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 r_1/2 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 + r_1/2). The problem is that Julie cannot know for sure what will be the one-period interest rate r_1/2 of next year. Thus, she can only estimate a return by guessing the expectation of r_1/2.

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 r_1/2 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…..

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Forward Pricing in Commodity Markets. Note Quote.

We use the Hilbert space

H_α := {f ∈ AC(R₊,C) : ∫^∞₀ |f′(x)|² e^αx dx < ∞}

where AC(R₊,C) denotes the space of complex-valued absolutely continuous functions on R₊. We endow H_α with the scalar product ⟨f,g⟩α := f(0) g^–(0) + ∫^∞₀ f′(x) g(x) e^αx dx, and denote the associated norm by ∥ · ∥_α. Filipović shows that (H_α, ∥ · ∥_α) is a separable Hilbert space. This space has been used in Filipović for term structure modelling of bonds and many mathematical properties have been derived therein. We will frequently refer to H_α as the Filipović space.

We next introduce our dynamics for the term structure of forward prices in a commodity market. Denote by f (t, x) the price at time t of a forward contract where time to delivery of the underlying commodity is x ≥ 0. We treat f as a stochastic process in time with values in the Filipović space H_α. More specifically, we assume that the process {f(t)}_t≥0 follows the HJM-Musiela model which we formalize next. The Heath–Jarrow–Morton (HJM) framework is a general framework to model the evolution of interest rate curve – instantaneous forward rate curve in particular (as opposed to simple forward rates). When the volatility and drift of the instantaneous forward rate are assumed to be deterministic, this is known as the Gaussian Heath–Jarrow–Morton (HJM) model of forward rates. For direct modeling of simple forward rates the Brace–Gatarek–Musiela model represents an example.

On a complete filtered probability space (Ω,{F_t}_t≥0,F,P), where the filtration is assumed to be complete and right continuous, we work with an H_α-valued Lévy process {L(t)}_t≥0 for the construction of H_α-valued Lévy processes). In mathematical finance, Lévy processes are becoming extremely fashionable because they can describe the observed reality of financial markets in a more accurate way than models based on Brownian motion. In the ‘real’ world, we observe that asset price processes have jumps or spikes, and risk managers have to take them into consideration. Moreover, the empirical distribution of asset returns exhibits fat tails and skewness, behavior that deviates from normality. Hence, models that accurately fit return distributions are essential for the estimation of profit and loss (P&L) distributions. Similarly, in the ‘risk-neutral’ world, we observe that implied volatilities are constant neither across strike nor across maturities as stipulated by the Black and Scholes. Therefore, traders need models that can capture the behavior of the implied volatility smiles more accurately, in order to handle the risk of trades. Lévy processes provide us with the appropriate tools to adequately and consistently describe all these observations, both in the ‘real’ and in the ‘risk-neutral’ world. We assume that L has finite variance and mean equal to zero, and denote its covariance operator by Q. Let f₀ ∈ H_α and f be the solution of the stochastic partial differential equation (SPDE)

df(t) = ∂xf(t)dt + β(t)dt + Ψ(t)dL(t), t≥0,f(0)=f₀ 

where β ∈ L ((Ω × R₊, P, P ⊗ λ), H_α), P being the predictable σ-field, and

Ψ ∈ L²_L(H_α) := ∪_T>0 L²_L,T (H_α)

where the latter space is defined as in Peszat and Zabczyk. For t ≥ 0, denote by U_t the shift semigroup on H_α defined by U_tf = f(t + ·) for f ∈ H_α. It is shown in Filipović that {U_t}_t≥0 is a C₀-semigroup on H_α, with generator ∂x. Recall, that any C₀-semigroup admits the bound ∥U_t∥_op ≤ Me^wt for some w, M > 0 and any t ≥ 0. Here, ∥ · ∥_op denotes the operator norm. Thus s → U_t−s β(s) is Bochner-integrable (The Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions). and s → U_t−s Ψ(s) is integrable with respect to L. The unique mild solution of SPDE is

f(t) = U_tf₀ + ∫^t₀ U_t−s β(s)ds+ ∫^t₀ U_t−s Ψ(s)dL(s)

If we model the forward price dynamics f in a risk-neutral setting, the drift coefficient β(t) will naturally be zero in order to ensure the (local) martingale property (In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings and only the current event matters. In particular, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values.) of the process t → f(t, τ − t), where τ ≥ t is the time of delivery of the forward. In this case, the probability P is to be interpreted as the equivalent martingale measure (also called the pricing measure). However, with a non-zero drift, the forward model is stated under the market probability and β can be related to the risk premium in the market. In energy markets like power and gas, the forward contracts deliver over a period, and forward prices can be expressed by integral operators on the Filipović space applied on f. The dynamics of f can also be considered as a model for the forward rate in fixed-income theory. This is indeed the traditional application area and point of analysis of the SPDE. Note, however, that the original no-arbitrage condition in the HJM approach for interest rate markets is different from the no-arbitrage condition. If f is understood as the forward rate modelled in the risk-neutral setting, there is a no-arbitrage relationship between the drift β, the volatility σ and the covariance of the driving noise L.