put-call parity states $$𝑝 + 𝑆 * e^{-r_{ccy1}*t} = 𝑐 + 𝐾*e^{-r_{ccy2}*t}$$ holds for market prices (pmkt and cmkt) and for Garman Kohlhagen prices (pbs and cbs) As a result, pmkt− pbs =cmkt− cbs.

When pbs = pmkt, it must be true that cbs = cmkt.

It follows that the implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity. Now the same strike means one is ITM, the other is OTM - thus the volatility smile for European call options should be exactly the same as that for European put options.

In a nutshell, FX options are quoted in At-the-money Delta neutral straddles (ATM DNS), as well as Risk Reversals (RR) and Butterflies (BF) for varying delta levels.

  • ATM determines the level,
  • RR the skew (how its tilted, towards OTM calls in the example below) and
  • BF the kurtosis (how pronounced the general wings are).

Using a somewhat simplified method from Malz, one can quickly demonstrate this with a few lines of code in Julia.

If you only have ATM quotes, you are kind of in the "Black Scholes" world where vol is known and constant. enter image description here

RR determines the skew. enter image description here

BF the kurtosis. enter image description here

and combined you get the full vol surface. enter image description here

Notation wise, these charts are a bit sloppy. I just used delta in terms of put, which is why it is from 0 to 1. However, as explained above, 10 delta put (10DP) is equal to 90 delta call (90DC). 10DP = 90DC. It's customary to use OTM quotes only as these are the main options of interest. 50 delta is roughly ATM.

Now, if there is a markup for additional risk aversion is a separate question. Probably yes. Driven by demand from option buyers.

Side remark 2) The reason Variance Swap rates are higher than realized variance is also demand and supply as well as market makers trying to earn a living with the product. Also, due to practical difficulties in replicating the actual log payout across strikes, (no continuum of strikes/ option prices) , the market for equity index varswaps usually trades at a basis to the replicating portfolio, which is generally already based on the ivol surface. Hence a double markup so to say.

Lastly, "this means implicitly, that we value OTM/ITM options higher?" may be a misunderstanding. (Deep) OTM is always cheaper compared to ATM in the sense that you will pay less for them (reflecting the fact that the strike is such that you will be less likely to exercise). However, higher IVOL will make them relatively more expensive. A bit like a single pint is cheaper than a keg, but the sum of pints will be more expensive than the barrel itself. In option terminology, if you were to use the same vol from (deep) OTM options for ATM options, they would be more expensive in absolute terms. You can see some actual quotes where I "stole" the FX vol surface explanation from.

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