# Financial Forward Rate “Strings” (Didactic 1) Imagine that Julie wants to invest \$1 for two years. She can devise two possible strategies. The first one is to put the money in a one-year bond at an interest rate r1. At the end of the year, she must take her money and find another one-year bond, with interest rate r1/2 which is the interest rate in one year on a loan maturing in two years. The final payoff of this strategy is simply (1 + r1)(1 + r1/2). The problem is that Julie cannot know for sure what will be the one-period interest rate r1/2 of next year. Thus, she can only estimate a return by guessing the expectation of r1/2.

Instead of making two separate investments of one year each, Julie could invest her money today in a bond that pays off in two years with interest rate r2. The final payoff is then (1 + r2)2. This second strategy is riskless as she knows for sure her return. Now, this strategy can be reinterpreted along the line of the first strategy as follows. It consists in investing for one year at the rate r1 and for the second year at a forward rate f2. The forward rate is like the r1/2 rate, with the essential difference that it is guaranteed : by buying the two-year bond, Julie can “lock in” an interest rate f2 for the second year.

This simple example illustrates that the set of all possible bonds traded on the market is equivalent to the so-called forward rate curve. The forward rate f(t,x) is thus the interest rate that can be contracted at time t for instantaneously riskless borrowing 1 or lending at time t + x. It is thus a function or curve of the time-to-maturity x2, where x plays the role of a “length” variable, that deforms with time t. Its knowledge is completely equivalent to the set of bond prices P(t,x) at time t that expire at time t + x. The shape of the forward rate curve f(t,x) incessantly fluctuates as a function of time t. These fluctuations are due to a combination of factors, including future expectation of the short-term interest rates, liquidity preferences, market segmentation and trading. It is obvious that the forward rate f (t, x+δx) for δx small can not be very different from f (t,x). It is thus tempting to see f(t,x) as a “string” characterized by a kind of tension which prevents too large local deformations that would not be financially acceptable. This superficial analogy is in the follow up of the repetitious intersections between finance and physics, starting with Bachelier who solved the diffusion equation of Brownian motion as a model of stock market price fluctuations five years before Einstein, continuing with the discovery of the relevance of Lévy laws for cotton price fluctuations by Mandelbrot that can be compared with the present interest of such power laws for the description of physical and natural phenomena. The present investigation delves into how to formalize mathematically this analogy between the forward rate curve and a string. We formulate the term structure of interest rates as the solution of a stochastic partial differential equation (SPDE), following the physical analogy of a continuous curve (string) whose shape moves stochastically through time.

The equation of motion of macroscopic physical strings is derived from conservation laws. The fundamental equations of motion of microscopic strings formulated to describe the fundamental particles derive from global symmetry principles and dualities between long-range and short-range descriptions. Are there similar principles that can guide the determination of the equations of motion of the more down-to-earth financial forward rate “strings”?

Suppose that in the middle ages, before Copernicus and Galileo, the Earth really was stationary at the centre of the universe, and only began moving later on. Imagine that during the nineteenth century, when everyone believed classical physics to be true, that it really was true, and quantum phenomena were non-existent. These are not philosophical musings, but an attempt to portray how physics might look if it actually behaved like the financial markets. Indeed, the financial world is such that any insight is almost immediately used to trade for a profit. As the insight spreads among traders, the “universe” changes accordingly. As G. Soros has pointed out, market players are “actors observing their own deeds”. As E. Derman, head of quantitative strategies at Goldman Sachs, puts it, in physics you are playing against God, who does not change his mind very often. In finance, you are playing against Gods creatures, whose feelings are ephemeral, at best unstable, and the news on which they are based keep streaming in. Value clearly derives from human beings, while mass, charge and electromagnetism apparently do not. This has led to suggestions that a fruitful framework to study finance and economy is to use evolutionary models inspired from biology and genetics.

This does not however guide us much for the determination of “fundamental” equa- tions, if any. Here, we propose to use the condition of absence of arbitrage opportunity and show that this leads to strong constraints on the structure of the governing equations. The basic idea is that, if there are arbitrage opportunities (free lunches), they cannot live long or must be quite subtle, otherwise traders would act on them and arbitrage them away. The no-arbitrage condition is an idealization of a self-consistent dynamical state of the market resulting from the incessant actions of the traders (ar- bitragers). It is not the out-of-fashion equilibrium approximation sometimes described but rather embodies a very subtle cooperative organization of the market.

We consider this condition as the fundamental backbone for the theory. The idea to impose this requirement is not new and is in fact the prerequisite of most models developed in the academic finance community. Modigliani and Miller [here and here] have indeed emphasized the critical role played by arbitrage in determining the value of securities. It is sometimes suggested that transaction costs and other market imperfections make irrelevant the no-arbitrage condition. Let us address briefly this question.

Transaction costs in option replication and other hedging activities have been extensively investigated since they (or other market “imperfections”) clearly disturb the risk-neutral argument and set option theory back a few decades. Transaction costs induce, for obvious reasons, dynamic incompleteness, thus preventing valuation as we know it since Black and Scholes. However, the most efficient dynamic hedgers (market makers) incur essentially no transaction costs when owning options. These specialized market makers compete with each other to provide liquidity in option instruments, and maintain inventories in them. They rationally limit their dynamic replication to their residual exposure, not their global exposure. In addition, the fact that they do not hold options until maturity greatly reduces their costs of dynamic hedging. They have an incentive in the acceleration of financial intermediation. Furthermore, as options are rarely replicated until maturity, the expected transaction costs of the short options depend mostly on the dynamics of the order flow in the option markets – not on the direct costs of transacting. For the efficient operators (and those operators only), markets are more dynamically complete than anticipated. This is not true for a second category of traders, those who merely purchase or sell financial instruments that are subjected to dynamic hedging. They, accordingly, neither are equipped for dynamic hedging, nor have the need for it, thanks to the existence of specialized and more efficient market makers. The examination of their transaction costs in the event of their decision to dynamically replicate their options is of no true theoretical contribution. A second important point is that the existence of transaction costs should not be invoked as an excuse for disregarding the no-arbitrage condition, but, rather should be constructively invoked to study its impacts on the models…..

# Diffeomorphic Lift Let G be a connected and simply connected Lie group, Γ ⊂ G an arbitrary totally disconnected subgroup. While it is possible to develop a general theory of fibre bundles and covering spaces in the diffeological setting, we shall directly prove some lifting properties for the quotient map π : G → G/Γ.

Lifting of diffeomorphisms: The factor space G/Γ is endowed with the quotient diffeology, that is the collection of plots α: U → G/Γ which locally lift through π to a smooth map U → G.

Proposition: Any differentiable map (in the diffeological sense) φ : G/Γ → G/Γ has a sooth lift φ : G → G, that is a C∞ map such that πφ = φπ.

Proof. By definition of quotient diffeology, the result is locally true, that is for any x ∈ G there exists an open neighbourhood Ux and a smooth map φx : Ux → G such that π ◦ φx = φ ◦ π on Ux. We can suppose that Ux is a connected open set.

Now we define an integrable distribution D on G × G in the following way. Since an arbitrary point (x, g) ∈ G × G can be written as (x, φx(x)h) for some h ∈ G, let

D(x,g) = {(v,(Rh ◦ φx)∗x(v)): v ∈ TxG} ⊂ T(x,g)(G × G).

The distribution D is well defined because two local lifts differ by some translation. In fact, let x, y ∈ G such that Ux ∩ Uy ≠ ∅. Then for any z ∈ Ux ∩ Uy and any connected neighbourhood Vz ⊂ Ux ∩ Uy, the local lifts φx, φy define the continuous map γ : Vz → Γ given by γ(t) = φx(t)−1φy(t). Since Vz is connected and the set Γ is totally disconnected, the map γ must be constant, hence φy = Rγ ◦ φx.

Moreover D has constant rank, and it is integrable, the integral submanifolds being translations of the graphs of the local lifts.

Let us choose some point x0 ∈ G such that [x0] = φ([e]), and let G~ be the maximal integral submanifold passing through (e, x0). We shall prove that the projection of G ⊂ G × G onto the first factor is a covering map. Since G is simply connected, it follows that G is the graph of a global lift.

Lemma: The projection p1: G~ ⊂ G × G → G is a covering map.

Proof. Clearly p1 is a differentiable submersion, hence an open map, so p1(G) is an open subspace of G. Let us prove that it is closed too; this will show that the map p1 : G~ → G is onto, because the Lie group G is connected.

Suppose x ∈ G is in the closure of p1(G), and let Ux be a connected open neighbourhood where the local lift φx is defined. Let y ∈ Ux ∩ p1(G), then (y, φx(y)h) ∈ G for some h ∈ G. This implies that the graph of Rh ◦ φx, which is an integral submanifold of D, is contained in G . Hence x ∈ p1(G).

It remains to prove that any x ∈ G has a neighbourhood Ux such that (p1)−1(Ux) is a disjoint union of open sets, each one homeomorphic to Ux by p1. It is clear that we can restrict ourselves to the case x = e. Let φe : Ue → G be a connected local lift of φ. We can suppose that φe(e) = x0.

Let U~e be its graph. Then U~e is an open subset of G~, containing (e, x0), with p1(U~e) = Ue. Let I be the non-empty set

I = {γ ∈Γ : (e,x0γ) ∈ G~} .

Then (p1)−1(U) is the disjoint union of the sets Rγ(U~e), γ ∈ I.

Corollary: Any diffeomorphism of G/Γ can be lifted to a diffeomorphism of G.

Corollary: Let U be a connected simply connected open subset of Rn, n ≥ 0. Any differentiable map (resp. diffeomorphism) U × G/Γ → U × G/Γ can be lifted to a C∞ map (resp. diffeomorphism) U × G → U × G.

# Diffeomorphism Diffeology via Leaves of Lie Foliation The notion of diffeological space is due to Jean-Marie Souriau.

Let M be a set. Any set map α: U ⊂ Rn → M defined on an open set U of some Rn, n ≥ 0, will be called a plot on M. The name plot is chosen instead of chart to avoid some confusion with the usual notion of chart in a manifold. When possible, a plot α with domain U will be simply denoted by αU.

A diffeology of class C on the set M is any collection P of plots α: Uα ⊂ Rnα → M, nα ≥ 0, verifying the following axioms:

1. (1)  Any constant map c: Rn → M, n ≥ 0, belongs to P;
2. (2)  Let α ∈ P be defined on U ⊂ Rn and let h: V ⊂ Rm → U ⊂ Rn beany C map; then α ◦ h ∈ P;
3. (3)  Let α: U ⊂ Rn → M be a plot. If any t ∈ U has a neighbourhood Ut such that α|Ut belongs to P then α ∈ P.

Usually, a diffeology P on the set M is defined by means of a generating set, that is by giving any set G of plots (which is implicitly supposed to contain all constant maps) and taking the least diffeology containing it. Explicitly, the diffeology ⟨G⟩ generated by G is the set of plots α: U → M such that any point t ∈ U has a neighbourhood Ut where α can be written as γ ◦ h for some C map h and some γ ∈ G.

A finite dimensional manifold M is endowed with the diffeology generated by the charts U ⊂ Rn → M, n = dimM, of any atlas.

Basic constructions. A map F : (M, P) → (N, Q) between diffeological spaces is differentiable if F ◦ α ∈ Q for all α ∈ P. A diffeomorphism is a differentiable map with a differentiable inverse.

Let (M,P) be a diffeological space and F : M → N a map of sets. The final diffeology FP on N is that generated by the plots F ◦ α, α ∈ P. A particular case is the quotient diffeology associated to an equivalence relation on M.

Analogously, let (N, Q) be a diffeological space and F : M → N a map of sets. The initial diffeology FQ on M is that generated by the plots α in M such that F ◦ α ∈ Q. A particular case is the induced diffeology on any subset M ⊂ N.

Finally, let D(M,N) be the space of differentiable maps between two diffeological spaces (M,P) and (N,Q). We define the functional diffeology on it by taking as a generating set all plots α: U → D(M,N) such that the associated map α~ : U × M → N given by α~ (t, x) = α(t)(x) is differentiable.

Diffeological groups.

Definition 2.3. A diffeological group is a diffeological space (G,P) endowed with a group structure such that the division map δ : G × G → G, δ(x, y) = xy−1, is differentiable.

A typical example of diffeological group is the diffeomorphism group of a finite dimensional manifold M, endowed with the diffeology induced by D(M,M). It is proven that the diffeomorphism group of the space of leaves of a Lie foliation is a diffeological group too.

Both constructions verify the usual universal properties.

Let (M, P), (N, Q) be two diffeological spaces. We can endow the cartesian product M × N with the product diffeology P × Q generated by the plots α × β, α ∈ P, β ∈ Q.

# Why Shouldn’t Philosophers Worry that the Detector Criterion is too Operationalist? Scattering Theory to the Rescue. Note Quote. It is not true that a representation (K,π) of U must be a Fock representation in order for states in the Hilbert space K to have an interpretation as particle states. Indeed, one of the central tasks of “scattering theory,” is to provide criteria – in the absence of full Fock space structure – for defining particle states. These criteria are needed in order to describe scattering experiments which cannot be described in a Fock representation, but which need particle states to describe the input and output states.

Haag and Swieca propose to pick out the n-particle states by means of localized detectors; we call this the detector criterion: A state with at least n-particles is a state that would trigger n detectors that are far separated in space. Philosophers might worry that the detector criterion is too operationalist. Indeed, some might claim that detectors themselves are made out of particles, and so defining a particle in terms of a detector would be viciously circular.

If we were trying to give an analysis of the concept of a particle, then we would need to address such worries. However, scattering theory does not end with the detector criterion. Indeed, the goal is to tie the detector criterion back to some other more intrinsic definition of particle states. The traditional intrinsic definition of particle states is in terms of Wigner’s symmetry criterion:

A state of n particles (of spins si and masses mi) is a state in the tensor product of the corresponding representations of the Poincaré group.

Thus, scattering theory – as originally conceived – needs to show that the states satisfying the detector criterion correspond to an appropriate representation of the Poincaré group. In particular, the goal is to show that there are isometries Ωin, Ωout that embed Fock space F(H) into K, and that intertwine the given representations of the Poincaré group on F(H) and K.

Based on these ideas, detailed models have been worked out for the case where there is a mass gap. Unfortunately, as of yet, there is no model in which Hin = Hout, which is a necessary condition for the theory to have an S-matrix, and to define transition probabilities between incoming and outgoing states. (Here Hin is the image of Fock space in K under the isometry Ωin, and similarly for Hout.)

Buchholz and collaborators have claimed that Wigner’s symmetry criterion is too stringent – i.e. there is a more general definition of particle states. They claim that it is only by means of this more general criterion that we can solve the “infraparticles” problem, where massive particles carry a cloud of photons.

The “measurement problem” of nonrelativistic QM shows that the standard approach to the theory is impaled on the horns of a dilemma: either

(i) one must make ad hoc adjustments to the dynamics (“collapse”) when needed to explain the results of measurements, or

(ii) measurements do not, contrary to appearances, have outcomes.

There are two main responses to the dilemma: On the one hand, some suggest that we abandon the unitary dynamics of QM in favor of stochastic dynamics that accurately predicts our experience of measurement outcomes. On the other hand, some suggest that we maintain the unitary dynamics of the quantum state, but that certain quantities (e.g. position of particles) have values even though these values are not specified by the quantum state.

Both approaches – the approach that alters the dynamics, and the approach with additional values – are completely successful as responses to the measurement problem in nonrelativistic QM. But both approaches run into obstacles when it comes to synthesizing quantum mechanics with relativity. In particular, the additional values approach (e.g. the de Broglie–Bohm pilot-wave theory) appears to require a preferred frame of reference to define the dynamics of the additional values, and in this case it would fail the test of Lorentz invariance.

The “modal” interpretation of quantum mechanics is similar in spirit to the de Broglie–Bohm theory, but begins from a more abstract perspective on the question of assigning definite values to some observables. Rather than making an intuitively physically motivated choice of the determinate values (e.g. particle positions), the modal interpretation makes the mathematically motivated choice of the spectral decomposition of the quantum state (i.e. the density operator) as determinate.

Unlike the de Broglie–Bohm theory, it is not obvious that the modal interpretation must violate the spirit or letter of relativistic constraints, e.g. Lorentz invariance. So, it seems that there should be some hope of developing a modal interpretation within the framework of Algebraic Quantum Field Theory…..