The Message Of The Greeks

Let’s imagine a pre-Black-Scholes world, where people have no access to analytical option pricing tools and therefore have no way of methodically visualizing the price function and the associated risk parameters (what we now call the greeks). They can only think about them and imagine them. Or perhaps, draw them down with a pen. Let’s then put ourselves in the shoes of pre-1973 option traders who lack a formal pricing model but would nonetheless like to have an idea as to the characteristics of the options that they are selling to their customers. In particular, they would like to know the kind of risks that they are facing as option writers and how the value of the option will change as these parameters change.

After thinking for a few minutes, the traders realize that one obvious risk is so-called spot risk, as clearly the value must change when the underlying itself changes. This is of course what we nowadays call delta. Given the importance of spot risk, it would also be critical to know how such risk changes with the underlying, or in other words, how exposed the value of the option is to jumps in the underlying. This is what we know as gamma. Additionally, it should be clear that the option’s value is affected by how volatile the underlying is, as this will determine the probability that the option ends in or out of the money. This is our famous vega. Finally, time is also an important risk factor, given the fact that less time means less chances for movement in the underlying, and movement is always assumed to be good for option holders given the asymmetric payoffs. This is theta.

So, let’s picture these ancient traders sitting at their favourite restaurant (apparently, pre-exchanges option traders on Wall Street conducted their businesses from a restaurant, with the phone booths acting as offices). They have pen and paper and are going to try to draw graphs representing the change in value of the option with respect to each of the main risk parameters described above.

By putting ourselves into the shoes of pre-1973 option traders we try to get a no-nonsense view of the risk parameters of an option from the perspective of model-deprived, purely practical minds. What we would obtain are results that could be labelled as “street-smart reasoning”, the kind of results obtained from somebody who is unaware of the existence of any analytical option pricing model and who does not have any other tools at his disposal than his logic and his understanding for what makes sense. In a way, such analysis would detail how the greeks should behave in practice in real life. After all, we are reading seasoned traders’ minds here.

Interestingly, if one embarks on such exercise (if one tries to draw the greeks based on intuition, on what makes sense) the resulting figures are extraordinarily similar to what we would obtain from Black-Scholes. It is tempting to conclude that both the no-nonsense ancient traders and the model would agree on the greeks (on the shape of the graph, not the scale as this would depend on the volatility parameter).

Is this meaningful? Can we see this as an endorsement of Black-Scholes? After all, what the old traders lacked was a way of putting precise numbers to the greeks, a way of quantifying the risk parameters. What Black-Scholes did was provide a way, for the first time in history (yes?), for traders to assign a specific number to the many risks they face under different scenarios. Armed with the model, traders could tell if each landmine along the way could kill them or just inflict harmless injuries. Most important, once you know how big a risk you are facing you can equip yourself with the necessary matching hedge. Traders now had a key analytical tool at their disposal that would allow them to run trading books.

Could the message from the greeks be that the Black-Scholes framework is indeed the right one, because its outputs would be found agreeable by pre-modelling traders? Does Black-Scholes have a trader’s mind embedded into it?.