# When The Market Comes to the Model

by R. Passov

Modeling in finance is done through the lens of mathematics. To put something into a model where you are not guided by observable constants, such as the speed of light, requires assumptions.

With so many models off the shelf a common understanding of assumptions is slipping by. If you go far enough back, most good finance text books bothered to explain the assumptions underlying the model. One such text – Modern Finance by Copeland and Weston – offers a comprehensive discussion of the assumptions necessary to argue that the world of asset pricing is mean-variant efficient (MVE.)

MVE underpins the Capital Asset Pricing Model (CAPM), the second most important model in all of finance; a model most students in business classes in western universities are exposed to; and something that simply can’t work. Much can be proven about what the model can say.

The most important of which is that there’s a certain portfolio of assets – the Efficient Frontier – that is better than all others.

But it turns out that while this portfolio can always be found in historical data, it can never be identified in the present.

But there are other models which can be derived from the same set of unrealistic assumptions. In 1997, The Nobel Prize committee awarded the prize in Economic Sciences to Robert Merton and Myron Scholes for their “… method to determine the value of derivatives,” – the Black Scholes Options Pricing Model (BS).

These two, along with Fisher Black who had passed prior to the award, solved the puzzle of pricing the right which affords its holder a specific time frame within which to purchase, for a set price, a risky asset. The right can be to buy (a call) or sell (a put) or otherwise manipulated in almost any fashion that mathematics allows, and still some form of the BS equation will arrive at a price.

The assumptions necessary for the options pricing model to mirror reality have never been met. And yet, pricing options and the reams of creative derivatives that spew forth is a several-hundred-trillion-dollar market.

The notional value of derivatives collapse, or ‘net,’ to a much smaller number as most activity is part of a giant zero-sum game. Still, options exist. Farmers have long since contracted in advance to sell yet-to-be harvested crop. It’s only in the past 45 years that a workable formula has been available to help someone negotiate a price.

The basic formula was derived in 1900 by a French mathematician.

Bachelier developed the idea of using a random walk, understood how to truncate the distribution so that expected future values were only those that could be realized (prices can’t be negatives) and knew the calculated future value would have to be brought back to the present. It’s in this last part that he got stuck. To determine the certain value today of an uncertain future value requires a ‘risk-adjusted’ interest rate.

Leaving his formula in need of such a rate meant that his derived option value was subject to the risk appetite of the particular user. Each practitioner would arrive at his or her own version of a rate and therefore of a value. No common price could rise above all others.

Black and Scholes, with Merton’s help, derived their formula (essentially the same as Bachelier’s) and then something else. They reasoned that under certain assumptions including zero transactions costs, a Replicating Portfolio of a long position in the stock, financed by borrowed money, could be adjusted on a continuous basis so as to mimic the value of the option on the underlying stock. (A similarly constructed portfolio could be managed to replicate the value of options on other commodities.)

Since the Replicating Portfolio could be created using ‘risk-less borrowing and lending’ it could be asserted that the value of the option is independent of an individual’s preference for risk.

This observation was a breakthrough. Since the option value is independent of risk preferences, pick a rate and get on with things. The chosen rate is Libor – the London Interbank Offer Rate – the rate at which major global banks lend to one another (currently being phased out after it was determined that, surprisingly, in the process of determining Libor, to gain advantage certain banks submitted faulty rate quotes).

The Replicating Portfolio became the tool for hedging. It facilitates market-making — the providing of prices by traders who agree to buy or sell the particular underlying option and then use some version of the Replicating Portfolio in an attempt to manage their position to a profit over time.

These tools, in turn, create trails that fit nicely into what computers are good at: tracking, calculating, displaying and so on. And since tools are addictive, the world of modern finance was born.

Donald MacKenzie, in a wonderful but oddly titled book – An Engine, not a Camera – follows the evolution of modern finance keenly noting where the process is ‘performative’ and where it’s not.

According to Mackenzie, the MVE view of the world, though still drained into most MBA students, is not performative. By this it’s meant that though there is a presumptive model for the price of risk under various assumptions, the market has not conformed to the model.

In one particularly exciting passage (if you’re into this stuff) MacKenzie writes of Black and Scholes deciding to sell tables of valuations for popular options then being traded on the Chicago Board of Exchange. When an early practitioner complained that the printed values were not matching the actual quoted prices, Black and Scholes explained how, if one believed in the modeled prices, to profit from the discrepancies – to play the ‘spread.’

Over time “… the academics won.” The observed prices, driven by the actions taken by those who followed instructions, converged to the modeled prices. Thus, options pricing as it is practiced, is performative: A concept of the imagination, once modeled, has become a game that at times helps, and at other times hurts, the real world.

Why is the CAPM, the capital asset pricing model built from MVE, not performative while options pricing models are? After all, both are imaginative concepts mapped into mathematics. And, it turns out, Black and Scholes derived their version of the option pricing model to be consistent with the assumptions implicit in CAPM.

MacKenzie argues that the actions a performative model describes have to be in existence independent of the model. It’s true that there were options on securities before there was a semblance of agreement on pricing. But in a similar fashion, the market for stocks precedes the CAPM.

And yet, in the case of the former, the market responded to the model in a much more impactful way than in the latter. We continue to use the lingo of CAPM, just not the model itself. The common terms ‘beta’ – a market-tracking asset – and ‘alpha’ – something that outperforms beta – come from CAPM. But it can be argued that the usage of these terms contradicts the model as, if you can get to the Efficient Frontier then by definition, you would not be able to construct a more efficient portfolio, i.e., alpha would not exist.

In part, what distinguishes the two can be found in the particulars of the questions they tackled: CAPM is an attempt to derive a market-price while the BS model and its variants, is a construct that constrains the free parameters in a modeling exercise such that the presumptive unknown is expressed through one variable – volatility (the square root of variance).

While the BS model is used to provide a window into how the market is expressing the unknown parameter, using the CAPM to price an asset requires forecasts: The expected return on the entire market, the variance of the entire market and the correlation of the asset in question with the expected return on the market. In short, CAPM does not obviate the need for naked prediction.

A stretched analogy could be horse racing: Once the process of odds making was agreed upon there could be a market in betting on horse racing even though (in a fair race) no one had any idea who would actually win. CAPM fails to offer tools necessary to construct a market in its unknowns.

But Mackenzie also notes the importance of calculative resources. Here he means “… the role of Black’s sheets … as material means of calculation, as aspects of distributed cognition, as ways of connecting the apparently abstract mathematics of the model to the sweaty, jostling bodies on exchange trading floors.”

In the early days, when transactions costs were high, options traders needed a ‘wide spread’ in order to profit from a position. The same calculative resources that facilitated the early sheets, over time, drove down transactions costs to the point where trading is almost instantaneous with costs measured in millionths of a penny.

Since the size of the traded spread is a function of the transactions costs of trading, market makers are continuously forced to make up in volume what they are losing in margin. This race is facilitated by the same calculative engines that allowed for the trading sheets – the massive investments in computing resources across all of Wall Street.

When Black’s trading sheets were distributed monthly, it was feasible to calculate prices by hand. Trading at the speed of light requires calculations be made at least as fast. As such, a race continues; a performative one where advances in trading capabilities demand advances in analytical techniques which in turn fuel an ever-increasing demand for enhanced trading capabilities.

Like other self-fulfilling prophecies noted within the social sciences, this goes on until, for one reason or another, the fault in a basic assumption is exposed at which point the edifice crashes. But unlike their run-of-the-mill cousins, performative markets are engineered to rise from their ashes.

So far, anyway.

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Variance: get a sample of changes in asset prices. Subtract the mean of the sample from each observation.  Square the result, sum up the squares then decide by the number of observations in the data set.