The following article was published in Financial Times on December 10, 2007 and written by Pablo Triana, Director of the Centre for Advanced Finance at IE. If you wish to learn more about our finance related programs, please visit http://www.miaf.ie.edu/.
The recent debate on the Black-Scholes-Merton options pricing model continues to make waves. Research by veteran option traders and authors Nassim Taleb and Espen Haug claims that BSM was not really an original invention, that its real-world popularity has been greatly exaggerated, and that there may be no mathematical models behind option prices, with simple supply-demand interaction claiming authorship instead.
Such bold statements, while a welcome contribution to the search for truth, may deprive us of something that seemed valuable. BSM, for all its flaws, offered quantifiable light where before there was only unknown darkness. By “losing” BSM, we would lose such certainty, albeit misplaced.
The most obvious certainty that the model offered was a precise number for the value of an option. This it did in a quite ingenious way. Everyone would tell you that an option should cost money because it gives you the right to enjoy a potentially large pay-out while limiting the possible losses if things do not go your way. But who ensures that that right has value per se? That value is supposed to come from the probability assigned to the option expiring in-the-money. So who guarantees that the probability of making money on the option is non-zero? Who can honestly claim to know the exact distribution governing financial assets? Pricing options based simply on probabilistic assumptions sounds a bit fishy.
What BSM innovatively did was price options through non-arbitrage arguments. It showed that (mathematically, at least) dealers could build a replicating portfolio of the underlying asset and borrowed money that would always mirror the value of the option. Non-arbitrage conditions guaranteed that the value of the option at any point should equal the value of the portfolio. Since the whole thing becomes riskless, we could assume a risk-free rate of discount, thus avoiding the need to guess the expected return of the underlying asset – a devilishly difficult task.
Providing a good rationale for why options should cost money and how much was not the only valuable quantitative contribution of BSM. The model also helped produce hard numbers for the risk parameters of options, the factors that can change the value of an option, the famous “Greeks”. These Greeks would tell you exactly by how much the value of the option would be modified if the underlying asset moved (“delta”), if the volatility moved (“vega”), if time to maturity moved (“theta”), or if the underlying asset jumped (“gamma”). This is highly valuable information for any options dealer or speculator, as the Greeks could now determine with Swiss-like precision your mark-to-market exposure. BSM provided option dealers with a compass to allow them to risk-manage their options business and, in principle, help them confidently expand to include new clients and new areas.
In an ideal world, BSM would have thus been a sensational contribution, worthy of the admiration of any derivatives pro. Unfortunately, the world is not ideal, or rather BSM is not an ideal fit for the real world. The model itself relies on assumptions regarding the market’s probability distribution that are plainly wrong (financial markets are not “normal”). More worrisome, the “dynamic hedging” behind the replication strategy is neither possible nor desirable because liquidity and transaction cost issues make the required continuous replicating impractical. Delta hedging may be a mathematical beauty, but it is not an implementable tool.
Obviously if there is no model (ie if option prices come from supply-demand or from static, not dynamic, replication) then there are no Greeks. Delta, gamma, vega, and theta are 100 per cent model-dependent; they have to be derived from a formula. There are no Greeks in supply-demand option pricing, so there are no precise numbers to guide dealers into how to risk-manage their positions.
Herein lies the tragedy. Quantitative finance produces mathematically beautiful devices that promise to tackle critical problems. Unfortunately, real life has a habit of relentlessly rendering such numerical applications useless. It would be great to be able to count on the certainties of BSM, but chaos-prone markets stubbornly refuse to allow their unpredictability and untameability to be ironed out. Perhaps the only certainty about the markets is that financial truth will always be determined by humans, however lacking in rigour, and not by precision-promising, nicely designed theoretical masterpieces like BSM.
To view the original article, please vist the FT website.