Serving the Quantitative Finance Community

 
User avatar
Marsden
Topic Author
Posts: 215
Joined: August 20th, 2001, 5:42 pm
Location: Maryland

Gedankenexperiment

June 9th, 2022, 3:06 pm

Given a company with enterprise value (EV) of 100, a single debt payment of 50 due in one year, 3.00% riskless interest rate, and implied volatility (IV) of the EV of 25%, this gives a market value (MV) of 51.49 for the company (using Black-Scholes pricing and assumptions in addition to the information given).

Pricing European call options on the MV with expiration immediately after the the debt payment in one year can be done in the same manner as determining the MV, albeit using the debt payment plus the strike price of the option in place of just the debt payment.

This gives the following values for the options on the MV:

C(20): 32.61
C(30): 24.15
C(40): 16.97
C(50): 11.35
C(60): 7.26
C(70): 4.46
C(80): 2.66

Taking another pass -- and ignoring what we know -- we can calculate the IVs of these options using just the MV as the stock price:

C(20): 63.60%
C(30): 56.80%
C(40): 52.50%
C(50): 49.60%
C(60): 47.45%
C(70): 45.75%
C(80): 44.45%

(Smile if you notice anything interesting about this.)

Geske and Zhou (2007) looked at the effect of leverage on S&P 500 Index Put Options' volatility (hope the link works):

https://docs.google.com/viewer?url=http ... F08-07.pdf
 
User avatar
Marsden
Topic Author
Posts: 215
Joined: August 20th, 2001, 5:42 pm
Location: Maryland

Re: Gedankenexperiment

July 1st, 2022, 2:30 pm

The cumulative standard normal function of -6 is very nearly 10^-9, so a 6 standard deviation event is about a one in a billion occurrence.

Given a debt free company for which the volatility of its stock price is 25% and given equity risk premium (as defined by μ-r+½σ²) of 5%, a -6 standard deviation event for the return on the stock over one year will be priced as a -5.8 standard deviation event according to the Black-Scholes model. A -5.8 standard deviation is about a one in 300 million occurrence.

So -- using less rounded numbers now -- the Black-Scholes price for a one year binary put with strike price equivalent to a -6 standard deviation event on the stock price will be about 3.36 times its expected value (ignoring interest rates in all this; assume 0).

On its face, this is a gamble any casino would be happy to offer at the price if there was a demand for it. The face of it isn't the whole story, of course, because in an actual casino game they would expect to have a billion plays with only one of them paying off; with this binary option there would almost always be no pay off for a billion buyers ... but then one time in a billion there'd be a billion payoffs all at once.

Still, a taste of the market for which you could withstand the worst case would be attractive.
 
User avatar
bearish
Posts: 4811
Joined: February 3rd, 2011, 2:19 pm

Re: Gedankenexperiment

July 4th, 2022, 7:25 pm

This is hardly hypothetical. Models in this spirit have been around since the 70’s in the academic literature, and I have a former colleague who ran a significant corporate bond portfolio for a large Chicago based hedge fund based on somewhat richer versions of such models. Extracting implied parameters from distributional assumptions that are false by construction is not all that enlightening.
 
User avatar
Marsden
Topic Author
Posts: 215
Joined: August 20th, 2001, 5:42 pm
Location: Maryland

Re: Gedankenexperiment

July 4th, 2022, 9:24 pm

What I had hoped to demonstrate with my 01-Jul comment, bearish, was that even within the fishbowl of Black-Scholes assumptions, a form of arbitrage is permitted.

And maybe "arbitrage" is not the right term; "disequilibrium" might be better.

Per Black-Scholes prices, given the assumptions laid out, 3.36 time expected value should be the equilibrium price for the binary put option described: all market participants should be no more inclined to buy any such option than they are to sell it.

Is it believable?

And, of course even longer odds and even higher relative prices can be found farther out in the tails.

It's one thing to say that Black-Scholes prices don't model the real world accurately, and so we therefore can expect that blindly accepting them will lead to miss-pricing. And no, that is not all that enlightening.

But that's not what I'm saying.
 
User avatar
bearish
Posts: 4811
Joined: February 3rd, 2011, 2:19 pm

Re: Gedankenexperiment

July 4th, 2022, 10:28 pm

No, that’s neither arbitrage nor disequilibrium. It may offend the senses of an actuary, and may appear to be (and almost always will be) a good bet, but that’s a totally different story. Of course, as you start relaxing the model assumptions, you quickly get into a territory where the model is more realistic as well as useless. There are very real trade-offs here, and a lot of pretty smart people have thought about them for a long time. A surprising number of said people are also quite well off, although there is all sorts of selection biases embedded in that statement.
 
User avatar
Marsden
Topic Author
Posts: 215
Joined: August 20th, 2001, 5:42 pm
Location: Maryland

Re: Gedankenexperiment

July 5th, 2022, 12:07 pm

A surprising number of said people are also quite well off, although there is all sorts of selection biases embedded in that statement.

Surprising for what reason? I speculate that a majority of the people who did well are following variations of the same strategy ... although I also speculate that if you sort through the rubble of those who did not do well, you will find a lot of them also following variations of the same strategy, just with unfortunate timing and possibly with not-very-good risk controls.

And then, all of what we do with our models and equations is based on rational pricing considerations, without going too much into what that really means. But there are irrational actors in the market as well, hopefully for their own sakes not too many, but with the advent of trading platforms like Robin Hood probably more than ever before.

So there will always be people who play state lotteries, despite the fact that the expected return is about 50 cents on the dollar; what they're buying, really, is the happy feeling between the time they buy their ticket and the time it expires worthlessly. A friend of mine once told me that if he won the grand prize from Publisher's Clearing House (for non-Americans, this is -- or maybe, was -- a heavily advertised free lottery scheme that tried to get you to buy magazine subscriptions that you didn't really want) AND two of the lower prizes of new cars, he'd give me one of the cars! I think this was shortly before he decided that totaling the car he had and that he couldn't really afford was one of the luckiest things that had ever happened to him ...

And there will always -- I guess -- be people who buy the warranties on cheap appliances, the ones offered at check out that make me shake my head in disbelief and ask, "How much???!" I'm not sure what these people are really buying, and for the record I think that buying a protective put probably almost always makes more sense than paying $8 for a warranty on a $40 appliance. But I guess, as with the Nigerian princes who can help me recover the fortune my distant uncle left me if I just forward some of my banking information to them, someone must be biting because they keep dropping their lines in the water.

Anyway, as casinos know, and as insurance companies know, there are pretty strong markets for things that casinos and insurance companies wouldn't sell if they thought they were clearly worth every bit of the price paid.

And even in financial markets, I think there is sometimes room to be "the house" rather than the individual gambler walking in with high hopes and a full wallet. The question I see is, will the house's edge be closer to the 50% of state lotteries or the scant few percent of most casino games?

In any case, that's what a simple actuary sees.

;-)
 
User avatar
katastrofa
Posts: 6913
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

Re: Gedankenexperiment

July 7th, 2022, 11:16 pm

I don’t believe in strategies (not to confuse with mathematics). I believe in opportunities (-;

I’m actually one of those people who buy extended warrranties/insurances on some products ( saved me a lot of money on electric toothbrushes!). A high risk averse investor must think in terms of worst case electric toothbrush scenarios as me. I guess another ocasión when protective puts become popular are unorthodox investors who need to consider the counterparts risk profile.

You’re an actuary! Explains why I don’t get you (-: (all actuaries and accountants I met felt like a distinct kind of people, not that it stopped me from liking or simply respecting them).
 
User avatar
Marsden
Topic Author
Posts: 215
Joined: August 20th, 2001, 5:42 pm
Location: Maryland

Re: Gedankenexperiment

July 8th, 2022, 1:09 pm

I submit that the world doesn't work that way, though.

Exhibit One is Hurricane Katrina, which leveled New Orleans in 2005.

For decades it had been noted that a major hurricane hitting New Orleans was one of the top two natural disaster threats to America (the other is a major earthquake hitting San Francisco), and that several billion dollars should be spent on flood control to limit the damage and loss of life ... but every year for decades it was decided that the hurricane was unlikely to hit THAT year, so we could probably spend the money on something else with immediate benefit, and not be sorry that we did.

And for decades that was a successful strategy, right up until it wasn't.

Is ignoring low frequency, high severity events rational?

Meh.

But it seems to be how the world works.
 
User avatar
katastrofa
Posts: 6913
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

Re: Gedankenexperiment

July 11th, 2022, 10:33 pm

"ignoring low frequency, high severity events" - if they only had a frequency! unless you mean it's given by the spacings between nontrivial zeros of the Rieman zeta function :-D (vide Polya and Hilbert's work).

Seriously - or not, I don't believe in rationality as discussed here either. No, I'm not a postmodernist. Rather there are pragmatic methodological approach to that: agents in the system have individual utility functions/rationalities, which can be mutually irrational... Many examples in Nature (or computer simulations) suggest that a population wouldn't survive without this kind of diversity (e.g. breeding in birds). That's why I can imagine many reasons why New Orleanians chose the short-term solutions, even if post mortem suggests they were all wrong.
When it comes to rare large events, for practical purposes I'd define them as the events which leave you scr*d no matter what precautions you took, "rational" or not. Take Fukushima: it was well secured against earthquakes - which aren't rare in Japan, but the Tohoku 2011 was exceltionaly strong and yet the construction withstood it. One could argue that it even withstood the tsunami as the safety systems kicked in as designed. The catastrophe ultimately happened because the whole surrounding area was destroyed by the elements, and it was impossible to reach the reactors on time with proper equipment (batteries for cooling them afair). Maybe all such things seem to have been possible to predict once they happen, but too much imagination might have paralysing effects - we can speak of similarly dangerous ubiquitous large idleness.
Anyway, maybe it's right to accept that under certain circumstances it's only rational to pray =^O.O^=~
 
User avatar
Marsden
Topic Author
Posts: 215
Joined: August 20th, 2001, 5:42 pm
Location: Maryland

Re: Gedankenexperiment

July 12th, 2022, 2:33 pm

Hey Kat.

I don't think that diversity of risk tolerance really matters, for the reason that, in a micro sense, everyone is a price-taker whose risk tolerance is reflected in his investment profile rather than anything else. So if you think that losing 10% of your wealth is completely unacceptable, you essentially load up on put options that prevent that, regardless of price. But you don't really affect the price.

In a macro sense, the entire market acts like it has a risk tolerance of one kind or another, and this DOES affect prices. The question is, what is the dominant risk tolerance that effectively sets market prices? Now, in general it could be just about anything; it could defy any of our notions of reasonable.

But, building upon your evolution example, a thought experiment. Suppose we accept that wealth has three different "races:" "dreamers," who bet most heavily on the best outcomes to the detriment of everything else (TTDOEE); "planners," who bet most heavily on the most likely outcomes TTDOEE; and "hoarders," who bet most heavily on the worst outcomes TTDOEE.

So, which "race" can we expect to dominate (and not that I have not addressed differences in perceptions of probability; let's assume those are uniform and the only thing that affects distribution of investment is risk tolerance)?

The first thing to be aware of is that, unlike most biological natural selection, whatever "race" of wealth dominates, that race is at a disadvantage to the others in accruing more wealth and expanding: if most of the market is making the same bets as you, it raises the price and lowers the expected return on those bets.

So in a market where "hoarders" dominate, "planners" and "dreamers" will flourish, relatively; and likewise if things are arranged otherwise.

(Of course this ignores changing risk tolerance based on experience.)

Anyway, can we expect that there will be a point of stability -- a "strange attractor," maybe -- in this system?
 
User avatar
katastrofa
Posts: 6913
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

Re: Gedankenexperiment

July 12th, 2022, 7:24 pm

Sounds like a prey-predator model. Stable points are either extinction or oscillations.
I think I meant that risk aversion is a derivative of a complex individual utility function - but the context is changing too quickly for me (-: