Black-Scholes Formula Is A BS, We Tell You Why
The valuation of options has always been standardized following formulas that treat all types of financial assets equally, regardless of the underlying. This means that the only metrics involved in their valuation (price) are the implied volatility, the strike, the risk-free interest rate, the current price of the underlying, as well as a stationarity variable (maturity or expiration date). This means that their valuation only assumes variables from the past, which do not necessarily influence the future (volatility and risk-free rate).
This has meant that historically this model has been heavily criticized:
As this type of derivative is priced, the price of the underlying would follow a Geometric Brownian Motion that could be synthesized into returns following a log-normal distribution.
What we have been able to verify is that reality does not exactly follow this type of distribution, but rather that the tails of this distribution are heavier (fat tails), which results in unlikely events (5–7 standard deviations) occurring more often than the model predicts.
In addition, financial option valuations do not include fundamental ratios derived from the underlying (% sales growth, Gross Margin, ROA, ROIC, EPS…). This means that two companies with the same implied volatility and no dividends have the same Price/Strike valuation for a given time to maturity, regardless of the underlying.
What we are going to try to see in this article is if this statement is really true or not. We are going to check the theoretical price calculated and the real price in periods of maximum market stress as was the case of the months of February and March 2020.
The underlying chosen is the ETF with the largest capitalization in the world $SPY, specifically the strike chosen will be 150 which is a very OTM strike and the expiration date 21JAN22*. The price of the contract over the last two years has been:
For the date 14Feb2020 we find that the real price is 0.66, while if we calculate by means of the model we find that the price is measured according to the distribution previously mentioned.
some_option = Option(european=False, kind=’put’, s0=337, k=150, t=685, sigma=0.288, r=0.0, dv=0) some_option.getPrice(method='BT',iteration = 10000)
However, when we checked the option price at one of the most critical times (16Mar2020), we found that the actual price stood at $12 per contract, while the theoretical price was:
some_option = Option(european=False, kind=’put’, s0=239.85, k=150, t=655, sigma=0.398, r=0.0, dv=0) some_option.getPrice(method='BT',iteration = 10000)
As we can see, the theoretical valuation is almost 15% below the real valuation.
In extreme market situations, the price of out-of-the-money options tends to grow above the theoretical range. Only one such event is necessary to achieve returns in excess of the long-term index.
It is difficult to sacrifice short-term returns for better long-term performance. But we must have a philosophy based on the ‘Roundabout Path’ which consists of getting rid of instant gratification in the short term, as most investors seek, for much greater gratification in the long term.
Our purpose is to convey to the individual investor, through our newsletter, simple strategies to have a covered, uncorrelated and asymmetric portfolio as possible so that any investor can implement this strategy for himself. One of the main strengths of this strategy is to be able to sleep peacefully each and every night and for this, we have between 40–60% of the portfolio invested in fixed income (other currencies, bonds, gold …) and the rest is invested in options or ETFs that make us benefit from any market situation
*Ideally, we would have liked a much closer expiration date, but we do not currently have the price history of expired option chains.
This article is for informational purposes only, it should not be considered Financial or Legal Advice. Not all information will be accurate. Consult a financial professional before making any major financial decisions.