### The word subprime itself denotes the likelihood of default. And of course this likelihood threatened, and then became certain for the Lehman Brothers

By Colls Ndlovu

Jointly developed by Fischer Black, Myron Scholes and Robert Merton, the Black-Scholes option pricing model describes how the price of a ﬁnancial derivative changes over time. It enables investors to accurately price a derivative and manage risk around such a derivative.

It is influenced by factors like time to maturity, volatility, forces of demand and supply as well as the credit risk profile of the issuer of such a derivative, inter alia.

The effect of this model has been the gargantuan growth of the derivative markets with an ever increasing array of newer and more complex products.

This has come with leaps of economic prosperity occasioned by slumps at certain intervals such as was witnessed in 2008 and is being experienced right now during Covid-19.

Derivatives have been a leading source of growth within the financial industry. By definition, a derivative is a financial instrument which derives its value from the value of an underlying asset.

Derivatives have been described as being not money, nor investments in stocks and shares, but as being investments in investments. That is to say, promises about promises. Typically, derivative traders use virtual money, that is, numbers in a computer.

The repercussions of derivative transactions ultimately result in either real profits or losses being realized at some point down the transaction chain. Real people then do incur real losses in real life.

That is the tragedy of derivatives. Within derivative transactions, things can look honky dory for a long time until some tragedy strikes and then that’s when reality hits the guts. When this happens, there are real debts that have to be repaid with real money.

Lehman Brothers collapse

There are collateral positions that have to be covered with actual tangible assets. Such a hard reality on the ground is what triggered the dramatic episode of 2008 that led to the contagious collapse of Lehman Brothers.

Fueled by low interest rates, enormous bonus payments, all this encouraged bankers to take ever larger risks on derivatives and eventually the whole matrix collapsed leaving its adherents in shocked disbelief.

The sheer complexity and opacity of the derivatives upon derivatives type of investments made the whole sector perplexingly difficult to decipher and comprehend. The more it became complex the more it became incomprehensible.

Likely to default debtors came into the scene and these were called the subprime mortgage borrowers. The word subprime itself denotes the likelihood of default. And of course this likelihood became certain to default in 2008.

The system seemed to be going until hard questions started being asked as to what the real value of such investments was.

That’s when things took a nasty twist for the worse. The fallout climaxed with the spectacular and unprecedented fall of Lehman Brothers.

When Lehman Brothers collapsed, it went with all manner of confidence. The global financial system was left writhing in agony.

Massive bailouts through the euphemistically named quantitative easing were dished out to all manner of financial institutions and shadow banks in fear of the impending financial Armageddon.

At the core of most derivative transactions was a mathematical equation called the Black-Scholes model. Simpler and straight forward derivatives like options and futures (especially commodities futures) are not really new financial products because they date back to the 1600s.

Smoke and mirrors

But as newer and more complex forms of derivatives arrived, traders saw an opportunity to turn smokes and mirrors into hard cash. Initially such trades seemed benign and innocuous. However, with time greed and avarice took their toll.

In particular, post-2000 the world’s ﬁnancial sector began to invent ever more complex variants of derivatives riding on the euphoria of free enterprise and less regulation powered by the greedy appetite of traders.

Derivatives became delinked from the underlying assets that underwrote the traditional derivatives. Derivatives started to be based on other derivatives instead of real assets. Banks were buying and selling bets on bets.

The hard reality of all this is that derivatives trades almost entirely on money that does not actually exist, that is to say, virtual money in a computer with no link to tangible assets.

The Black-Scholes formula has its origin in 1973 when it was introduced by Fischer Black and Myron Scholes while Robert Merton prepared the underlying mathematics.

The equation entails time, price, riskfree interest rate, volatility, inter alia. It expresses the rate of change of the price of the derivative, with respect to time, as a linear combination of three terms: “the price of the derivative itself, how fast that changes relative to the stock price, and how that change accelerates”.

A number of assumptions were made in order to arrive at the Black–Scholes equation (no transaction costs, no limits on short-selling, and that it is possible to lend and borrow money at a known, ﬁxed, risk-free interest rate).

Standard deviation

A closer look reveals that the Black-Scholes equation traces its origin to Bachelier’s thesis which assumes that market prices behave statistically like Brownian motion, in which both the rate of drift and the market volatility are constant.

Some sort of a constant random stochastic movement. It hinges on the movement of the mean with volatility simply denoting standard deviation, that is to say, a measure of average divergence from the mean.

Typically, there are two types of options: a put and a call option. A put option has the buyer of the option purchasing the right to sell a commodity or a ﬁnancial instrument at a speciﬁed time for an agreed price. A call option confers the right to buy instead of sell at a specified time for an agreed price.

The Black–Scholes equation has explicit solutions for put and call options.

The Black-Scholes formulas enable traders and investors to calculate the prices of the options. It brings a degree of rationality to the derivative market under normal market conditions.

It enables investors to calculate the value of options anywhere along the maturity curve.

The formula standardizes the calculation of the option prices. It provides a degree of both rationality and certainty insofar as pricing is concerned. But that ends with normal market conditions. Prof Scholes was awarded the Nobel prize together with Prof Robert Merton for it (Prof Black had already died by then).

Initially, the equation was applied conservatively. But with time as confidence grew in the market, caution was thrown to the wind. The possibility of the model going bananas was inconceivable.

Dichotomous Investment

It was regarded as a recipe for making everything turn to gold. A hedge fund called LTCM was the first casualty. LTCM had a dichotomous investment strategy that was designed to protect investors when the market went down or up.

Its strategy was based on mathematical models, including the Black–Scholes equation inclusive of arbitrage techniques which exploit price disparities of bonds and the value that can actually be realised. At inception, LTCM generated returns in excess of 40% up to 1998.

Thereafter, it lost US$4.6 billion within 4 months. LTCM nearly collapsed and was rescued by the US Federal Reserve which persuaded its major creditors to bail it out to the tune of US$4 billion.

In 2000, LTCM was wound up. Credit default swaps and collateralised debt obligations have been sigled out as being the main causes of toxicity within the financial industry.

While a credit default swap has been defined as some form of financial insurance within the derivatives markets collateralised debt obligation is based on a collection of assets, usually mortgage backed securities.

These two derivatives were and are traded by banks and related speculators while their pricing is done using the Black–Scholes equation.

The central weakness of the Black-Scholes equation lies in its assumptions that do not stand the test of reality. As they say, a hypothesis remains just that: a hypothesis. This miscalculation of risk remained the Achilles heel of the Black-Scholes equation and ultimately of the financial industry up until hell broke loose in September 2008.

Financial voodoo

It was therefore no wonder that an equation that belonged to the realm of classical continuum mathematics, rooted in differential equations of mathematical physics, brought about the biggest financial collapse in history in 2008.

The differential equations in mathematics have infinity divisibility, time ﬂows continuously, and variables change smoothly.

This works in mathematical physics but not in finance where human behavior rears its head. Some of the weaknesses of the Black–Scholes equation arise from its reliance on traditional economics assumptions of perfect information, perfect rationality, market equilibrium, the law of supply and demand.

These assumptions are taken as apriori axioms and are never queried by orthodox economists. Of course complexity and opacity also played a big role in obfuscating derivatives and making them incomprehensible and hence difficult to manage.

The too-big-to-fail mantra also allowed big banks to take undue risks.

Whether the Black-Scholes equation contributed to crisis or not is neither here nor there. To the extent that the equation contributed to the crash, that could be so only because it was abused by those employing it.

If a trader loses money, his computer is never blamed.

The same affliction should apply to the Black-Scholes equation. If anything, the financial sector needs more, robust and better mathematical models to use mathematics intelligently, rather than as some financial voodoo.

*Colls Ndlovu, a currency expert, is an award-winning economist and central banker, and is the inventor of the NCX Currency Index. He can be contacted on collsndlovu@gmail.com*

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