SpecMat Options Tutorials.
Let’s introduce the logic, then back up and unpack it.
Person: How does the package of the BSM operator chase delta-zero?
Speculative Materialist: By becoming delta-neutral, by always desiring to move yet faster towards zero.
P: But what about stochastization and nonlinearity? Won’t prices move, move randomly, and won’t volatility jump around?
SM: Well, yes, an operator needs to continuously recalibrate the delta of the package back towards its becoming-zero.
P: What’s recalibration? How does an operator continuously recalibrate?
SM: By dynamically replicating.
P: What’s dynamic replication? How does an operator dynamically replicate?
SM: With the Greeks: with delta (∆), gamma (Γ), vega (V), theta (Θ), and so on.
P: But what are the Greeks?
A standard financial economic definition of the Greeks is a map of the relation of the price of an option with respect to a variety of parameters; by the term ‘relation’ we should understand ‘speed’, or ‘rate of change’. A speculative materialist’s brain often wishes to explode when ruminating on the ontological status of the Greeks. For us, they explicate the intensive differential relations between an option price and any number of parameters which would remain quite operative, but inaccessible to examination, strictly virtual, and thus unactualized were we lacking analytic recourse to their concepts.
We’ve conveyed our intention to tweak the Greeks, and before that to probe their peculiar ontological status for dromocratic purposes. However, let us first observe that risk has strong consequences for valuation and cash flow; but that risk is multidimensional, and dimension is not an economic invariant. In financial economics, each Greek letter (e.g. delta (Δ), gamma (Γ), theta (Θ), vega (V), etc.) purports to explicate a different dimension of risk –and its associated cash flow; so that an operator managing her Greeks will always be seeking to hedge her exposure to the multidimensional risks whose differential relations and amounts of relations perpetually supervene on her portfolio, affecting its actual value, and therefore it’s return.
First, then, delta (Δ). Delta is first among Greeks, literally. It’s the first mathematical derivative of the product with respect to the referent, and the most important parameter to begin to understand dynamic replication by manner of continuous recalibration. BSM’s first analytic virtue is to provide a formula for knowing delta.
Hull defines delta as ‘the rate of change of the option price with respect to the price of the underlying.’ Taleb defines delta as ‘the sensitivity of the option price to the change in the underlying price.’ More generally, this means that delta maps the speed of change between the option price and referent price. Like all Greeks, delta is an intensive economic property –which importantly, more generally means its concept denotes the speed of spread between valuations of the derivative expressed, represented, metricized, and territorialized as price, and valuations of its referent expressed, represented, metricized, territorialized as price. It is common to depict such expressions of delta in either a % or in total amounts. So for example, as Taleb explains, ‘a 50% delta is supposed to mean that the derivative is half as sensitive as the [referent] asset’ –this is delta in % terms; or ‘that one needs two dollars in face value of the derivative to replicate the behavior of one dollar of [referent] asset.’
Standard industry definitions of delta will typically observe, as we have above, that it’s the first mathematical derivative of the product with respect to the referent asset. This means delta provides the primary numerics of the hedge ratio for any price moves between the referent and its derivative, when the two are combined to comprise some amount in a package. For this reason ‘delta’ and ‘hedging’ are terms which, while the former is somehow both more than but also only a subclass of the latter (for there is more than one way to hedge), are often combined so that ‘delta-hedging’, ‘hedging’, or ‘hedging with delta’ intends to denote that any infinitely small price moves in an option are offset by price moves in its referent, and vice versa.
And in a simple, linear, Euclidean world –which is to say in a world without drift, a world wherein the distribution of variations are normal –and not that there’s no volatility, but rather volatility is steady, oscillating, constant, its variations invariable, and thus globally predictable; in such a linear world, delta would obtain a value, that value would remain unchanged, and in turn would cause the efficient, effective, and sufficient achievement of a riskless package with delta-hedging: an operator would use BSM to tell her the number of units of stock (or other referent) relative to options she should hold to deterritorialize her package from a continual barrage of inundating, multidimensional risks; and constructing her riskless package by way of BSM would merely be a matter of static, linear, delta hedging, in a fixed, Euclidean, linear world.
At this time we remind our reader of D&G’s principle of asignifying rupture –one of six principles endemic to the rhizome model of economy outlined in Chapter 1. Asignifying rupture marks nonlinearity, nonlinearity has deep ontological-economic significance, and options are thoroughly nonlinear. The first good observation an options pricing manual will make is that options are nonlinear assets, and therefore the trajectories of an option’s constitutive parameters orbit amidst a nonlinear world. The second thing your manual will tell you is that this requires dynamic hedging, which in turn requires consideration of some second-order derivatives. We can and will be observing the epistemological significance of this deep ontological fact as well –albeit, as we will see, the issue obtains a qualitatively different profundity in dromocracy. For rather than seeking to mitigate, negotiate, or in some way tame nonlinearity for sedentary distributive purposes, the denizens of a dromocracy embrace its dynamics, urge its vortical propulsion, stoke its nomadic distributive thrust.
So we will demonstrate that dynamic delta-hedging consists of hedging on an ongoing and interminable, rather than static or one-time basis. There is neither true nor final ‘being’ to delta, but only ever its becoming. To dynamically-hedge with second-order derivatives is to continuously recalibrate, which renders the latter a financial economic method for interminably deterritorializing one’s package always back towards a zero that it will only ever asymptotically approach. Dynamic replication by method of continuous recalibration is then, for us, understood as a matter of interminable deterritorialization, a pure activity of intermittency isomorphic to the becoming that is the Cantor set –for it is infinitely-becoming zero, yet now comprises a substance unto itself. And insofar as dynamic replication requires second-order derivatives, such as for instance gamma, theta, vega, and rho; and that these Greeks lend us analytic purview into the multidimensionality of risk; this means that continuous recalibration provides us with a practical method for interminable deterritorialization, yes, but moreover is one whose technology provides for an operator, a set of operators, or indeed clusters of exotic operators to comingle with deterministic chaos, to drink from its open wealth –this is our interest in and wager on technology of continuous recalibration for the institutional and behavioral purposes of dromocracy.
 Taleb’s analysis of the Greeks are generally good (see Chapters 7-11), but his definition is characteristically careless. He invokes the term ‘sensitivity’ to denote the speed, or rate of change (‘“The Greeks”…denote the sensitivity of the option price with respect to several parameters’ pg. 10). Hull is less analytically-expansive, (see Chapter 17), but his definition is perhaps more robust (‘Each Greek letter measures a different dimension to risk in an option position and the aim of the trader is to manage the Greeks so that all the risks are acceptable.’ pg. 357)
 It should be observed that in any economy an operator is always virtually reading and writing their options to, among, and with others, and is therefore always faced with the actual problem of managing her multidimensional risk –the hedging of which may be easy or hard to come by, had cheaply or at great expense. Bob Meister has illustrated that the virtual exchange of options were already occurring between Adam Smith’s yeoman farmer and his financier (See his “Liquidity”, unpublished draft, Sept. 2013). The difference, for dromocracy, is that operators are actually buying and selling and swapping options, creating and destroying them, and all of their multidimensional and virtual but very real risks. In dromocracy, operators trade standardized options, yes, but significantly more so exotic and hyperexotic options as well. Moreover, because a war machine economy is equitable (but not equal), clusters provide operators a permanent set of impermanent venues for hedging their exposures to risk easily and cheaply. And because, as we will also see, by manner of universal synthetic CDO everyone is leveraged on top of everyone else –a leverage, natural and infinite– every operator is simultaneously an entire financial institution unto herself, together with her incessantly-changing cluster, and universally as an One.
 (‘A delta is expressed as the first mathematical derivative of the product with respect to the underlying asset. [This] means that it is the hedge ratio of the asset for an infinitely small move. Somehow, when the portfolio includes more than one option, with a combination of shorts and longs, delta and hedge ration start parting ways.’) Taleb pg. 115
 Hull pg. 360
 Taleb pg. 10
 Taleb pg. 115