I would like someone to discuss the following hypothesis:
The Black-Scholes formula is not a valid optionpricing model.
When backtesting S&P stock options using B&S and the real volatility (= standard deviation) ex post the costs exceed the payoffs by 4 percent, using the VIX (=Volatility of the S&P500) by 26 percent.
The method is: buy fictitious call options day by day over 15 years at the money and at the price of fair value - compare the sum with the cumulated payoffs. The rationale:
The payoffs should somehow match the amounted procurement costs at least.
(For puts it's even worse - 18 / 46 percent overpricing.)
Any comment appreciated.
It's not "expected" payoffs, my fictitious options are exercised - ex post by backtesting.
One would pay for illusion only as long as there is hope.
The seller doesn't need to consider individual risk of an option:
He can compensate call and put risks.
Straddle selling is no risk.
Of course, individual hedging is costly.
Do we share, Joakim, the view that B&S is a slightly(?) overpricing formula?
I haven't see any paper yet coming to that conclusion. I think the formula is inadequate.
And: Selling straddles is no risk - buying is. No need for hedging/protection.
And: there is no historical quotation (stock, index, exchante rate) until now with returns distributed normally: leptocurciv, skew, fat tails.
To re-fame the flame we could look back: W. Stuetzel* (1925 - 1987) postulated that economic goods don't have a genuine (or fix) value. Values would rather depend on the situation of someone evaluating that good and his relation to it, would be "relative". W. Engels (1933 - 1995) spoke of "fields of determination". Today we prefer (perfect, efficient...) markets where prices are established by demand and supply. Or the Arrow-Debreu model.
The B&S-formula to determine "fair values" uses market data of the underlying (course, interest rate, volatility). However, there is no rationale why the volatility might influence the option value: The swell doesn't change the depth of the sea. And there is a kind of flip-flop to "compute" the price with an implied volatility computed with the same formula reversely.
*Spätestens Wolfgang Stützel postulierte, dass es den „wahren“ Wert nicht gäbe, sondern "der Wert immer von der Situation des Bewertenden und von dessen Verhältnis zu dem Vermögensgegenstand abhängig ist."
Dazu Wolfram Engels ist „der Wert eines ... Wirtschaftsgutes immer relativ und hängt grundsätzlich von dem Entscheidungsfeld ab, in dem bewertet wurde.“
George, nothing against simplified models. However, when a model is based on equilibrium like the B&S but reality isn't it should be not of practical use.
Accounting standards IFRS 9+13 allow B&S in a vague reference "models like B&S"... allowing "creative accounting".
I am nut sure how the US accg standards FASB formulate the corresponding matter.
George, I red the Rubén Figueroa, Maria do Rosário Grossinho paper [0] including ref's [1] and [6]. There is a small intersection only. I am not primarily focussing on tx-costs.
May basic approach is to (try a) proof the B&S model experimentally: Bying 15 years, day by day options on a broad portfolio like the S&P 500 at fair value (B&S) and at-the-money the sum should match the amount of payoffs of he options when exercised. Otherwise fair values would be fancy prices. Thus far it’s pure economic reasoning.
The findings are the sum of call prices is 3% higher than the total payoffs when using the real volatility ex post and 30% when using the S&P volatility index VIX (for puts 19% rsp. 46%).
Another experiment would be to model premature executing/selling to cover the case that an option would have more (or at least some) value (say more than x $) during leadtime instead of final execution time. I am working on that.
The ref papers [1] and [6] with the idea of upper and lower boundaries seem similar to the value-at-risk. However, volatility is not considered at all.
I think still the B&S model does’nt reflect reality and VIX is not predictive.
Added a working paper that contains more detailed experiments applying the BS-formula to S&P and VIX data. 2 questions:
o Does the revers application of B&S induce a systematic error with a tendency to increase the index volatility?
o The payoffs of options on a broad portfolio like S&P don't match their fair value at emission time. Is the B&S formula still applicable - even for financial reports and balance sheets?
It is very difficult to accurately estimate the standard deviation and hence your problem.
B-S model is a very robust model and provides solution many varieties of problems.
You should work on estimating the standard deviation of the underlying asset to test your assertion.
Option traders are not using the BS-model to price options, they use the model to calculate implied volatilities, and these numbers are used to trade in calls and puts. I strongly advise you to spend some time on a trading floor, that helps in clarifying how the BS-model is really used! And, as a first start, the BS-model as a pricing instrument, if you really would like to price with it, for instance an over the counter product, is not a totally bad idea, as the model is quite robust. Just read for instance work by Mark Davis!
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