Overvalued, undervalued options

[Kavita.bhangdia]

Hi David,
How do we find the shape on the implied vol curve based on the shape of distribution of underlying..

Thanks
Kavita

 

[David Harper CFA FRM]

Hi @Kavita.bhangdia Given your post title (overvalued, undervalued options) I am not entirely sure what you are asking me, are we synched on the underlying concept which is illustrated by Hull Fig 20.1 and Fig 20.2 (below)?. Typically, we infer the shape of the implied volatility smile from (i) an option pricing model (Black Scholes) and (i) the set of observed, traded option prices at various strike prices (the x-axis on Figure 20.1). As a pricing model, BSM returns call price, c, as function of volatility, σ, and the other inputs; c = f[σ, ...]. We can goal-seek to solve for implied vol as a function of the observed call price: σ(implied) = f[c, ...]. The implied vol simile is the function of σ(implied) at various strike prices. Our baseline is the price distribution assumed by BSM, the BSM model assumes prices are lognormal (because it assumes continuous returns are normal). If the implied volatility plot happens to be flat, we can infer the prices are lognormally distributed. If it exhibits the smile as shown on the left, the implied distribution has heavier tails than the lognormal.

To your question, say we observe a 1-year $100 ATM (S = K =100) put option trading at $6.936 while the Rf rate is 2.0%; its implied volality is 20.0%. I got that by plugging σ =20% into BSM and it returns a price of $6.936. If I lower the strike to 80, the associated put price is $0.959 and N(-d2) = 0.1323. This tells me that conditional on a lognormal price distribution there is a 13.23% probability the price will finish below $80 (imagine the left tail of Fig 20.2 above); ie, N(-d2) is the prob of a Euro put option being exercised. But let's say I fit an actual return distribution and I observe this CDF probability is actually 18.0%. Then I can goal-seek to find the volatility that produces N(-d2) equal to 0.18, which is about 23.5% (as we expect, higher than 20%). So I am still goal-seeking but against N(-d2) rather than the option price. The whole issue has a broad and deep literature history, for example, see the end of this CFA research monograph "Option-Implied Risk-Neutral Distributions and Risk Aversion" at http://trtl.bz/option-implied-distributions
I hope that's a helpful start!

 

[Kavita.bhangdia]

Yes actually I wanted to ask something about overvalued options, but then my daughter interrupted and then I completely lost that thread in my head and started this... Apologies for the confusion...

Actually in your problem set I think you have questions on what will. Be smile if distribution is lepto/Plato kurtotic etc

What I understand is that BSM assumes lognormal distribution where the smile would be a straight line. But because we see a smile/skew, it means that is it not lognormal but in stead has heavier tails...

How do we incorporate other shapes of distribution.??

Thanks
Kavita

 

[Mkaim]

Yes actually I wanted to ask something about overvalued options, but then my daughter interrupted and then I completely lost that thread in my head and started this... Apologies for the confusion...

Actually in your problem set I think you have questions on what will. Be smile if distribution is lepto/Plato kurtotic etc

What I understand is that BSM assumes lognormal distribution where the smile would be a straight line. But because we see a smile/skew, it means that is it not lognormal but in stead has heavier tails...

How do we incorporate other shapes of distribution.??

Thanks
Kavita

David gives a great presentation in the topic review video, and it addresses your question. The link is below, watch it from the 2:19 to 12:08 mark. If you still have issues/questions then perhaps you should consider reading Hull, chapter 17, it has only 10 pages. However, I must say that I like David's summary in this review video much better as he relates the end points of the smile/frown to the corresponding distributions. Also, keep in mind that if asset returns are normally distributed then asset prices are lognormally distributed. Hope this helps.

https://learn.bionicturtle.com/topic/market-risk-focus-review-video-1-of-2/

 

[ShaktiRathore]

Hi,
u can refer to these links if you like:
https://forum.bionicturtle.com/threads/volatility-smiles.5472/#post-18996
https://forum.bionicturtle.com/thre...ding-to-the-volatility-smile.7681/#post-28727

 

[David Harper CFA FRM]

Thank you @Mkaim and @ShaktiRathore

@Kavita.bhangdia Just as lognormal price property is inherent to the BSM, you would incorporate a different distribution by using a pricing model that reflects your desired distribution; or more likely, that reflects the assumptions you can life with, within a pricing model. That's the thing about implied volatility: it's really model-implied volatility. A flat implied vol curve tells you: we observe prices, across the strikes, that are totally consistent with the price distribution (or expected distribution) inherent to whatever model we use to price. As really all we are doing to retrieve implied volatility is transforming a market price into a model input, we can choose literally any model; e.g., GARCH. Although, from what i understand, the typical extensions here relate less to the lognormal price distribution and tend to remedy the BSM assumption that volatility is constant. This is the onerous assumption. John Hull and many others have developed stochastic volatility option pricing models. Thanks,