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

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

dlogP(t,s) = f(t,x) dt − ∫0xdy 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

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

where Cov [dtf(t,x),dtf(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) − ∫0x dyEt,dtf(t,y) + 1/2 ∫0x dy ∫0x dy′ Cov dtf(t,y) dtf(t,y′)] –

0x dy [dtf(t,y) − Et,dtf(t,y)] —– (3)

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

Et,dtf(t,x) ≡ Et [dtf(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)+ ∫0x dy Et,dtf(t,y)/dt − 1/2 ∫0x dy ∫0x dy′ c(t,y,y′) + o(1) —– (5)

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

c(t,y,y′)dt = Cov [dtf(t,y)dtf(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,

Et,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 Et,dtf(t,y)/dt. Taking the derivative with respect dt to x of the no-arbitrage condition (5), we obtain

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

Fig-3-Realizations-of-the-fractional-Brownian-sheet-on-the-plane-with-graph-dimensions

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).

Financial Forward Rate “Strings” (Didactic 1)

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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…..

Quantum Field Theory and Evolution of Forward Rates in Quantitative Finance. Note Quote.

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Applications of physics to finance are well known, and the application of quantum mechanics to the theory of option pricing is well known. Hence it is natural to utilize the formalism of quantum field theory to study the evolution of forward rates. Quantum field theory models of the term structure originated with Baaquie. The intuition behind quantum field theory models of the term structure stems from allowing each forward rate maturity to both evolve randomly and be imperfectly correlated with every other maturity. This may also be accomplished by increasing the number of random factors in the original HJM towards infinity. However, the infinite number of factors in a field theory model are linked via a single function that governs the correlation between forward rate maturities. Thus, instead of estimating additional volatility functions in a multifactor HJM framework, one additional parameter is sufficient for a field theory model to instill imperfect correlation between every forward rate maturity. As the correlation between forward rate maturities approaches unity, field theory models reduce to the standard one1 factor HJM model. Therefore, the fundamental difference between finite factor HJM and field theory models is the minimal structure the latter requires to instill imperfect correlation between forward rates. The Heath-Jarrow-Morton framework refers to a class of models that are derived by directly modeling the dynamics of instantaneous forward-rates. The central insight of this framework is to recognize that there is an explicit relationship between the drift and volatility parameters of the forward-rate dynamics in a no-arbitrage world. The familiar short-rate models can be derived in the HJM framework but in general, however, HJM models are non-Markovian. As a result, it is not possible to use the PDE-based computational approach for pricing derivatives. Instead, discrete-time HJM models and Monte-Carlo methods are often used in practice. Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle.

A Lagrangian is introduced to describe the field. The Lagrangian has the advantage over Brownian motion of being able to control fluctuations in the field, hence forward rates, with respect to maturity through the addition of a maturity dependent gradient as detailed in the definition below. The action of the field integrates the Lagrangian over time and when exponentiated and normalized serves as the probability distribution for forward rate curves. The propagator measures the correlation in the field and captures the effect the field at time t and maturity x has on maturity x′ at time t′. In the one factor HJM model, the propagator equals one which allows the quick recovery of one factor HJM results. Previous research has begun with the propagator or “correlation” function for the field instead of deriving this quantity from the Lagrangian. More importantly, the Lagrangian and its associated action generate a path integral that facilitates the solution of contingent claims and hedge parameters. However, previous term structure models have not defined the Lagrangian and are therefore unable to utilize the path integral in their applications. The Feynman path integral, path integral in short, is a fundamental quantity that provides a generating function for forward rate curves. Although crucial for pricing and hedging, the path integral has not appeared in previous term structure models with generalized continuous random processes.

Notation

Let t0 denote the current time and T the set of forward rate maturities with t0 ≤ T . The upper bound on the forward rate maturities is the constant TFR which constrains the forward rate maturities T to lie within the interval [t0, t0 + TFR].

To illustrate the field theory approach, the original finite factor HJM model is derived using field theory principles in appendix A. In the case of a one factor model, the derivation does not involve the propagator as the propagator is identically one when forward rates are perfectly correlated. However, the propagator is non trivial for field theory models as it governs the imperfect correlation between forward rate maturities. Let A(t,x) be a two dimensional field driving the evolution of forward rates f (t, x) through time. Following Baaquie, the Lagrangian of the field is defined as

Definition:

The Lagrangian of the field equals

L[A] = -1/2TFR  {A2(t, x) + 1/μ2(∂A(t,x)∂x)2} —– (1)

Definition is not unique, other Lagrangians exist and would imply different propagators. However, the Lagrangian in the definition is sufficient to explain the contribution of field theory ∂A(t,x)∂x  that controls field fluctuations in the direction of the forward rate maturity. The constant μ measures the strength of the fluctuations in the maturity direction. The Lagrangian in the definition implies the field is continuous, Gaussian, and Markovian. Forward rates involving the field are expressed below where the drift and volatility functions satisfy the usual regularity conditions.

∂f(t,x)/∂t = α (t, x) + σ (t, x)A(t, x) —– (2)

The forward rate process in equation (2) incorporates existing term structure research on Brown- ian sheets, stochastic strings, etc that have been used in previous continuous term structure models. Note that equation (2) is easily generalized to the K factor case by introducing K independent and identical fields Ai(t, x). Forward rates could then be defined as

∂f(t, x)/∂t = α (t, x) + ∑i=1K σi(t, x)Ai(t, x) —– (3)

However, a multifactor HJM model can be reproduced without introducing multiple fields. In fact, under specific correlation functions, the field theory model reduces to a multifactor HJM model without any additional fields to proxy for additional Brownian motions.

Proposition:

Lagrangian of Multifactor HJM

The Lagrangian describing the random process of a K-factor HJM model is given by

L[A] = −1/2 A(t, x)G−1(t, x, x′)A(t, x′) —– (4)

where

∂f(t, x)/∂t = α(t, x) + A(t, x)

and G−1(t, x, x′)A(t, x′) denotes the inverse of the function.

G(t, x, x′) = ∑i=1K σi(t, x) σi(t, x’) —– (5)

The above proposition is an interesting academic exercise to illustrate the parallel between field theory and traditional multifactor HJM models. However, multifactor HJM models have the disadvantages associated with a finite dimensional basis. Therefore, this approach is not pursued in later empirical work. In addition, it is possible for forward rates to be perfectly correlated within a segment of the forward rate curve but imperfectly correlated with forward rates in other segments. For example, one could designate short, medium, and long maturities of the forward rate curve. This situation is not identical to the multifactor HJM model but justifies certain market practices that distinguish between short, medium, and long term durations when hedging. However, more complicated correlation functions would be required; compromising model parsimony and reintroducing the same conceptual problems of finite factor models. Furthermore, there is little economic intuition to justify why the correlation between forward rates should be discontinuous.