Mania of the Revisionary Narratives. Note Quote.

For if Lacan is either symptom or agent of a theoretical turn, it is far from the “care of the self” imagined by this proposition because the French return to Freud explodes any ready notion of self-care. It also removes the props for identity politics. Poststructural psychoanalysis has been the key provocation of a turn to the identity-destabilizing work of the unconscious that, along with an unlikely ally in historicism, has galvanized the transition from transparent to unstable, internally divided, and overdetermined identity categories. The tense debates of the 1980s and 1990s between feminism and poststructuralism have without much fanfare yielded to a tacit consensus that, rather than invalidating politically engaged analysis, psychologically and historically mobile conceptualizations of gender make intellectual and political alliances possible across previously hostile discursive terrains. As self-difference opens the door to other differences, theorizations that emanate from one racial or sexual or class turf are more likely to provoke new questions than old accusations from competing grounds. We are just at the beginning of a generative process that encompasses not only the particularization that results from historical refinement and nuancing but also the elaboration of revisionary narratives: what happens when the dark plantation son retells the story of the primal horde, or when the racial shadow falls across the mirror stage, or the queer encounters and reforms the melancholic? Fracturing the subject has also poked holes in the walls that have divided psychoanalysis and history, launching a potentially interminable analysis.

No-Arbitrage & Conditional Drift from the Covariance of Fluctuations. (Didactic 4)

From P(t,s) = exp {−∫₀^s−t f(t,x)dx}, we get

d_t logP(t,s) = f(t,x) dt − ∫₀^xdy dt f(t,y) —– (1)

where x ≡ s − t. We need the expression of dP(t,s)/P(t,s) which is obtained from (1) using Ito’s calculus. In order to get Ito’s term in the drift, recall that it results from the fact that, if f is stochastic, then

d_tF(f) = ∂F/df dtf + 1/2 ∫ dx ∫ dx′ ∂²F/∂f(t,x)∂f(t,x′) Cov [d_tf(t,x), d_tf(t,x′)] —– (2)

where Cov [d_tf(t,x),d_tf(t,x′)] is the covariance of the time increments of f(t,x). Using this Ito’s calculus, we obtain

dP(t,s)/P(t,s) = [dt f(t,x) − ∫₀^x dyE_t,dtf(t,y) + 1/2 ∫₀^x dy ∫₀^x dy′ Cov d_tf(t,y) d_tf(t,y′)] –

∫₀^x dy [d_tf(t,y) − E_t,dtf(t,y)] —– (3)

We have explicitly taken into account that d_tf(t,x) may have in general a non-zero drift, i.e. its expectation

E_t,dtf(t,x) ≡ E_t [d_tf(t,x)|f(t,x)] —– (4)

conditioned on f(t,x) is non-zero. The no-arbitrage condition for buying and holding bonds implies that PM is a martingale in time, for any bond price P. Technically this amounts to imposing that the drift of PM be zero:

f(t,x) = f(t,0)+ ∫₀^x dy E_t,dtf(t,y)/dt − 1/2 ∫₀^x dy ∫₀^x dy′ c(t,y,y′) + o(1) —– (5)

assuming that d_tf(t,x) is not correlated with the stochastic process driving the pricing kernel and using the definitions

c(t,y,y′)dt = Cov [d_tf(t,y)d_tf(t,y′)] —– (6)

and r(t) = f (t, 0). In (5), the notation o(1) designs terms of order dt taken to a positive power. Expression (5) is the fundamental constraint that a SPDE for f (t, x) must satisfy in order to obey the no-arbitrage requirement. As in other formulations, this condition relates the drift to the volatility.

It is useful to parametrize, without loss of generality,

E_t,dtf(t,x)/dt = ∂f(t,x)/∂x + h(t,x) —– (7)

where h(t, x) is a priori arbitrary. The usefulness of this parametrization (7) stems from the fact that it allows us to get rid of the terms f(t,x) and f (t,0) in (5). Indeed, they cancel out with the integral over y of E_t,dtf(t,y)/dt. Taking the derivative with respect dt to x of the no-arbitrage condition (5), we obtain

h(t, x) = −1/2 ∫₀^x [c(t, y, x) + c(t, x, y) —– (8)

In sum, we have the following constraint that the SPDE for f (t, x) must satisfy

E_t [d_tf(t,x)|f(t,x)] = dt ∂f(t,x)/∂x −1/2 ∫₀^x dy (Cov[d_tf(t,y) d_tf(t,x)] + Cov[d_tf(t,x) d_tf(t,y)]) —– (9)

which, for symmetric covariance, lead to

E_t [d_tf(t,x)|f(t,x)] = dt ∂f(t,x)/∂x − ∫₀^x dy Cov[d_tf(t,y) d_tf(t,x)] —– (10)

This expression (10) is reminiscent of the fluctuation-dissipation theorem in the Langevin formulation of the Brownian motion, linking the drag coefficient (analog to the l.h.s.) to the integral over time of the correlation function of the fluctuation forces (analog of the second term of the r.h.s.). In the usual fluctuation-dissipation theorem, the drag coefficient is determined self-consistently as a function of the amplitude and correlation of the fluctuating force in order to be compatible with the equilibrium distribution. Similarly, here the no-arbitrage condition determines self-consistently the conditional drift from the covariance of the fluctuations. In contrast, notice however that the same quantity f enters in both sides of (9) and (10). Thus, the no-arbitrage condition imposes a condition on the structure of the partial differential equations that govern the dynamical evolution of the forward rates f(t,x).

 

 

 

Hyperbolic Brownian Sheet, Parabolic and Elliptic Financials. (Didactic 3)

Financial and economic time series are often described to a first degree of approximation as random walks, following the precursory work of Bachelier and Samuelson. A random walk is the mathematical translation of the trajectory followed by a particle subjected to random velocity variations. The analogous physical system described by SPDE’s is a stochastic string. The length along the string is the time-to-maturity and the string configuration (its transverse deformation) gives the value of the forward rate f(t,x) at a given time for each time-to-maturity x. The set of admissible dynamics of the configuration of the string as a function of time depends on the structure of the SPDE. Let us for the time being restrict our attention to SPDE’s in which the highest derivative is second order. This second order derivative has a simple physical interpretation : the string is subjected to a tension, like a piano chord, that tends to bring it back to zero transverse deformation. This tension forces the “coupling” among different times-to-maturity so that the forward rate curve is at least continuous. In principle, the most general formulation would consider SPDE’s with terms of arbitrary derivative orders. However, it is easy to show that the tension term is the dominating restoring force, when present, for deformations of the string (forward rate curve) at long “wavelengths”, i.e. for slow variations along the time-to-maturity axis. Second order SPDE’s are thus generic in the sense of a systematic expansion.

In the framework of second order SPDE’s, we consider hyperbolic, parabolic and elliptic SPDE’s, to characterize the dynamics of the string along two directions : inertia or mass, and viscosity or subjection to drag forces. A string that has “inertia” or, equivalently, “mass” per unit length, along with the tension that keeps it continuous, is characterized by the class of hyperbolic SPDE’s. For these SPDE’s, the highest order derivative in time has the same order as the highest order derivative in distance along the string (time-to-maturity). As a consequence, hyperbolic SPDE’s present wave-like solutions, that can propagate as pulses with a “velocity”. In this class, we find the so-called “Brownian sheet” which is the direct generalization of Brownian motion to higher dimensions, that preserves continuity in time-to-maturity. The Brownian sheet is the surface spanned by the string configurations as time goes on. The Brownian sheet is however non-homogeneous in time-to-maturity.

If the string has no inertia, its dynamics are characterized by parabolic SPDE’s. These stochastic processes lead to smoother diffusion of shocks through time, along time-to-maturity. Finally, the third class of SPDE’s of second-order, namely elliptic partial differential equations. Elliptic SPDE’s give processes that are differentiable both in x and t. Therefore, in the strict limit of continuous trading, these stochastic processes correspond to locally riskless interest rates.

The general form of SPDE’s reads

A(t,x) ∂²f(t,x)/∂t² + 2B(t,x) ∂²f(t,x)/∂t∂x + C(t,x) ∂²f(t,x)/∂x² = F(t,x,f(t,x), ∂f(t,x)/∂t, ∂f(t,x)/∂x, S) —– (1)

where f (t, x) is the forward rate curve. S(t, x) is the “source” term that will be generally taken to be Gaussian white noise η(t, x) characterized by the covariance

Cov η(t, x), η(t′, x′) = δ(t − t′) δ(x − x′) —– (2)

where δ denotes the Dirac distribution. Equation (1) is the most general second-order SPDE in two variables. For arbitrary non-linear terms in F, the existence of solutions is not warranted and a case by case study must be performed. For the cases where F is linear, the solution f(t,x) exists and its uniqueness is warranted once “boundary” conditions are given, such as, for instance, the initial value of the function f(0,x) as well as any constraints on the particular form of equation (1).

Equation (1) is defined by its characteristics, which are curves in the (t, x) plane that come in two families of equation :

Adt = (B + √(B2 − AC))dx —– (3)

Adt = (B − √(B2 − AC))dx —– (4)

These characteristics are the geometrical loci of the propagation of the boundary conditions.

Three cases must be considered.

• When B² > AC, the characteristics are real curves and the corresponding SPDE’s are called “hyperbolic”. For such hyperbolic SPDE’s, the natural coordinate system is formed from the two families of characteristics. Expressing (1) in terms of these two natural coordinates λ and μ, we get the “normal form” of hyperbolic SPDE’s :

∂²f/∂λ∂μ = P (λ,μ) ∂f/∂λ +Q (λ,μ) ∂f/∂μ + R (λ,μ)f + S(λ,μ) —– (5)

The special case P = Q = R = 0 with S(λ,μ) = η(λ,μ) corresponds to the so-called Brownian sheet, well studied in the mathematical literature as the 2D continuous generalization of the Brownian motion.

• When B² = AC, there is only one family of characteristics, of equation

Adt = Bdx —– (6)

Expressing (1) in terms of the natural characteristic coordinate λ and keeping x, we get the “normal form” of parabolic SPDE’s :

∂²f/∂x² = K (λ,μ)∂f/∂λ +L (λ,μ)∂f/∂x +M (λ,μ)f + S(λ,μ) —– (7)

The diffusion equation, well-known to be associated to the Black-Scholes option pricing model, is of this type. The main difference with the hyperbolic equations is that it is no more invariant with respect to time-reversal t → −t. Intuitively, this is due to the fact that the diffusion equation is not conservative, the information content (negentropy) continually decreases as time goes on.

• When B² < AC, the characteristics are not real curves and the corresponding SPDE’s are called “elliptic”. The equations for the characteristics are complex conjugates of each other and we can get the “normal form” of elliptic SPDE’s by using the real and imaginary parts of these complex coordinates z = u ± iv :

∂²f/∂u² + ∂²f/∂v² = T ∂f/∂u + U ∂f/∂v + V f + S —– (8)

There is a deep connection between the solution of elliptic SPDE’s and analytic functions of complex variables.

Hyperbolic and parabolic SPDE’s provide processes reducing locally to standard Brownian motion at fixed time-to-maturity, while elliptic SPDE’s give locally riskless time evolutions. Basically, this stems from the fact that the “normal forms” of second-order hyperbolic and parabolic SPDE’s involve a first-order derivative in time, thus ensuring that the stochastic processes are locally Brownian in time. In contrast, the “normal form” of second-order elliptic SPDE’s involve a second- order derivative with respect to time, which is the cause for the differentiability of the process with respect to time. Any higher order SPDE will be Brownian-like in time if it remains of order one in its time derivatives (and higher-order in the derivatives with respect to x).

Conjuncted: Whats Right-Wing With Negri? Note Quote.

Already with his concept of the socialised worker, Negri had rejected the central pillar of Marx’s economics – the relationship between value and labour. As the whole of society becomes a social factory, so the duration of labour becomes unquantifiable and it becomes impossible to reduce specific forms of labour into abstract socially necessary labour. As the 1980s and 1990s unfolded Negri underpinned his new politics with reference to two fashionable right wing theories – the idea of a ‘weightless economy’ developing out of a high tech ‘third industrial revolution’ and, more recently, extreme versions of globalisation theory depicting the death of the nationstate. Today Negri claims that ‘immaterial labour’ has taken the place of industrial labour as the hegemonic form of production that other forms of labour tend towards. Negri’s descriptions of contemporary production will seem unfamiliar to most workers: ‘A gigantic cultural revolution is under way. Free expression and the joy of bodies, the autonomy, hybridisation and the reconstruction of languages, the creation of new singular mobile modes of production—all this emerges, everywhere and continually’.

[Global corporations are anxious to include] difference within their realm and thus aim to maximise creativity, free play and diversity in the corporate workplace. People of all different races, sexes and sexual orientations should potentially be included in the corporation; the daily routine of the workplace should be rejuvenated with unexpected changes and an atmosphere of fun. Break down the old boundaries and let 100 flowers bloom!

Exploitation, in the Marxist sense of the pumping of unpaid surplus labour out of workers, has ended. Exploitation today means capturing the creative energies of a joyous, cooperating multitude – who may be inside or outside of the workplace. The domination of dead labour, such as machinery or computers, over living is finished because living (for Negri, intellectual) labour is now dominant. The tool of production is now the brain. Paul Thompson explains how Negri’s thinking parallels right wing accounts of the economic changes since the 1970s:

This appears to be remarkably similar to knowledge economy arguments, which we might briefly summarise in the following way. In the information age, capital and labour are said to have been displaced by the centrality of knowledge; brawn by brain; and the production of goods by services and manipulation of symbols. As a commodity, knowledge is too complex, intensive and esoteric to be managed through command and control. The archetypal worker in the new economy makes his or her living from judgement, service and analysis… As none of this is calculable or easily measured, it is the inherent property of the producer… This shifts the power balance to the employee, an increasing proportion of whom fall into the category of mobile, self-reliant and demanding ‘free workers’.

Thompson goes on to provide a detailed critique of the idea of immaterial labour. Even at the most immaterial end of the labour market, intellectual property regimes allow the commodification of knowledge. And such workers are still subject to exploitation and control centred upon the workplace.

Far from the workplace ceasing to be the centre of capital accumulation for the ruling class, it plays an increasingly important role in a world of labour intensification and tightening managerial control. The workplace is still the point at which fixed capital necessary for the production of most goods and services is centralised. And it is still the site where surplus value is extracted from workers – the central obsession of capitalists and states – and thus the point at which those opposed to the rule of capital should concentrate their efforts. Just like his vision of the weightless economy, Negri’s account of
globalisation is almost entirely unsupported by empirical evidence. He writes that:

large transnational corporations have effectively surpassed the jurisdiction and authority of nation-states…the state has been defeated and corporations now rule the earth!

Negri’s Dismissive Approach to Re-engaging Growing Ideological Opposition to Capitalism. Note Quote.

Negri’s politics are shaped by the defeat of the movement of the 1960s and 1970s. His borrowed economic theory was shaped by the triumphalism following the restructuring of US capitalism in the 1980s and the collapse of the Stalinist regimes. Having created a Marxism gutted of its central emphasis on the working class, he filled this empty shell with the poststructuralist philosophy developed by a generation of disappointed post-1968 French intellectuals.

Atilio Boron argues that Hardt and Negri’s increasing reliance on poststructuralist philosophers flows from a shared backdrop of trying to come to terms with working class defeat and capitalist hubris. Faced with a system that appears, for the time being, unbeatable:…a series of theoretical and practical consequences emerge that…are neatly reflected in the postmodern agenda. On the one hand, an almost obsessive interest in the examination of the social forms that grow in the margins or in the interstices of the system; on the other hand, the search for those social forces that at least for now could commit some sort of transgression against the system, or could promote some type of limited and ephemeral subversion against it.

This concern with subversion and transgression is indeed characteristic of many of the autonomist movements with which Negri is associated. But for Negri, with the rise of post-industrial production and the multitude, the potential for postmodern subversion has spread across the whole social terrain, and across the globe. One might expect Hardt and Negri to explain what such a confrontation would look like. However, what we instead get is a retreat into philosophy and descriptions of the multitude that the authors themselves admit are merely ‘poetic’.

Hardt and Negri also borrow from the poststructuralists, especially Deleuze and Guattari, an eclectic form of expression known as ‘assemblage’.

Timothy Brennan writes in his Italian Ideology:

It expresses itself as a gathering of substantively incompatible positions. In Empire’s assemblage, the juxtaposition of figures whose political views are mutually hostile to one another…is presented as the supersession of earlier divisions in pursuit of a more supple and inclusive combination.

So, in Empire, philosophers such as Michel Foucault or Baruch Spinoza and revolutionaries such as Rosa Luxemburg rub shoulders with Bill Gates, former US labour secretary Robert Reich and St Francis of Assissi. This form of expression evolved as a rejection of attempts at a ‘grand narrative’ such as Marxism that could hope to explain and help transform the world, or of an agency such as the working class that could carry through such a transformation. For Hardt and Negri this method mirrors the multitude that they describe—a series of heterogeneous, isolated subjects, coming together to fleetingly act in common. Indeed they have gone so far as to say that the struggles of the multitude have become ‘incommunicable’ and lack a ‘common enemy’.

Their assertion would be contested by most of those who have attended the great international gatherings and protests of the anti-capitalist movement since Seattle. Here opposition to neo-liberalism and war have become common themes. The world working class may have been traumatised by the impact of neo-liberalism and the defeat of the movements of the 1960s and 1970s. But, rather than celebrating the much-exaggerated demise of the working class, the challenge today is to re-engage the growing ideological opposition to capitalism with the potential power that workers still hold. Negri is dismissive of such a project, but offersanothing substantial in its place.

His faux pas—over neo-liberalism, the EU constitution and the war in Iraq—stem from his failure to come to terms with either the defeats of the past or the nature of contemporary capitalism. Almost every assertion in his recent writings vanishes into thin air once subjected to even a cursory empirical examination. As for strategy, Multitude ends:

We can already recognise that today time is split between a present that is already dead and a future that is already living – and that yawning abyss between them is becoming enormous. In time, an event will thrust us like an arrow into that living future. This will be the real political act of love.

With an upsurge of the Techno-Commercial Right in the world, multinationals and Commodity Trading firms and HFTs and states wreaking havoc, and global warming (believe it or not!) threatening our very survival as a species, waiting for an act of political love to save us sounds like bad advice.

For No Arbitrage Opportunities to Exist, Investment Strategy Must be a Martingale. (Didactic 2)

We postulate the existence of a stochastic discount factor (SDF) that prices all assets in this economy and denote it by M. This process is also termed the pricing kernel, the pricing operator, or the state price density. It is well known that assuming that no dynamic arbitrage trading strategies can be implemented by trading in the financial securities issued in the economy is roughly equivalent to the existence of a strictly positive SDF. For no arbitrage opportunities to exist, the product of M with the value process of any investment strategy must be a martingale. Technically, recall that a martingale is a family of random variables ξ(t) such that the mathematical expectation of the increment ξ(t₂)−ξ(t₁) (for arbitrary t₁ < t₂), conditioned on the past values ξ(s) (s ≤ t₁), is zero. The drift of a martingale is thus zero. This is different from the Markov process, which is better known in the physical community, defined by the independence of the next increment on past values.

Under an adequate definition of the space of admissible trading strategies, the product MV is a martingale, where V is the value process of any admissible self-financing trading strategy implemented by trading on financial securities. Then,

V(t) = E_tV(s) M(s)/M(t) —– (1)

where s is a future date and E_t[x] denotes the mathematical expectation of x taken at time t. In particular, we require that a bank account and zero-coupon discount bonds of all maturities satisfy this condition.

A security is referred to as a (floating-rate) bank account, if it is “locally riskless”. Thus, the value at time t, of an initial investment of B(0) units in the bank account that is continuously reinvested, is given by the following process

B(t) = B(0) exp{∫₀^t r(s)ds} —– (2)

where r(t) is the instantaneous nominal interest rate.

We further assume that, at any time t, riskless discount bonds of all maturity dates trade in this economy and let P (t,s) denote the time t price of the s maturity bond. We require that P (s,s) = 1, that P (t,s) > 0 and that ∂P(t,s)/∂s exists.

Instantaneous forward rates at time t for all times-to-maturity x > 0, f(t,x), are defined by

f(t,x)=−∂logP(t,t+x)/∂x —– (3)

which is the rate that can be contracted at time t for instantaneously riskless borrowing or lending at time t + x. We require that the initial forward curve f (0,x), for all x, be continuous.

Equivalently, from the knowledge of the instantaneous forward rates for all times- to-maturity between 0 and time s − t, the price at time t of a bond with maturity s

can be obtained by

P(t,s) = exp {−∫₀^s−t f(t,x)dx} —– (4)

Forward rates thus fully represent the information in the prices of all zero-coupon bonds. The spot interest rate at time t, r(t), is the instantaneous forward rate at time t with time-to-maturity 0,

r(t) = f(t,0) —– (5)

For convenience, we model the dynamics of forward rates. Clearly, we could as well model the dynamics of bond prices directly, or even the dynamics of the yields to maturity of the zero-coupon bonds. We use forward rates with fixed time-to-maturity rather than fixed maturity date. The model of HJM starts from processes for forward rates with a fixed maturity date. This is different from what we do. If we use a “hat” to denote the forward rates modeled by HJM,

f^ˆ(t,s) = f (t,s−t) —– (6)

or, equivalently,

f^ˆ(t,x) = f(t,t+x) —– (7)

for fixed s. Brace and Musiela define forward rates in the same fashion. Miltersen, Sandmann and Sondermann, and Brace, Gatarek and Musiela use definitions of forward rates similarly, albeit for non-instantaneous forward rates. Modelling forward rates with fixed time-to-maturity is more natural for thinking of the dynamics of the entire forward curve as the shape of a string evolving in time. In contrast, in HJM, forward rate processes disappear as time reaches their maturities. Note, however, that we still impose the martingale condition on bonds with fixed maturity date, since these are the financial instruments that are actually traded.