Volatility Smile

[jcjc0602]

Hi David, I have two questions about the volatility smile?
1. Crashophobia. In Hull's book, it means that traders are concerned about the possibility of another crash similar to October 1987, and so increase the price of out of the money put (lower strike price), therefore increasing the implied volatility of it. If another crash happens, the put might be in the money. That makes sence. But the increase of the implied vol of OTM put will also increase the implied vol of the corresponding call. My question is, however, if traders are worried about that the crash can make a OTM put ITM, then increasing the price of that put, how would they price the corresponding call ? The crash will make the call become OTM and the price of that call shouldn't be increased, but the implied vol is increased because of the put-call parity. What's your opinion? Thanks

2. A company's stock is selling for $4 and without outstanding debt. Analysts consider the liquidation value of the company to be at least $300,000 and there are 100,000 shares outstanding. What volatility smile would you expect?-- another problem in Hull's book.
The solution says that the company's stock should be at least $3, then a thinner left tail and fatter right tail than lognormal distribution can be concluded. I don't know how can these to points be connected?

Thanks for your help!

 

[David Harper CFA FRM]

Hi jcjc0602,

question: where do you get Hull answers for the assignment questions (my solutions doesn't include those)?

I think your questions are great. Hull's explain of crashophobia/leverage have long baffled me; e.g.,
http://forum.bionicturtle.com/viewthread/2168/#4532
... i keep assuming i must misunderstand but yet nobody's been able to clarify.

1. Re crashophobia: I understand that this theory applies, just as you suggest, to OTM PUTs. I totally agree per put-call parity this implies that ITM calls (i.e., @ lower strike = OTM put/ITM calls) need to have higher implied volatilities. I have searched my library for explains/reconciliations but have found none... all refer to the OTM put

2. Well, first, I continue to find Hull's leverage (as opposed to crashophobia) rationale inconsistent. He says "we can expect the volatility of equity to be a decreasing function of price and is consistent with Fig 18.3 and 18.4." The figures show implied vol as a decreasing function of strike price and an INCREASING function of stock price; i.e., equity is a CALL option on firm assets, such that Fig 18.3 (IMO) suggest volatility is an increasing function of stock price (?!) ... unless you let the smile rise up on the right-hand side

But, your answer is consistent with a portion of his text and here is the only reconcile i can find:
If we imagine 18.3 extended to the right, we a true "smile" such that at high strike/low stock prices the curve bends up again. Then, for OTM call (ITM put), the right-hand SEGMENT is indeed "implied vol decreasing with higher stock price" or "increasing as equity is underwater or increasingly OTM call."

Going further, if we forget about crashophobia and go only on his text (and follow a Merton model), in regard to equity as a call option, we can imagine a "smirk" that is low implied volatility on the left (i.e., ITM call. equity has positive value) and high implied vol on the right (OTM call. Equty is negative). This smirk is consistent with "thin left tails [i.e., low implied vol per ITM call = equity positive] and heavy right tails [high implied vol per OTM call = equity negative]."
... it's true, your question starts with ITM call ($3 implies ITM. I am not sure the question is aware that $4 > $3 is a market vs book issue. Either implies ITM call)... i can't reconcile that aspect (essentially, I cannot perfectly connect the points either!)... however, part of my issue is: the smile/smirk is independent of the current incidental OTM/ITM, so i don't quite see how the assumptions impact the answer (i.e., the smile spans all scenarios. Put another way, if smirk is such and such, changing assumptions merely locates a different point on the same smirk, I'd think)

hope that helps, if only to basically agree with you! David

 

[David Harper CFA FRM]

.... you know, I am even further baffled by Hull's leverage argument on a simpler level: the firm's capital structure is independent of the option instrument's features. The firm can be highly leveraged or not, and in either case, OTM or ITM call options can be issued on the shares. The firm can be highly leveraged and we can still buy a deeply ITM call. It seems to me firm leverage speaks to a shift in the overall smile; e.g., high leverage implies upward shift ... now i can't connect firm capital structure with non-parallel smile shifts...yea, every single time i try to understand Hull's leverage argument, I come out worse

David

 

[jcjc0602]

Hi David, thanks for your response. Your answers give me a lot of thoughts.

I think the question baffling you is interesting and is not simpler. Did Hull's book mention smile shift? Could you explain it a liitle bit more? Thanks

 

[jcjc0602]

btw, if u need assignment solution, I can email you one.

 

[David Harper CFA FRM]

Hull doesn't mention a vol smile shift . Here is what he says about leverage (18.3 7th Ed):

"One possible explanation for the smile in equity options concerns leverage. As a company's equity declines in value, the company's leverage increases. This means that the equity becomes more risky and its volatility increases. As a company's equity increases in value, leverage decreases. The equity then becomes less risky and its volatility decreases. This argument shows that we can expect the volatility of equity to be a decreasing function of price and is consistent with Figures 18.3 and 18.4. Another explanation is "crashophobia" (see Business Snapshot 18.2)."

... equity is a call option. However, it seems to me that it's a fallacy to apply this to the smile b/c it concerns the company's capital structure independently of options issued on the equity: a highly leveraged company should have more volatile equity; but options issued on this volatile equity can be OTM/ATM/ITM

...re the assignment solutions: thanks but i'd prefer to my own (legal) copy, was curious who publishes?

David

 

[[email protected]]

Re: Crashophobia and Put/Call Parity

I think the below extract from the ft blog sheds some light on vol skew and why implied vols on calls of equal strikes to that of OTM puts are lower.

http://ftalphaville.ft.com/blog/2011/02/10/484911/skewed-the-cboes-new-fear-indicator/

"Skew is not unique to, but is most commonly observed in, equity options. Société Générale labels it “crashphobia”.

“Lab studies,” SG avers, “suggested that losses are twice as powerful emotionally and psychologically as gains. Reflecting this fear of loss, equity puts are consequently better bid than calls.”

More people want to buy defensive put options for strikes below the present price of an index or stock than want to buy bullish calls. This means the option writers can demand higher premiums for the former.

Plug those premiums into the standard option pricing model and it spits out a reading that the implied volatility of the out-of-the-money puts is higher than that of calls with strikes equally far above today’s price.

In other words, the dear pricing of these popular puts implies that the option writer is at greater risk of having to pay out on the puts than on the calls. The model reads that as saying the underlyings are more volatile on the downside than on the upside.

Let’s use some real numbers. Under the put-call parity theory, at-the-money calls and puts on the same instrument should have the same implied volatility.

For example, a March at-the-money call on the S&P at 1290 has implied vol of 14.7% – and so does the equivalent put.

But further from today’s price, divergence sets in. A 10% out-of-the-money call, on the S&P hitting 1420 in March, has implied vol of 15.0% – only slightly dearer than the at-the-money call. This is one side of the volatility smile.

The opposite put, on the S&P sinking to 1170 by March, has implied vol of 24.6% – a much steeper increase from the at-the-money, giving a skewed smile. Yet both contracts put the writer at risk of a payout if the S&P budges by 10%.

Quite something, eh? Equity option investors are a fearful lot.

Take the vol of the out-of-the-money call, 15%, and subtract it from the vol of the opposite put, 24.6%, and you get 9.6%. That’s today’s (January 25) three month skew on the S&P.

Now watch how that number changes day by day. If it rises, investors are growing more fearful. If it falls, the opposite is true.

 

[Rblc]

Here's the article in question if you have reached your view limit on FT