Interest rates in quantum finance: The Wilson ... - APS Link Manager

PHYSICAL REVIEW E 80, 046119 共2009兲

Interest rates in quantum finance: The Wilson expansion and Hamiltonian Belal E. Baaquie* Department of Physics and Risk Management Institute, National University of Singapore, S117542 Singapore 共Received 16 March 2009; revised manuscript received 24 August 2009; published 26 October 2009兲 Interest rate instruments form a major component of the capital markets. The Libor market model 共LMM兲 is the finance industry standard interest rate model for both Libor and Euribor, which are the most important interest rates. The quantum finance formulation of the Libor market model is given in this paper and leads to a key generalization: all the Libors, for different future times, are imperfectly correlated. A key difference between a forward interest rate model and the LMM lies in the fact that the LMM is calibrated directly from the observed market interest rates. The short distance Wilson expansion 关Phys. Rev. 179, 1499 共1969兲兴 of a Gaussian quantum field is shown to provide the generalization of Ito calculus; in particular, the Wilson expansion of the Gaussian quantum field A共t , x兲 driving the Libors yields a derivation of the Libor drift term that incorporates imperfect correlations of the different Libors. The logarithm of Libor ␾共t , x兲 is defined and provides an efficient and compact representation of the quantum field theory of the Libor market model. The Lagrangian and Feynman path integrals of the Libor market model of interest rates are obtained, as well as a derivation given by its Hamiltonian. The Hamiltonian formulation of the martingale condition provides an exact solution for the nonlinear drift of the Libor market model. The quantum finance formulation of the LMM is shown to reduce to the industry standard Bruce-Gatarek-Musiela-Jamshidian model when the forward interest rates are taken to be exactly correlated. DOI: 10.1103/PhysRevE.80.046119

PACS number共s兲: 89.65.⫺s

I. INTRODUCTION

Interest rates are the return earned on cash deposits. The two main international currencies are the United States dollar and the European Union euro. Cash fixed deposits in these currencies account for almost 90% of the simple interest rates that are traded in the capital markets. Cash deposits in U.S. dollar as well as the British pound earn interest at the rate fixed by Libor and deposits in euro earn interest rates fixed by Euribor. A brief discussion of these instruments is given below for motivating the study of interest rates 关1兴. A. Libor

The interest rates offered by commercial banks for cash time deposits are often based on Libor, the London interbank offered rate 关2兴. Libor is one of the main instruments for interest rates in the debt market and is widely used for multifarious purposes. Libor was launched on 1 January 1986 by British Bankers’ Association. Libor is a daily quoted rate based on the interest rates at which commercial banks are willing to lend funds to other banks in the London interbank money market. The minimum deposit for a Libor has a par value of $1 000 000. Libor is a simple interest rate for fixed bank deposits and the British Bankers’ Association has daily Libor quotes for loans in the money market of the following duration: overnight 共24 h兲; one and two weeks; and 1, 3, 4, 5, 6, 9, and 12 months. Libors of longer duration are obtained from the interest rate swap market and are quoted for future loans of duration from 2 to 30 years. A Libor zero coupon yield curve is constructed from the swap market and is

*[email protected] 1539-3755/2009/80共4兲/046119共23兲

quoted by vendors of financial data. The Libor market is active in maturities ranging from a few days to 30 years, with the greatest depth in the 90 and 180 day time deposits. Libor is defined for duration of cash deposits that are integral multiples of ᐉ, where ᐉ is called the tenor of the deposit; in particular, L共t , Tn兲 is the forward simple interest rate, fixed at time t, for a future cash deposit from time Tn to Tn + ᐉ. Libor time is defined by Tn = nᐉ; both calendar time and future time are defined on a time lattice as shown in Fig. 1. The three-month Libor is the benchmark rate that forms the basis of the Libor derivative market. All Libor swaps, futures, caps, floors, swaptions, and so on are based on the three-month deposit. The term Libor will be taken to synonymous with the three-month Libor. In 1999 the open positions on Eurodollar futures had a par value of about U.S. $750 billion and had grown tremendously since then. The Chicago Mercantile Exchange 共CME兲 Libor futures represent one-month Libor rates on a $3 million deposit. In 2008, CME had Eurodollar futures and options on Libor with open interest of over 40 million Libor contracts and an average daily volume of 3.0 million. Libor is among the world’s most liquid short term interest rate

FIG. 1. Libor calendar and future time lattice; the tenor 共future time lattice spacing兲 is given by ᐉ = 90 days. 046119-1

©2009 The American Physical Society


BELAL E. BAAQUIE 6.0

9

5.5

8

5.0

Euribor (percentage)

Interest Rate in %

10

7 6 5 4 3

0

100

200

300

400

500

600

700

800

4.5 4.0 3.5 3.0 2.5 2.0 1.5

900

0

Trading Days

200

400

600

800

1000

1200

1400

Time series FIG. 2. Daily Libor forward interest rates L共t , t + 7 years兲, L共t , t + 6 years兲, … L共t , t + 1 year兲, and L共t , t + 0.25 years兲 with calendar time being t 苸关1996, 1999兴.

future contracts. Interest rate swaps, with Libor taken as the floating rate, currently trade on the interbank market for maturities of up to 50 years. Market data on Libor futures are given for daily time t in the form of L共t , Ti − t兲, with fixed dates of maturity Ti 共March, June, September, and December兲 and are shown in Fig. 2. B. Euribor

Euribor 共euro interbank offered rate兲 is the benchmark rate of the euro money market, which has emerged since 1999. Euribor is simple interest on fixed deposits in the euro currency; the duration of the deposits can vary from overnight, weekly, monthly, and three monthly out to long duration deposits of 10 years and longer. Euribor is sponsored by the Financial Markets Association and by the European Banking Federation, which represents 4500 banks in the 24 member states of the European Union and in Iceland, Norway, and Switzerland. Euribor is the rate at which euro interbank term deposits are offered by one prime bank to another. The choice of banks quoting for Euribor is based on market criteria. These banks are of first class credit standing and are selected to ensure that the diversity of the euro money market is adequately reflected, thereby making Euribor an efficient and representative benchmark. All the features discussed for Libor can also be applied to Euribor. Euribor was first announced on 30 December 1998 for deposits starting on 4 January 1999. Figure 3 shows daily values for Euribor forward interest rate on 90 day deposits for deposits one, two, and three years in the future. Since its launch, Euribor has been actively traded on the option markets and is the underlying rate of many derivative transactions both over the counter and exchange traded. Euribor is one of the most liquid global interest rate instruments, second only to Libor. The Euribor zero coupon yield curve, based on the rates, is contracted in the Euribor swap market and extends out to 50 years in the future. The term Libor is used for generic interest rate instruments and includes both Libor and Euribor.

FIG. 3. Euribor maturing one, two, and three years in the future from 26 May 1999 to 17 May 2004. II. LIBOR MARKET MODEL

From Figs. 2 and 3, it is clear that interest rates L共t , T兲, which at every instant t form a curve extending in future time out to almost 50 years, evolve in a random manner. It is hence natural to take the interest rates to be a random function; in other words, L共t , T兲 is taken to be a random twodimensional stochastic field. The observed market behavior of interest rates is obtained by averaging over all possible values of the stochastic field. Since the functional averaging is identical to the functional averaging over all possible configurations of a quantum field, interest rates L共t , T兲 are modeled as a two-dimensional Euclidean quantum field 关3兴. Interest rate instruments can be modeled using either the zero coupon bonds B共t , T兲 or the simple interest Libor L共t , T兲. Both these approaches are, in principle, equivalent but are quite different from an empirical, computational, and analytical point of view. B共t , T兲 is the price of a zero coupon bond—at time t—that pays a prefixed amount, say 1 euro, at future time T. Recall L共t , T兲 is the simple interest, agreed at time t, for a fixed deposit from future time T to T + ᐉ. The forward interest rate, denoted by f共t , x兲, is the interest rate, fixed at time t, for an instantaneous loan at some future time x ⬎ t; note that x and t refer to future and calendar time, respectively. One can take the view that there exists one set of underlying forward interest rates f共t , x兲 that can be used for modeling both L共t , T兲 and zero coupon bonds B共t , T兲. Forward interest rates are strictly positive, that is, f共t , x兲 ⱖ 0. The positivity of f共t , x兲 is intuitively obvious and also required by absence of arbitrage. The Heath-Jarrow-Morton 共HJM兲关4兴 model of f共t , x兲—and its quantum finance generalization 关2兴—goes a long way in accurately modeling interest rate instruments. However, the HJM model and its quantum finance generalization have one serious shortcoming: both allow f共t , x兲 to be negative with a finite probability, which, in turn, implies that simple interest rate L共t , T兲 has a finite probability of being negative. Giving up f共t , x兲 ⱖ 0 does not pose a very serious problem for the bond sector since B共t , T兲 is strictly positive even for those configurations for which f共t , x兲 ⱕ 0. However, for the interest rate sector of the debt market, a model that allows Libor to be negative can yield results that allow for arbitrage

046119-2


INTEREST RATES IN QUANTUM FINANCE: THE WILSON…

and hence are not permissible as a consistent model for interest rate instruments. One needs to go beyond modeling f共t , x兲 and instead develop a model based directly on Libor L共t , T兲. The Libor market model 共LMM兲 aims at modeling interest rates in terms of debt instruments that are directly traded in the financial markets. In particular, forward interest rates f共t , x兲 are not directly traded, but instead what are traded are 共a兲 Libor and Euribor for fixed time deposits and 共b兲 zero coupon bonds B共t , T兲 as well as coupon bonds. LMM takes the traded values of Libor L共t , T兲 to be the main ingredient in modeling interest rates—instead of deriving Libor from an underlying Libor forward interest rate model. In the LMM all Libors are strictly positive: L共t , T兲 ⬎ 0. Zero coupon bonds and the Libor forward interest rates are both derived from Libor instead from f共t , x兲. Strictly positive Libor has the added advantage that all zero coupon bonds and hence coupon bonds as well are all strictly positive. The LMM approach was pioneered by Bruce-GatarekMusiela 共BGM兲关5兴 and Jamshidian 关6兴, with many of its subsequent developments discussed by Rebonata 关7,8兴. One of the biggest achievements of the LMM is a derivation of Black’s formula for pricing interest rate caplets from an arbitrage free model—something that many experts thought was not possible. Various extensions of the LMM have been made; Anderson and Andresean 关9兴 and Joshi and Rebonata 关10兴 incorporated stochastic volatility into the LMM whereas Labordere 关11兴 combined LMM with the stochastic alpha, beta, rho 共SABR兲 model 关12兴. The calibration and applications of BGM-Jamshidian model have been extensively studied 关7,13兴. In the BGM-Jamshidian approach, similar to the HJM modeling of the forward interest rates, all Libors for different future times are exactly correlated. In contrast, in the quantum finance formulation, Libors are driven not by white noise but rather by the two-dimensional stochastic field A共t , x兲. The values of all Libor instruments are given by averaging A共t , x兲 over all its possible values. Hence, A共t , x兲 is mathematically equivalent to a two-dimensional quantum field. The equal time Wilson expansion 关14,15兴 of the bilinear product of the quantum field A共t , x兲 is discussed in Sec. V and provides a generalization of Ito calculus. The key link in deriving the quantum finance version of the LMM and in, particular, of the Libor drift is the singular property of the bilinear product of Gaussian quantum field A共t , x兲. The quantum finance generalization of the Libor market model contains crucial correlation terms reflecting the imperfect correlation of the different Libors and avoids systematic errors that arise from the assumption of perfectly correlated Libors. A derivation of Libor drift independent of the Wilson expansion and based on the Hamiltonian formulation of the Libor market model is given in Sec. XV. The LMM is driven by f共t , x兲, the Libor forward interest rates, which is distinct from both the empirical forward interest rates and the bond forward interest rates. It is shown that f共t , x兲 has a nonlinear evolution equation with both its drift and volatility being stochastic. Libor forward interest

FIG. 4. Libor future time.

rates are strictly positive and nonsingular, being finite for all calendar and future time. An efficient description of Libor instruments is obtained by doing a nonlinear change of independent variables from f共t , x兲 to L共t , Tn兲 and then to logarithmic Libor field, namely, ␾共t , x兲. It is shown that, when the limit of perfectly correlated Libor is taken, the quantum finance LMM reduces to the BGM-Jamshidian model and, in turn, yields—in the limit of zero Libor tenor 共ᐉ → 0兲—the HJM model for the bond forward interest rates.

III. LIBOR AND ZERO COUPON BONDS

In terms of Libor forward interest rates f共t , x兲 Libor zero coupon bond B共t , T兲 is defined as

再冕冎冋再冕冎册 T

B共t,T兲 = exp −

dxf共t,x兲

共1兲

dxf共t,x兲 − 1 .

共2兲

t

and Libor L共t , Tn兲 is given by L共t,Tn兲 =

1 exp ᐉ

Tn+ᐉ

Tn

Libor zero coupon bonds B共t , T兲 are not actual instruments traded in the market but rather a way of encoding the discounting of future cash flows consistent with all the Libors. The price of a traded zero coupon treasury bond B共t , T兲 is not equal to a Libor bond B共t , T兲, but these differences are small and will be ignored. From Eq. 共2兲, Libor is given by L共t,Tn兲 =

B共t,Tn兲 − B共t,Tn+1兲 ᐉB共t,Tn+1兲

and which yields B共t,Tn + ᐉ兲 =

B共t,Tn兲 . 1 + ᐉL共t,Tn兲

共3兲

Equation 共3兲 provides a recursion equation that allows one to express B共t , Tn兲 solely in terms of L共t , Tn兲. Recall that Libors are only defined for discrete future time given by Libor future time T = Tn = nᐉ, n = 0 , ⫾ 1 , ⫾ 2 , . . . , ⫾ ⬁. Libor future time lattice is shown in Fig. 4. Hence, from Eq. 共3兲 k

B共t,Tk+1兲 =

1 B共t,Tk兲 , = B共t,T0兲兿 1 + ᐉLk共t兲 n=0 1 + ᐉLn共t兲

where Ln共t兲 ⬅ L共t , Tn兲. Bonds B共t , T0兲 that have time t not at a Libor time ᐉk cannot be expressed solely in terms of Libor rates. Consider only zero coupon bonds that are issued at Libor time, say T0, and mature at another Libor time Tk+1; since B共T0 , T0兲 = 1, B共T0 , Tk+1兲 can be expressed entirely in terms of Libor as follows:

046119-3


BELAL E. BAAQUIE

E关R共t兲R共t⬘兲兴 = ␦共t − t⬘兲.

E关R共t兲兴 = 0, B (T0 , Tn +1 )

F (t0, T0 , Tn +1 )

FIG. 5. The zero coupon bond B共T0 , Tn+1兲 is issued at T0 and expires at Tn + ᐉ. Its forward Libor bond price F共t0 , T0 , Tn+1兲 is given at present 共Libor兲 time t0 = T−k. k

B共T0,Tk+1兲 = 兿

n=0

1 . 1 + ᐉLn共T0兲

The volatility functions ␴共t , x兲 , ␥k共t兲 are deterministic. The drift ␣共t , x兲 is deterministic in the HJM model, whereas Libor drift ␨k共t兲 depends on Libors Lk共t兲 for the BGMJamshidian model. Future time x and Tk have been introduced in both the HJM and BGM-Jamshidian models only in the drift and volatility of the interest rate term structure. A single white noise R共t兲 drives the entire forward interest rate curve and leads, as follows, to perfectly correlated rates: E

共4兲 E

Let us present time be t0 = T−k. Suppose a zero coupon bond B共T0 , Tn + ᐉ兲 is going to be issued at some future time T0 ⬎ t0 = T−k, with expiry at time Tn + ᐉ; the zero coupon bond and its forward price are defined for Libor time and shown in Fig. 5. The forward bond price is the price—at present time t0—of a zero coupon bond that is to be issued at some future time T0 ⬎ t0. From Eq. 共4兲, the forward bond price is given by

=

B共t0,Tn+1兲 B共t0,T0兲

再

n

兿 i=−k

n

=兿 i=0

冎再

1 / 1 + ᐉL共t0,Ti兲

−1

1

兿 i=−k 1 + ᐉL共t0,Ti兲

1 . 1 + ᐉLi共t0兲

冎

共5兲

IV. LIBOR MARKET MODEL AND QUANTUM FINANCE

The Libor market model is defined in the framework of quantum finance by defining the time evolution of the Libor rates L共t , T兲. Modeling in finance widely uses the concept of stochastic differential equations. The bond forward interest rates f B共t , x兲 and Libor Lk共t兲, as defined by the HJM and BGMJamshidian models, respectively, are both expressed as functions of white noise and given by

⳵ f B共t,x兲 = ␣共t,x兲 + ␴共t,x兲R共t兲 ⳵t 1 ⳵ Lk共t兲 = ␨k共t兲 + ␥k共t兲R共t兲 Lk共t兲 ⳵ t

册

⳵ f B共t,x兲/⳵ t − ␣共t,x兲 ⳵ f B共t⬘,x⬘兲/⳵ t⬘ − ␣共t⬘,x⬘兲 = ␦共t − t⬘兲, ␴共t,x兲 ␴共t⬘,x⬘兲

冋

−1

L 共t⬘兲 ⳵ Lk⬘共t⬘兲/⳵ t⬘ − ␨k⬘共t⬘兲 L−1 k 共t兲 ⳵ Lk共t兲/⳵ t − ␨k共t兲 k⬘ ␥k共t兲 ␥k⬘共t⬘兲

= ␦共t − t⬘兲.

Forward bond price and Libor

F共t0,T0,Tn + ᐉ兲 ⬅

冋

共HJM model兲,

共8兲

Note that the right-hand side of above equations is independent of x , x⬘ and Tk , Tk⬘, respectively, showing perfect correlation in future time. The quantum finance model of the bond forward interest rates f B共t , x兲 given in 关2兴 and its HJM limit have a major unavoidable side effect: there is a finite probability that f B共t , x兲 can take negative values. Since, empirical forward interest rates can never be negative, the Libor market model takes the view that for the debt market one should replace the bond forward interest rates f B共t , x兲 by strictly positive Libor forward interest rates f共t , x兲. These rates are used for modeling all interest rate instruments and, in particular, yield all L共t , T兲 as always being positive. In the Libor market model, market interest rates L共T0 , Tn兲 and coupon and zero coupon bonds B共TN兲 and B共Tn , TN兲—given at Libor times Tn , TN, respectively—are expressed solely in terms of Libor L共T0 , Tn兲, as in Eq. 共4兲, without any direct reference to the underlying Libor forward interest rates f共t , x兲. Moreover, positive Libor rates automatically yield coupon and zero coupon bonds that are strictly positive, as seen in Eq. 共4兲. Only the quantum finance differential formulation of the Libor forward interest rates is the main focus of this paper; a similar generalization of the bond forward interest rates and of the HJM model has been extensively discussed in 关2兴. The BGM-Jamshidian model of the LMM is generalized by “promoting” white noise R共t兲 to a two-dimensional quantum field A共t , x兲 and yields the following:

⳵ f共t,x兲 = ␮共t,x兲 + v共t,x兲A共t,x兲, ⳵t

共6兲 f共t,x兲 = f共t0,x兲 +

冕

t

dt␮共t,x兲 +

t0

共BGM-Jamshidian model兲,

册

冕

共9兲

t

dtv共t,x兲A共t,x兲. 共10兲

t0

In terms of the Libors, the quantum LMM is given by

共7兲 where R共t兲 is Gaussian white noise, 046119-4

1 ⳵ L共t,Tk兲 = ␨k共t兲 + L共t,Tk兲 ⳵ t

冕

Tk+1

Tk

dx␥共t,x兲A共t,x兲.

共11兲



In the LMM formulation of Libor forward interest rates f共t , x兲 both the drift ␮共t , x兲 and volatility v共t , x兲 are stochastic and depend on f共t , x兲; in particular, Libor volatility v共t , x兲 ⬀ 关1 − exp兵−ᐉf共t , x兲其兴. The dynamics of the quantum field A共t , x兲 is given by the “stiff Lagrangian” 关16兴,

The singular piece of R2共t兲 is deterministic and to leading order is equal to 1 / ⑀; all the random terms that occur for R2共t兲 are finite as ⑀ → 0. For Gaussian quantum fields such as A共t , x兲, which have a quadratic action, one can give differential formulation of the theory of forward interest rates, as given in Eq. 共9兲, and that is similar to the HJM and BGM formulations. This is possible because the full content of a Gaussian 共free兲 quantum field, as discussed in Appendix A, is encoded in its propagator. Similar to white noise, the correlation function E关A共t , x兲A共t⬘ , x⬘兲兴 is infinite for t = t⬘ 共equal calendar time兲. The product of nonlinear 共non-Gaussian兲 quantum fields is the subject matter of the short “distance” Wilson expansion 关14兴. The singular product of two Gaussian quantum fields is the simplest case of the Wilson expansion and the singularity, similar to Eq. 共16兲, is expressed as follows:

L关A兴 = −

再

冉

1 ⳵ A共t,z兲 1 A2共t,z兲 + 2 2 ␮ ⳵z z = 共x − t兲␯ ;

冊冉 2

+

1 ⳵2A共t,z兲 ␭4 ⳵ 2z

冊冎

z 苸关0,⬁兴.

2

,

共12兲

The quantum field A共t , z兲 satisfies the Neumann boundary conditions

冏

⳵ A共t,z兲 ⳵t

冏

= 0. z=0

The provenance of the stiff quantum field A共t , z兲 is from the market behavior of interest rates 关16兴; to explain the empirical behavior of both Euribor and Libor, one needs to introduce remaining market time z = 共x − t兲␯ as well as strongly correlate the field A共t , z兲 using fourth-order derivatives 关17兴. The Libor market model is quantized in Appendix A using the Feynman path integral. The correlation function of the two-dimensional quantum field A共t , x兲 is given in Eq. 共A6兲 by E关A共t,x兲兴 = 0,

共13兲

E关A共t,x兲A共t⬘,x⬘兲兴 = ␦共t − t⬘兲D共x,x⬘ ;t兲,

共14兲

M v共x,x⬘ ;t兲 = v共t,x兲D共x,x⬘ ;t兲v共t,x⬘兲.

共15兲

1 A共t,x兲A共t,x⬘兲 = D共x,x⬘ ;t兲 + O共1兲. ⑀

The correlation of A共t , x兲A共t , x⬘兲 is singular for t = t⬘—very much like the singularity of white noise R共t兲. All the fluctuating components, which are contained in A共t , x兲A共t , x⬘兲, are regular and finite as ⑀ → 0. Since, for each x and each t, A共t , x兲 is an integration variable one may question as to how can one assign it a deterministic numerical value as in Eq. 共17兲. What Eq. 共17兲 means is that in any correlation function, wherever a product of fields is at the same time, namely, A共t , x兲A共t , x⬘兲, then—to leading order in ⑀—the product can be replaced by the deterministic quantity D共x , x⬘ ; t兲 / ⑀. In terms of symbols, Eq. 共17兲 states the following:

As expected, the Libor forward interest rates are imperfectly correlated,

冋

equal time

册

E关A共t1,x1兲A共t2,x2兲 . . . A共t,xn兲A共t,xn+1兲 . . . A共tN,xN兲兴

⳵ f共t,x兲/⳵ t − ␮共t,x兲 ⳵ f共t⬘,x⬘兲/⳵ t⬘ − ␮共t⬘,x⬘兲 E = ␦共t v共t,x兲 v共t⬘,x⬘兲 − t⬘兲D共x,x⬘ ;t兲

1 = E关A共t1,x1兲A共t2,x2兲 . . . A共tn−1,xn−1兲D共xn,xn+1 ;t兲 ⑀

共imperfectly correlated兲.

⫻ A共tn+2,xn+2兲 . . . A共tN,xN兲兴 + O共1兲. As discussed in 关17兴, one can choose the normalization of ␴共t , x兲 so that D共x , x ; t兲 = 1 / ⑀ and which yields from Eq. 共17兲

V. WILSON EXPANSION OF QUANTUM FIELD A(t , x)

Modeling in finance widely uses the concept of stochastic differential equations. The time derivative of various quantities such as a security S共t兲 is generically expressed as follows: dS共t兲 = ␮共t兲 + ␴共t兲R共t兲. dt The HJM and BGM-Jamshidian interest rate models are examples of stochastic differential equations, as can be seen from Eqs. 共6兲 and 共7兲. Ito’s stochastic calculus, for discrete time t = n⑀, is a result of the following identity 关2兴: E关R共t兲R共t兲兴 = ␦共t − t⬘兲 ⇒ R2共t兲 =

1 + O共1兲. ⑀

共17兲

共16兲

A2共t,x兲 =

1 + O共1兲, ⑀2

共18兲

showing even more clearly the similarity of Eqs. 共16兲 and 共18兲. The HJM model is a special case of quantum finance, given by taking the limit A共t,x兲 → R共t兲,

D共x,x⬘ ;t兲 → 1 ⇒ E关A共t,x兲A共t⬘,x⬘兲兴

→ E关R共t兲R共t兲兴 = ␦共t − t⬘兲.

共19兲

Since Eqs. 共16兲 and 共17兲 have a similar singularity structure, one expects that there should be a natural generalization of Ito calculus for Gaussian quantum fields. The singularity of the equal time quadratic product of the quantum field, in

046119-5


BELAL E. BAAQUIE

placed on either the drift ␮共t , x兲 and/or the volatility v共t , x兲, both of which can be arbitrary nonlinear functions of f共t , x兲. The bond portfolio Xn is a martingale, as discussed in Eq. 共C6兲 of Appendix C, if and only if E FIG. 6. Libor time lattice for the forward bond numeraire B共t , TI+1兲 with 共i兲 TI+1 ⬍ Tn and 共ii兲 TI+1 ⬎ Tn.

particular, leads to a differential formulation of the martingale condition for discounted zero coupon bonds and is discussed in Appendix B.

冋册 ⳵ Xn共t兲 ⳵t

The random terms in Eq. 共21兲 are proportional to A共t , x兲. Since, from Eq. 共13兲, E关A共t , x兲兴 = 0, the martingale condition given in Eq. 共21兲 requires that the drift—namely, terms independent of A共t , x兲—must be zero and yields

冕

Tn

1 2

dx␮I共t,x兲 =

TI+1

VI. LIBOR MARTINGALES AND FORWARD BOND NUMERAIRE

L共t,Tn兲B共t,Tn+1兲 B共t,TI+1兲

共martingales兲.

共20兲

Note that, for n = I, the portfolio XI共t兲 is equal to LI共t兲 ⬅ L共t , TI兲; hence, for the forward bond numeraire given by zero coupon bond B共t , TI+1兲, the Libor rate L共t , TI兲 is a martingale. As shown in 共i兲 and 共ii兲 in Fig. 6, time Tn can be either less than, equal to, or greater than TI. In terms of the Libor forward interest rates, from Eqs. 共1兲 and 共2兲, the martingale is

再冕

ᐉXn共t兲 = exp −

Tn

冎再冕

dxf共t,x兲 − exp −

TI+1

Differentiating portfolio Xn共t兲 using Eqs. 共9兲 and 共14兲 and the rules derived in Appendix B yields the following: ᐉ

⳵ Xn共t兲 ⳵t

冋冕

Tn

dx␮共t,x兲 +

= −

TI+1

冕冋冕冕

Tn

−

Tn+1

TI+1

Tn+1

+

TI+1

冕

dxdx⬘M v共x,x⬘ ;t兲

TI+1

册冋冕

dxv共t,x兲A共t,x兲 exp −

TI+1

+

1 2

Tn

1 dx␮共t,x兲 − 2

冕

Tn

dxf共t,x兲

TI+1

Tn+1

册冋冕

dxv共t,x兲A共t,x兲 exp −

Tn+1

TI+1

dxdx⬘M v共x,x⬘ ;t兲.

TI+1

x

dx⬘M v共x,x⬘ ;t兲.

共22兲

VII. TIME EVOLUTION OF LIBOR

The drift term, as given in Eq. 共22兲, is expressed in terms of the Libor forward interest rate volatility function v共t , x兲. The main theoretical objective of the Libor market model is to completely remove v共t , x兲 from the Libor evolution equation. More specifically, the objective is to express the drift of the Libor rates in terms of deterministic Libor volatility ␥共t , x兲关defined later in Eq. 共26兲兴. Consider the definition of Libor given in Eq. 共2兲 and choose the drift ␮ to be equal to the ␮I given in Eq. 共22兲. Equation 共9兲 and the Wilson expansion for A共t , x兲 given in Eq. 共17兲 yield ᐉ

再冕冕冕

⳵ L共t,Tn兲 = exp ⳵t +

Tn+1

dxf共t,x兲

Tn

冎冋冕

Tn+1

dx␮I共t,x兲

Tn

Tn+1

1 2

dxM v共x,x⬘ ;t兲

Tn

Tn+1

+

册

dxv共t,x兲A共t,x兲 .

Tn

共23兲

The drift for ⳵L共t , Tn兲 / ⳵t, from Eq. 共22兲, has the following simplification,

冕

册

Tn+1

Tn

dx␮I共t,x兲 +

冕冋冕冕冕冕冕 Tn+1

=

Note that in obtaining ⳵Xn共t兲 / ⳵t no condition has been 046119-6

dx⬘ +

dx

TI+1

冕

x

Tn

册

dx⬘ M v共x,x⬘ ;t兲

Tn

dx⬘M v共x,x⬘ ;t兲

Tn

Tn+1

Tn


x

dx

Tn

=

Tn+1

Tn

TI+1

Tn+1

册

冕

Tn

+

dxf共t,x兲 .

1 2

dx

Tn


TI+1

Tn

TI+1

dxf共t,x兲 .

TI+1

冕

␮I共t,x兲 =

冎

Tn+1

冕

The martingale condition given above is satisfied by choosing the following value for drift:

The factor for discounting of the future cash flows of traded instruments is of fundamental importance in finance and is called a numeraire 关2兴. A wide class of numeraires can be used to render all traded assets into martingales. Choose the zero coupon bond B共t , TI+1兲, with fixed index I, as the forward bond numeraire. The combination L共t , Tn兲B共t , Tn+1兲, from Eq. 共2兲, is equivalent to a portfolio of zero coupon bonds and hence is a traded asset. By a suitable choice of the drift, all traded assets can be made into martingales. In particular, all instruments Xn共t兲, defined below, for n = 0 , ⫾ 1 , ⫾ 2 , . . . , ⫾ ⬁ are martingales 关18兴; in other words, for all n Xn共t兲 ⬅

共21兲

= 0.




冕冕冕冕 Tn+1

+

=

Tn+1

dx

Tn

Tn

Tn+1

Tn+1

dx

v共t,x兲 ⯝ 关1 − e−ᐉf共t,x兲兴␥共t,x兲.


The following are the two limiting cases:

dx⬘M v共x,x⬘ ;t兲,

共24兲

Tn

TI+1

and yields, from Eqs. 共2兲, 共23兲, and 共24兲, the following,

⳵ L共t,Tn兲 = ⳵t

冋冕

Tn+1

dx

冕

Tn+1

冕

Tn+1


Tn

TI+1

+

v共t,x兲 =

dxv共t,x兲A共t,x兲

Tn

册

关1 + ᐉL共t,Tn兲兴 . ᐉ 共25兲

冕

Tn+1

dx␥共t,x兲A共t,x兲. 共26兲

Tn

Volatility ␥共t , x兲 is a deterministic function—independent of L共t , Tn兲. The drift ␨共t , Tn兲 is a nonlinear function of L共t , Tn兲 that is determined by the martingale condition. Volatility ␥共t , x兲 and drift ␨共t , Tn兲 are discussed in Secs. VIII and IX, respectively. VIII. VOLATILITY ␥(t , x) FOR POSITIVE LIBOR

A key assumption of the Libor market model is that the Libor volatility function ␥共t , x兲 is a deterministic function that is independent of the Libor rates. The main results of this section is to give an explicit derivation for Eq. 共26兲 and verify that ␥共t , x兲 is, in fact, a deterministic function. The market value for ␥共t , x兲—for Libor and Euribor—has been obtained in 关17兴. As it stands, Eq. 共25兲 for ⳵L共t , Tn兲 / ⳵t does not imply that the Libor interest rates L共t , Tn兲 are strictly positive. Libors are strictly positive only if Eq. 共26兲 holds; namely, if there exists a Libor volatility function ␥共t , x兲 such that, from Eqs. 共25兲 and 共26兲,

冕

Tn+1

dxv共t,x兲A共t,x兲 =

Tn

⇒ v共t,x兲 =

ᐉL共t,Tn兲 1 + ᐉL共t,Tn兲

ᐉL共t,Tn兲 ␥共t,x兲, 1 + ᐉL共t,Tn兲

冕

Tn+1

dx␥共t,x兲A共t,x兲

Tn

x 苸关Tn,Tn+1兲.

ᐉ␥共t,x兲f共t,x兲, ᐉf共t,x兲 Ⰶ 1

␥共t,x兲,

ᐉf共t,x兲 Ⰷ 1.

冎

共29兲

For small values of f共t , x兲, the volatility v共t , x兲 is proportional to f共t , x兲. It is known 关19兴 that Libor forward interest rates f共t , x兲 with volatility v共t , x兲 ⯝ f共t , x兲 are unstable and diverge after a finite time. However, in Libor market model, when the Libor forward rates become large, that is ᐉf共t , x兲 Ⰷ 1, the volatility v共t , x兲 becomes deterministic and equals ␥共t , x兲. It is shown in Appendix E that Libor forward interest rates f共t , x兲 are never divergent and Libor dynamics yields finite f共t , x兲 for all future and calendar times.

Note that, as expected, the drift is zero for n = I, making XI共t兲 = L共t , TI兲 a martingale. Libor drift ␨共t , Tn兲 and volatility ␥共t , x兲 are defined as follows: 1 ⳵ L共t,Tn兲 = ␨共t,Tn兲 + L共t,Tn兲 ⳵ t

再

IX. LIBOR DRIFT FOR MARTINGALES ␹n(t)

The main motivation for introducing the Libor market model is to have manifestly positive interest rates and bonds. To ensure that the Libor rates L共t , Tn兲 are always positive, it is sufficient to show that they are the exponential of real variables. To obtain positive Libor rates requires a nontrivial drift; a quantum finance derivation of the drift term is given in this section and generalizes earlier results of the BGMJamshidian approach. The drift ␨共t , Tn兲 in Eq. 共26兲 is chosen to make ␹n共t兲—given in Eq. 共20兲—a martingales for all n. One needs to express the Libor drift ␨共t , Tn兲 solely in terms of Libor volatility function ␥共t , x兲. The Libor drift term ␨共t , Tn兲 is defined, from Eqs. 共25兲 and 共26兲, as follows:

␨共t,Tn兲 =

关1 + ᐉL共t,Tn兲兴 ᐉL共t,Tn兲 ⫻

冕

Tn+1

dxv共t,x兲

TI+1

冕

Tn+1

dx⬘D共x,x⬘ ;t兲v共t,x⬘兲.

Tn

共30兲 The Libor forward interest rate volatility function v共t , x兲 needs to be expressed in terms of the Libor volatility function ␥共t , x兲. To do so, a recursion equation is obtained from Eq. 共27兲 in the following manner. Multiply both sides of Eq. 共27兲 by A共t , x⬘兲v共t , x⬘兲 and use Eq. 共17兲, namely,

共27兲

1 A共t,x兲A共t,x⬘兲 = D共x,x⬘ ;t兲. ⑀

共28兲

This removes the quantum field from Eq. 共27兲 and, by equating the 1 / ⑀ term from both sides of the resulting equation, one obtains

In the Libor market model, v共t , x兲 yields a model of the Libor forward interest rates with stochastic volatility. Equation 共27兲 can be viewed as fixing the volatility function v共t , x兲 of the forward interest rates f共t , x兲 so as to ensure that ␥共t , x兲 is deterministic and leads to all Libor L共t , Tn兲 being strictly positive. To have a better understanding of v共t , x兲 consider the limit of ᐉ → 0, which yields 兰TTn+1dxf共t , x兲 ⯝ ᐉf共t , x兲. From Eqs. 共2兲 n and 共28兲

冕

Tn+1

dxv共t,x兲D共x,x⬘ ;t兲v共t,x⬘兲

Tn

=


冕

Tn+1

dx␥共t,x兲D共x,x⬘ ;t兲v共t,x⬘兲. 共31兲

Tn

Since the dynamics of L共t , Tn兲 is being analyzed, we integrate variable x⬘ from Tn to Tn+1 and obtain

046119-7


BELAL E. BAAQUIE

冕冕 Tn+1

Tn+1

dx

Tn

=

Calendar time

dx⬘v共t,x兲D共x,x⬘ ;t兲v共t,x⬘兲

Tn


冕

Tn+1

dxv共t,x兲␻n共t,x兲,

T0

共32兲

M γ ( x, x’; t )

Tn

where ␻n共t , x兲 is defined by

␻n共t,x兲 =

冕

Tn+1

t

dx⬘D共x,x⬘ ;t兲␥共t,x⬘兲.

t0

Hence, from Eqs. 共32兲 and 共30兲

冕

␨共t,Tn兲 =

dxv共t,x兲␻n共t,x兲.

共34兲

=

冕

Tn+1

冕

dxv共t,x兲␻n共t,x兲 =

Tn

兺

m=0

ᐉL共t,Tm兲 1 + ᐉL共t,Tm兲

n

冕

Tm+1

dx␥共t,x兲␻n共t,x兲

Tm

ᐉL共t,T 兲

m ⌳mn共t兲, 兺 1 + ᐉL共t,T m兲 m=0

共38兲

Tn


冕

冕冕冕冕冕 Tm+1

⌳mn共t兲 ⬅ Tn+1

dx␥共t,x兲␻n共t,x兲

Tm

dxdx⬘M ␥共x,x⬘ ;t兲,

Tm+1

Tn

=

M ␥共x,x⬘ ;t兲 ⬅ ␥共t,x兲D共x,x⬘ ;t兲␥共t,x⬘兲.

共36兲

The recursion equation dxv共t,x兲␻n共t,x兲 =

t

冕冕 Tn

dxv共t,x兲␻n共t,x兲

+


dx⬘M ␥共x,x⬘ ;t兲.

共39兲

Tn

There are three cases for ␨共t , Tn兲 corresponding to Tn = TI, Tn ⬎ TI, and Tn ⬍ TI, as shown in Fig. 6. Case 共i兲 Tn = TI. From Eq. 共34兲

␨共t,TI兲 = 0.

t

Tn+1

Tn+1

dx

Tm

dx⬘␥共t,x兲D共x,x⬘ ;t兲␥共t,x⬘兲

Tn

Tm+1

=

Tn+1

dx

Tm

where

冕

Future time

where, as shown in Fig. 7, the Libor propagator is given by

dx␥共t,x兲D共x,x⬘ ;t兲␥共t,x⬘兲.

共35兲

Tn+1

T n+1

dxv共t,x兲␻n共t,x兲 n

=

dxv共t,x兲D共x,x⬘ ;t兲␥共t,x⬘兲


Tn

T m+1

T0

Integrating x⬘ from Tn to Tn+1 yields Tn+1

Tn+1

=

Tn

Tm

T0

冕

The drift obtained in Eq. 共34兲 still depends on the volatility function v共t , x兲. To express this integral solely in terms of the volatility function ␥共t , x兲 one has to carry out a calculation similar to the one used in obtaining Eq. 共34兲. Multiplying both sides of Eq. 共27兲, this time by A共t , x⬘兲␥共t , x⬘兲, and using Eq. 共17兲 yield the following:

冕

t0

FIG. 7. Libor propagator ⌳mn共t兲 yields nontrivial and imperfect correlation between the different Libors.

TI+1

Tn+1

x’

共33兲

Tn

Tn+1

x

共37兲

Case 共ii兲 Tn ⬎ TI. Equation 共38兲 yields the following:

Tn

yields, from Eqs. 共35兲, the following:

冕 t

Tn+1

dxv共t,x兲␻n共t,x兲 =

冕

Tn


冕

= Tn+1

冕冕

Tn+1


TI+1


t

+

␨共t,Tn兲 =

n

=

For simplicity, let time t = T0; recursing above equation yields, using Eq. 共33兲, the following: 046119-8

dxv共t,x兲␻n共t,x兲 −

T0

dx␥共t,x兲␻n共t,x兲.

Tn

Tn+1

冕

TI+1


T0

ᐉL共t,T 兲

m ⌳mn共t兲. 兺 1 + ᐉL共t,T m兲 m=I+1

Case 共iii兲 Tn ⬍ TI. From Eq. 共38兲, one has the following:



␨共t,Tn兲 =

冕

Tn+1


TI+1

=−

冋冕

TI+1

dxv共t,x兲␻n共t,x兲 −

T0

Tn+1


T0

I

=−

冕

册

ᐉL共t,T 兲

m ⌳mn共t兲. 兺 1 + ᐉL共t,T m兲 m=n+1

Collecting the results from above yields 关18,20兴

␨共t,Tn兲 =

冦

n

ᐉL共t,T 兲

m ⌳mn共t兲, 兺 m=I+1 1 + ᐉL共t,Tm兲

Tn ⬎ TI

0,

Tn = TI I

−

ᐉL共t,Tm兲 ⌳mn共t兲, Tn ⬍ TI , 兺 m=n+1 1 + ᐉL共t,Tm兲

冧

+

L共T0,Tn兲 = L共t0,Tn兲e␤共t0,T0,Tn兲+Wn ,

冕

TI+1


共42兲

⳵ ln L共t,Tn兲 1 = lim 关ln L共t + ⑀,Tn兲 − ln L共t,Tn兲兴 ⳵t ⑀→0 ⑀

冋

1 ⳵ L共t,Tn兲 1 ⳵ L共t,Tn兲 ⑀ − L共t,Tn兲 ⳵ t 2 L共t,Tn兲 ⳵ t + O共⑀兲 ⇒

册

T0

1 Wn = − q2n + 2

共44兲

dt␨共t,Tn兲,

q2n =

冕

T0

dt⌳nn共t兲,

t0

冕冕 T0

Tn+1

dt


Tn

t0

Libor is proportional to the exponential of real quantities, namely, a real drift ␨共t , Tn兲 − q2n / 2 and a real valued 共Gaussian兲 quantum field A共t , x兲. Hence, Libor dynamics leads to positive Libor—as given in Eq. 共44兲.

dx␥共t,x兲A共t,x兲. 共41兲

TI

冕

t0

XI. LOGARITHMIC LIBOR RATES ␾(t , x)

Tn

The results obtained express the time evolution of Libor completely in terms of volatility ␥共t , x兲, which is a function that is empirically studied in 关17兴. Libor drift ␨共t , Tn兲 is fixed by Eq. 共40兲 and is a nonlinear and nonlocal function of all Libors. In the Libor evolution equations given in Eq. 共41兲, all reference to the volatility function v共t , x兲 of the Libor forward interest rates f共t , x兲 has been removed—as indeed was the whole purpose of the derivations of Sec. IX—with the drift being completely expressed in terms of Libor L共t , Tn兲 and its volatility ␥共t , x兲. Equation 共41兲 needs to be integrated to confirm that Libor dynamics yields positive valued Libors. Let T0 ⬎ t0 be the two points on the Libor time lattice. From Eqs. 共17兲 and 共41兲, the differential of logarithmic Libor is given by

=

␤共t0,T0,Tn兲 =

共40兲

In particular, since ␨共t , TI兲 = 0, Libor L共t , TI兲 has a martingale evolution given by

⳵ L共t,TI兲 = L共t,TI兲 ⳵t

共43兲

where

As stated in Eq. 共26兲, Libor dynamics is given by Tn+1

1 dx␥共t,x兲A共t,x兲 − ⌳nn共t兲. 2

Integrating above equation over time yields

X. LIBOR DYNAMICS

冕

冕

Tn+1

Tn

where ⌳mn共t兲 is given in Eq. 共39兲. Equation 共40兲 is the main result of the Libor market model and the equation incorporates imperfect correlation of Libor thus generalizing the earlier BGM-Jamshidian result.

1 ⳵ L共t,Tn兲 = ␨共t,Tn兲 + L共t,Tn兲 ⳵ t

FIG. 8. Simple interest earned over Libor time interval T0 to T2. Simple interest earned over the two subintervals T0 to T1 and from T1 to T2 must be equal to the interest earned from T0 to T2.

Since ␥共t , x兲, the volatility of L共t , Tn兲, is deterministic it is convenient to change variables from f共t , x兲 to L共t , Tn兲. Equation 共44兲 shows that Libor L共t , Tn兲 is a positive random variable. A change of variables to logarithmic coordinates shows the structure of the Libor market model more clearly. Let ␾共t , x兲 be a two-dimensional quantum field; define a change of variables by

再冕

ᐉL共t,Tn兲 = exp

Tn+1

冎

dx␾共t,x兲 ⬅ e␾n共t兲 .

Tn

共45兲

From its definition, ␾共t , x兲 has dimensions of 1/time and can be thought of as the effective logarithmic Libor interest rates. Consider, at some time t, a contract for a deposit to be made from future time T0 to T2; the principal plus simple interest earned, at time T2, is given by 1 + 共T2 − T0兲L共t , T0 , T2兲. This amount must be equal to that earned by first depositing the principal at time T0, then rolling over, at T0 + ᐉ = T1, the deposit and interest earned, and collecting the principal and interest at time T2 = T1 + ᐉ 共see Fig. 8兲. For there to be no arbitrage opportunities the two procedures must be equal, namely 关21兴, 1 + 共T2 − T0兲L共t,T0,T2兲 = 关1 + ᐉL共t,T0兲兴关1 + ᐉL共t,T1兲兴 ⇒ 共T2 − T0兲L共t,T0,T2兲 = e␾0共t兲+␾1共t兲 + e␾0共t兲

2

+ e␾1共t兲 ⇒ e␾0共t兲+␾1共t兲

冋冕

⳵ ln L共t,Tn兲 = ␨共t,Tn兲 ⳵t

T2

= exp

T0

046119-9

dx␾共t,x兲

册


BELAL E. BAAQUIE

FIG. 9. The dependence of ␾共t , x兲 on f共t , x兲.

= ᐉL共t,T0兲ᐉL共t,T1兲.

共46兲

exp兵兰TTm+ᐉdx␾共t , x兲其, n

similar to Eq. 共46兲, is related to Here, the future Libor rate L共t , Tn , Tm+1兲. The integral of ␾共t , x兲 over many Libor future time intervals yields the following: exp

再冕

Tm+ᐉ

Tn

冎

m

dx␾共t,x兲 = 兿关ᐉL共t,Ti兲兴. i=n

The Libor forward interest rates f共t , x兲 are related to logarithmic Libor by Eq. 共2兲, exp

再冕

Tn+1

dxf共t,x兲

Tn

冎

再冕

= 1 + ᐉL共t,Tn兲 ⇒ exp

再冕

Tn+ᐉ

冎

dxf共t,x兲

Tn

Tn+ᐉ

= 1 + exp

Tn

冎

Hence, there is no domain where both the quantum fields f共t , x兲 and ␾共t , x兲 take small values and consequently there is no consistent scheme for simultaneously defining a perturbation expansion, in powers of ␾共t , x兲 and f共t , x兲, for both the quantum fields. In summary, one can define a perturbation expansion for either f共t , x兲 or ␾共t , x兲 but not simultaneously for both the fields. Furthermore, as can be seen from Fig. 10, for Eq. 共47兲 to have real values for both f共t , x兲 and ␾共t , x兲, the following is required: − ⬁ ⱕ ␾共t,x兲 ⱕ + ⬁.

0 ⱕ f共t,x兲 ⱕ + ⬁,

dx␾共t,x兲 .

The definition of ␾共t , x兲 depends on the tenor, and for the benchmark case is taken to be ᐉ = 90 days 共three months兲. For Libor, the tenor is always finite, being a minimum of overnight 共24 h兲. The logarithmic Libor ␾共t , x兲 is well defined for any nonzero tenor ᐉ. For the limit of zero tenor, let ᐉ = ⑀ → 0; from defining Eq. 共2兲, it follows that, since f共t , x兲 is always finite, 1 1 + ⑀ f共t,x兲 = 2 + ⑀␾共t,x兲 ⇒ ␾共t,x兲 = 关f共t,x兲 − 1兴 → − ⬁. ⑀ In other words, the zero tenor limit is singular for ␾共t , x兲; however, for finite tenor ᐉ ⫽ 0, the field ␾共t , x兲 is always well defined. Since the interest derivative market is based on the threemonth Libor, let ᐉ = 1 / 4 year; one can approximately evaluate the integral and obtain the following: exp兵ᐉf共t,x兲其 ⯝ 1 + exp兵ᐉ␾共t,x兲其.

FIG. 10. The dependence of f共t , x兲 on ␾共t , x兲.

Note that ␾共t , x兲 is the natural quantum field for developing a Feynman perturbation expansion for Libor instruments as it is a flat degree of freedom taking values on the real line. The dynamics of Libor L共t , Tn兲 is specified in Eq. 共41兲 and yields, from Eq. 共43兲, the following defining equation for ␾共t , x兲:

⳵ ⳵t

Tn+1

dx␾共t,x兲 =

Tn

⳵ ln关ᐉL共t,Tn兲兴 = ␨共t,Tn兲 ⳵t +

冕

Tn+1

Tn

1 dx␥共t,x兲A共t,x兲 − ⌳nn . 2 共48兲

The drift ␨共t , Tn兲 for forward bond numeraire B共t , TI+1兲 is given by Eqs. 共39兲 and 共40兲. Integrating Eq. 共48兲 from calendar Libor time t0 to T0 yields

冕

Tn+1

Tn

共47兲

Equation 共47兲 is plotted in Fig. 9. For f共t , x兲 Ⰶ ln共2兲 / ᐉ ⬃ 400% / year, the value of ␾共t , x兲 ⯝ 0; furthermore, for ␾共t , x兲 Ⰶ 1 the value of f共t , x兲 ⯝ 0. Only when both the rates f共t , x兲 and ␾共t , x兲 are large they are approximately equal.

冕

dx␾共T0,x兲 =

冕冕

Tn+1

dx␾共t0,x兲 +

Tn

冕

T0

t0

Tn+1

+

Tn

冋

dt ␨共t,Tn兲

册

1 dx␥共t,x兲A共t,x兲 − ⌳nn . 2 共49兲

Exponentiating Eq. 共49兲 yields Eq. 共44兲 as expected.

046119-10



1 ⳵ ␾共t,x兲 = ␳共t,x兲 − ⌳共t,x兲 + ␥共t,x兲A共t,x兲, 2 ⳵t

共56兲

⬁

⌳共t,x兲 = 兺 Hn共x兲⌳n共t,x兲, n=0 ⬁

␳共t,x兲 = 兺 Hn共x兲␳n共t,x兲.

共57兲

n=0

FIG. 11. The characteristic function Hn共x兲 for the Libor interval 关Tn , Tn+1兲.

It is convenient to separate out a “kinetic” drift − 21 ⌳共t , x兲 that does not depend on the Libors, with the remaining drift ␳共t , x兲 being a nonlinear and nonlocal function of the Libors.

Dropping the 兰TTn+1dx integration from both sides of Eq. n 共48兲 yields, for x 苸关Tn , Tn+1兲, the following time evolution for logarithmic Libor: 1 ⳵ ␾共t,x兲 = − ⌳n共t,x兲 + ␳n共t,x兲 + ␥共t,x兲A共t,x兲, 2 ⳵t ⌳n共t,x兲 =

冕

Tn+1

dx⬘M共x,x⬘ ;t兲.

共50兲

XII. LAGRANGIAN OF LOGARITHMIC LIBOR ␾(t , x)

Equation 共56兲 encodes a change of variables relating two quantum fields ␾共t , x兲 and A共t , x兲 and is given by A共t,x兲 =

共51兲

Tn

1 ˜␳共t,x兲 ⬅ ␳共t,x兲 − ⌳共t,x兲. 2

The function ␳n共t , x兲 is defined as follows:

␨共t,Tn兲 =

冕

Tn+1

dx␳n共t,x兲.

共52兲

Tn

Hence, from Eqs. 共40兲 and 共52兲, for x 苸关Tn , Tn+1兲,

␳n共t,x兲 =

冦

n

兺

m=I+1

e␾m共t兲 ⌳m共t,x兲, 1 + e␾m共t兲

0, −

Tn ⬎ TI Tn = TI

I

e

␾m共t兲

兺 ␾ 共t兲 ⌳m共t,x兲, m=n+1 1 + e m

Tn ⬍ TI .

冧

⇒

冕

再

1, Tn ⱕ x ⬍ Tn+1 0, x苸 ” 关Tn,Tn+1兲

Tn+1

冎

dxHm共x兲 = ␦m−n

共59兲

The Lagrangian and action for the Gaussian “stiff” quantum field A共t , x兲, after doing an integration by parts in Eq. 共12兲, are given by

共53兲

冕

S关A兴 =

T

共54兲

Z=

冕

DAeS关A兴 .

The Lagrangian and action for logarithmic Libor quantum field ␾共t , x兲 are given by

⫻

and it is shown in Fig. 11. The characteristic function has the following important properties: ⬁

冋

冋

⳵ ␾共t,x⬘兲/⳵ t − ˜␳共t,x⬘兲 , ␥共t,x⬘兲

S关␾兴 =

f共x兲 = 兺 Hn共x兲f n共x兲,

册册

1 ⳵ ␾共t,x兲/⳵ t − ˜␳共t,x兲 ⫻ D−1共t,x,x⬘兲 2 ␥共t,x兲

L关␾兴 = −

共55兲

冕冕 ⬁

dt

t0

n=0

L关A兴.

The semi-infinite trapezoidal domain T is given in Fig. 16. The partition function is given by the Feynman path integral

Tn

f共x兲 = f n共x兲,

共58兲

1 L关A兴 = − A共t,x兲D−1共t,x,x⬘兲A共t,x⬘兲, 2

To write Eq. 共50兲 in a more compact form, define the characteristic function Hn共x兲 for the Libor time interval 关Tn , Tn+1兲 given by Hn共x兲 =

⳵ ␾共t,x兲/⳵ t − ˜␳共t,x兲 , ␥共t,x兲

⬁

dxdx⬘L关␾兴.

共60兲

t

The Neumann boundary conditions A共t , x兲 yields the following boundary conditions on ␾共t , x兲关2兴:

x 苸关Tn,Tn+1兲.

Hence, from Eqs. 共50兲 and 共29兲, for arbitrary future time x, 046119-11

⳵ ⳵x

冏冋

⳵ ␾共t,x兲/⳵ t − ˜␳共t,x兲 ␥共t,x兲

册冏

= 0. x=t

共61兲


BELAL E. BAAQUIE

It is shown in Appendix F that the Jacobian of the functional change of variables given in Eq. 共58兲 is a constant, independent of ␾共t , x兲 even though the transformation in Eq. 共58兲 in nonlinear due to the nonlinearity of Libor drift ␳共t , x兲. A constant Jacobian leads to ␾共t , x兲 being flat variables, with no measure term in the path integral. Flat variables have a well-defined leading order Gaussian path integral that generates a Feynman perturbation expansion for all financial instruments, thus greatly simplifying all calculations that are based on ␾共t , x兲. Hence, up to an irrelevant constant, the logarithmic Libor path integral measure is given by

冕

DA =

冕

冕

⬁

⬁

D␾ = 兿

兿 x=t

t=t0

+⬁

d␾共t,x兲.

−⬁

E关ᐉL共T0,TI兲兴 = ᐉL共t0,TI兲

冕

D␾eS关␾兴 =

冕

1 Z

冕

DAO关A兴eS关A兴 =

DAeS关A兴

1 Z

冕

L共t0,TI兲 = E关L共T0,TI兲兴,

冕

冕冕冕再冕 TI+1

dx␾共T0,x兲 =

TI

TI+1

TI

T0

TI+1

dt

+

t0

TI

t0

dx␥共t,x兲A共t,x兲

TI

eS关A兴 = ᐉL共t0,TI兲

共martingale兲.

dx␾共t0,x兲

⳵ f共t,x兲 e␾n共t兲 = ⳵t 1 + e␾n共t兲

冎

1 dx␥共t,x兲A共t,x兲 − ⌳II共t兲 . 2

Changing path integration variables from ␾共t , x兲 to A共t , x兲 and using the generating functional given in Eq. 共A5兲 yield

Tn+1

dx

Tn

⳵ ␾共t,x兲 e␾n共t兲 = ⳵t 1 + e␾n共t兲

冕冋 Tn+1

Tn

共63兲

冕

共64兲

册

1 dx − ⌳共t,x兲 + ␳共t,x兲 + ␥共t,x兲A共t,x兲 . 2

From Eqs. 共9兲 and 共65兲

⳵ f共t,x兲 = ␮共t,x兲 + v共t,x兲A共t,x兲 ⇒ ⳵t ⫻

冕冋 Tn+1

Tn

冕

Tn+1

冕

Tn+1

dx␮共t,x兲 =

Tn

册

dxv共t,x兲 =

e␾n共t兲 1 + e␾n共t兲

冕

Tn+1

共65兲

e␾n共t兲 1 + e␾n共t兲

1 dx − ⌳共t,x兲 + ␳共t,x兲 , 2

Tn

共66兲

␥共t,x兲.

共67兲

Tn

The result for v共t , x兲 has been obtained earlier in Eq. 共28兲; the value of ␮ is a new result. Writing the drift and volatility in terms of f共t , x兲 yields, from Eqs. 共2兲 and 共57兲, the following: f n共t兲 ⬅

For Libor L共t , TI兲, the drift is zero ␳I = 0 and hence, from Eq. 共52兲, ␨共t , TI兲 = 0; Eqs. 共45兲 and 共49兲 yield ln ᐉL共T0,TI兲 =

dx

Tn

T0 ⬎ t0 .

D␾eS关␾兴L共T0,TI兲.

Tn+1

⫻

The path integral for the right-hand side, from Eq. 共62兲, is given by the expectation value of a financial instrument O = L共T0 , TI兲; hence

冕

册冎

TI+1

dt

The dynamics of logarithmic Libor given in Eq. 共56兲 also defines the dynamics of the quantum field f共t , x兲. Differentiating Eq. 共2兲 and using Eqs. 共45兲 and 共56兲 yield the following:

D␾O关␾兴eS关␾兴 . 共62兲

Consider the n = I special case of the portfolio; from Eq. 共40兲, drift ␨共t , TI兲 = 0 and hence the instrument XI共t兲 = L共t , TI兲 is a martingale. The integral formulation of the martingale condition states that the present value of a martingale instrument is the conditional expectation value of its future value; in other words, the martingale condition is given by

1 Z

T0

The calculation confirms that XI共t兲 = L共t , TI兲 is in fact a martingale. The path integral derivation of the martingale condition for the Libor ␹I共T0兲 = L共T0 , TI兲 cannot be applied for the general case of ␹n共t兲. The reason being that, in general, drift ␳n is a nonlinear function of the quantum field A共t , x兲; evaluating the drift requires that a nonlinear path integral be evaluated exactly—something that is computationally intractable.

Path integral derivation of Libor martingale

E关L共T0,TI兲兴 =

再冕冋冕

DA exp

XIII. DYNAMICS OF LIBOR FORWARD INTEREST RATES f(t , x)

and the expectation value of a financial instrument O is given by E关O兴 =

冕

1 − ⌳II共t兲 2

The partition function for ␾ is Z=

1 Z

⫻

冕

Tn+1

Tn

dxf共t,x兲,

e␾n共t兲 = 1 − e−f n共t兲 , 1 + e␾n共t兲

冋

册

共68兲

1 ␮共t,x兲 = u共t,x兲 − ⌳共t,x兲 + ␳共t,x兲 , 2

共69兲

v共t,x兲 = u共t,x兲␥共t,x兲,

共70兲

⬁ where u共t , x兲 = 兺n=0 Hn共x兲关1 − e−f n共t兲兴. The drift ␮共t , x兲 and volatility v共t , x兲 are both functions of only exp兵−f n共t兲其, which, in turn, is the forward price of a zero coupon bond.

046119-12


INTEREST RATES IN QUANTUM FINANCE: THE WILSON… XIV. HAMILTONIAN FORMULATION OF MARTINGALE CONDITION

The Hamiltonian is a differential operator that acts on an underlying state space 关22兴. Financial instruments are elements of the state space and the evolution is determined by a Hamiltonian. As discussed in Sec. VI, choose the zero coupon bond B共t , TI+1兲 to be the forward bond numeraire; then, from Eq. 共20兲, for every n Xn共t兲 ⬅

L共t,Tn兲B共t,Tn+1兲 B共t,TI+1兲

共martingale兲.

Recall Libor time Tn can be less than, equal to, or greater than TI—as shown in Fig. 6. The drift ␳共t , x兲 is fixed in the Hamiltonian framework by imposing the martingale condition on Xn共t兲, namely, that 关2兴 H共t兲关Xn共t兲兴 = H共t兲

冋

册

L共t,Tn兲B共t,Tn+1兲 = 0. B共t,TI+1兲

dent derivation in this section based directly on the Hamiltonian formulation of the martingale condition given in Eq. 共71兲. The derivation for Libor drift ␳共t , x兲 for the Libor market model given in Sec. IX was quite circuitous; the Libor forward interest rates f共t , x兲 were used as scaffolding and it was not clear why one could not evaluate Libor drift directly using only Libor L共t , Tn兲. The result is also quite opaque, with the drift having summations and minus signs that do not have a clear explanation. In contrast, the Hamiltonian framework yields a clear and transparent derivation of Libor drift directly using Libor variables L共t , Tn兲 and the result is intuitively clear. The logarithmic Libor Hamiltonian is derived in Appendix H and is given by Eq. 共H8兲; for notational convenience, a kinetic piece is subtracted out from the drift ␳共t , x兲. Hence 关23兴

共71兲

H共t兲 = −

The instrument Xn共t兲 is an element of the Libor interest rate state space, which is discussed in Appendix G, and H共t兲 is the Hamiltonian. The differential formulation of the martingale given in Eq. 共71兲 states that for the Hamiltonian evolving interest rate instruments to be free from arbitrage opportunities, it must annihilate Xn共t兲. For notational ease, write Xn共t兲 =

B共t,Tn+1兲 L共t,Tn兲B共t,Tn+1兲 . ⬅ Ln B共t,TI+1兲 B共t,TI+1兲

Let t be a Libor time; from Eq. 共4兲, the zero coupon bond is given by n

B共t,Tn+1兲 = 兿

k=0

冉

冊

1 . 1 + ᐉLk

XI共t兲 = LI = L共t,TI兲. 共ii兲 For n ⬎ I, n

Xn共t兲 = Ln

兿 k=I+1

冉

共iii兲 For n ⬍ I, I

Xn共t兲 = Ln

兿

k=n+1

冊

再

再

共1 + ᐉLk兲 = Ln exp

冎

共74兲

兺

k=n+1

1 ␦2 + ␦␾共x兲␦␾共x⬘兲 2

冕

关⌳共t,x兲

x

␦ , ␦␾共x兲 +⬁

⬅

x

共76兲

dx.

t

The martingale condition given in Eq. 共71兲 requires the calculation of ␦ / ␦␾ acting on Xn共t兲, which in turn needs the following computation. The definition of logarithm Libor given in Eq. 共45兲, namely,

再冕

ᐉL共t,Tn兲 ⬅ ᐉLn = exp

Tn+1

冎

dx␾t共x兲 ⬅ e␾n ,

Tn

yields

1 = Ln exp − 兺 ln共1 + ᐉLk兲 . 1 + ᐉLk k=I+1

I

x,x⬘

M共x,x⬘ ;t兲

冕冕

共73兲

n

冕

− ␳共t,x兲兴

共72兲

The following are the three cases for Xn共t兲: 共i兲 For n = I,

1 2

␦ L = Hk共x兲Lk , ␦␾共x兲 k

共77兲

where the characteristic function is given in Eq. 共54兲 and is shown in Fig. 11. For the case of n = I, from Eq. 共73兲, XI共t兲 = LI = L共t , TI兲; by inspection, it can be easily seen that ␳I共t , x兲 = 0.

冎

A. Case (i) n ⬎ I

ln共1 + ᐉLk兲 .

For the case of Xn共t兲, n ⬎ I, Eqs. 共74兲 and 共77兲 yield

共75兲

XV. HAMILTONIAN DERIVATION OF LIBOR MARKET MODEL DRIFT

Libor market model drift ␳共t , x兲 was derived in Sec. IX using the Wilson expansion. The drift is given an indepen-

n

1 ␦Xn共t兲 e␾kHk共x兲 = Hn共x兲 − 兺 ␾k . Xn共t兲 ␦␾共x兲 k=I+1 1 + e

共78兲

Note that the summation term above is due to the discounting by the forward numeraire B共t , TI+1兲. The second functional derivative yields

046119-13


BELAL E. BAAQUIE

冋

n

␦2Xn共t兲 1 e␾ jH j共x兲 = Hn共x兲 − 兺 ␾j Xn共t兲 ␦␾共x兲␦␾共x⬘兲 j=I+1 1 + e n

−

兺

k=I+1 n

−

兺

j=I+1 n

+

兺

j=I+1

e␾kHk共x⬘兲 1 + e ␾k

册

册冋

冕

Hn共x⬘兲

冋册

2

n

dx␳n共t,x兲 =

Tn

Xn共t兲

冋册

n

1 e␾ j = 兺 2 j=I+1 1 + e␾ j

⌳ jj −

n

兺

j=I+1

兺 j=I+1

冕

n

dx

冕

共80兲

Tn+1

n

dx⬘M共x,x⬘ ;t兲.

共88兲

Tj

n=0

冕

Tn+1

dx␳n共t,x兲.

共81兲

e␾ j

兺 ␾ ⌳ jn . j=I+1 1 + e

共82兲

j

Hence

␳n共t,x兲 =

n

n

1 1 A jk = 兺 A jk + 兺 A jj 2 j,k=I+1 2 j=I+1

共84兲

applied to Eq. 共83兲 leads to the cancellation of all the terms on the right-hand side of Eq. 共80兲 and yields the final result, H关Xn共t兲兴 = 0

共martingale兲.

Hence, from Eqs. 共81兲 and 共82兲, Libor drift is given by

共85兲

兺

m=I+1

e␾m共t兲 ⌳m共t,x兲, 1 + e␾m共t兲

0,

n=I I

−

n⬎I

e␾m共t兲

兺 ␾ 共t兲 ⌳m共t,x兲, m=n+1 1 + e m

n ⬍ I.

冧

共89兲

XVI. LIBOR MARKET MODEL: HAMILTONIAN, LAGRANGIAN, AND A(t , x)

共83兲

The remarkable identity

冦

n

The result for Libor drift obtained in Eq. 共89兲 agrees exactly with the result obtained in Eq. 共40兲 using the Wilson short distance expansion.

j

n

e␾ j e␾ j e ␾k ␨ = 兺 j ␾j 兺 ␾k ⌳ jk . 1 + e␾ j j=I+1 1 + e k=I+1 1 + e

兺兺

冕

T j+1

⬁

Inspecting the result in Eq. 共80兲 leads to the following ansatz:

j=I+1 k=I+1

兺

e␾ j共t兲 1 + e␾ j共t兲

␳共t,x兲 = 兺 Hn共x兲␳n共t,x兲,

dx⬘M共x,x⬘ ;t兲

Tn

j

共87兲

j

j=n+1

where

␨n ⬅ ␨共t,Tn兲 =

n

dx⬘M共x,x⬘ ;t兲

Tj

e␾ j共t兲

I

and, as in Eq. 共52兲,

兺 j=I+1

T j+1

兺 ␾ 共t兲 ⌳ j共t,x兲. j=I+1 1 + e

␳n共t,x兲 = −

Tn

Tm

n

冕

The exact results given in Eqs. 共82兲, 共87兲, and 共88兲 yield Libor drift as follows:

Tm+1

␨n =

e␾ j共t兲

A derivation similar to case 共i兲 yields the result for Xn共t兲, n ⬍ I. One needs to keep track of the relative negative sign in ␹n, given in Eqs. 共74兲 and 共75兲 arising from the difference in the discounting factor. The following is the final result:

where ⌳mn =

dx⬘M共x,x⬘ ;t兲. 共86兲

Tj

B. Case (ii) n ⬍ I

e␾ j ⌳ jn 1 + e␾ j

e␾ j ␨j , 1 + e␾ j

T j+1

dx

j

=

n

−

Tn+1

兺 ␾ 共t兲 j=I+1 1 + e

␳n共t,x兲 =

1 e ␾ j+␾k ⌳ jk + ␨n + 兺 2 j,k=I+1 关1 + e␾ j兴关1 + e␾k兴 n

冕冕

Therefore, for Tn ⱕ x ⬍ Tn+1, the drift is given by

H j共x兲H j共x⬘兲. 共79兲

n

2

j

Tn

On applying logarithmic Libor Hamiltonian on Xn共t兲, n ⬎ I, Eqs. 共77兲–共79兲 yield, after a few obvious cancellations, H关Xn共t兲兴

e␾ j共t兲

兺 ␾ 共t兲 j=I+1 1 + e ⫻

e␾ jH j共x兲H j共x⬘兲 1 + e␾ j e␾ j 1 + e␾ j

Tn+1

The Hamiltonian formulation of the martingale condition, given in Eq. 共71兲, yields the Libor drift in a fairly transparent and direct manner compared to the rather roundabout approach adopted in Sec. IX. The summation that appears in the drift term is due to expressing the ratio B共t , Tn+1兲 / B共t , TI+1兲 as a product of Libor variables L共t , Tk兲. The relative minus sign in the summation term of the drift for n ⬍ I and n ⬎ I arises from the ratio B共t , Tn+1兲 / B共t , TI+1兲 being the product of 1 + ᐉL共t , Tk兲 or of its inverse. The derivation of Libor drift given in Sec. IX follows the general spirit of the BGM-Jamshidian approach— generalized to the case of quantum finance by the use of the

046119-14



Wilson expansion. The martingale condition was first expressed in terms of the Libor forward interest rates f共t , x兲; one then did a change of variables and re-expressed the drift in terms of the Libor variables. To carry out this change of variables for the quantum finance case, the Wilson expansion for the velocity quantum field A共t , x兲 was crucial in capturing the nontrivial correlation terms. In contrast, in the Hamiltonian derivation of Libor drift, there is no need to employ the A共t , x兲 field and all the correlation effects are produced by the Hamiltonian. The Libor Hamiltonian is expressed directly in terms of log Libor variables ␾共t , x兲, making no reference to f共t , x兲. The martingale condition is expressed directly in terms of the Hamiltonian H and leads to an exact derivation of Libor drift. The fact that the Jacobian of the transformation from A共t , x兲 to ␾共t , x兲 is a constant, as shown in Appendix F, is essential for obtaining the logarithmic Libor Hamiltonian; a nontrivial Jacobian would give rise to new terms. The Hamiltonian derivation of Libor drift leads to some general conclusions in the context of the Libor market model. The extension of the martingale condition from the case for equity to that of interest rate instruments is nontrivial due to 共a兲 the need to treat the discounting factor as being stochastic and 共b兲 the time-dependent state space 关2兴. The derivation of the nontrivial and nonlinear drift of the Libor market model verifies the correctness of the martingale condition stated in Eq. 共71兲. The Hamiltonian derivation of Libor drift provides independent proof of the correctness of the earlier derivation of Libor drift in Sec. IX, which crucially hinged on the Wilson expansion. The Hamiltonian result shows that the Wilson short distance expansion for a Gaussian quantum field is the correct generalization of Ito’s calculus and opens the way to applications of A共t , x兲 in theoretical finance.

of the Libor market model are nonlinear and nonsingular. The Libor forward interest rates f共t , x兲 are related to logarithmic Libor by Eq. 共2兲,

XVII. SUMMARY

Quantum finance provides a natural and mathematically tractable formulation of the Libor market model. Gaussian white noise R共t兲, in effect, is “promoted” by quantum finance to a quantum field A共t , x兲 that drives the time evolution of Libor. In particular, the Libor market model has been given three different and consistent formulations, namely, employing A共t , x兲, L关␾兴, and H共t兲—thus displaying the versatility and flexibility of quantum finance. In an economy where Libor rates are perfectly correlated across different maturities, a single volatility function is sufficient. In contrast, Libor term structure that is imperfectly correlated introduces many new features. In the quantum finance approach, due to the specific properties of Gaussian quantum fields, the entire Libor volatility function can be taken directly from the market and thus incorporates many key features of the market in a parameter free manner. The Gaussian quantum field A共t , x兲 has a quadratic action and hence one can obtain a differential formulation of the Libor market model. The singular quadratic product A共t , x兲 has a “short distance” Wilson expansion that generalizes the results of Ito calculus and yields a derivation of Libor drift. It is seen that the underlying Libor forward interest rates f共t , x兲

再冕

Tn+ᐉ

exp

Tn

冎

再冕

Tn+ᐉ

dxf共t,x兲 = 1 + exp

Tn

冎

dx␾共t,x兲 . 共90兲

The transformation from ␾共t , x兲 to f共t , x兲 is nonlinear and nonlocal and is well defined only for strictly positive f共t , x兲, that is, for f共t , x兲 ⱖ 0. In particular, this means that f共t , x兲 cannot be a Gaussian quantum field. The linear approximation that treats f共t , x兲 as a Gaussian quantum field needs to be carefully examined to ascertain whether it can be applied to the interest rate 共Libor and Euribor兲 market. Note the transformation given in Eq. 共90兲 is completely general and only requires that f共t , x兲 ⱖ 0; for example, one could take f共t , x兲 ⬃ e␰共t,x兲, where ␰共t , x兲 is any real quantum field and this yields strictly positive Libors. The Libor market model makes a very specific choice for f共t , x兲, namely, one for which the logarithmic Libor quantum field ␾共t , x兲 has deterministic volatility given ␥共t , x兲. This choice for f共t , x兲 leads to the field ␾共t , x兲 being a “flat” quantum field that takes values on the entire real line. The leading order effects of ␾共t , x兲 are described by a Gaussian quantum field; the nonlinear terms are all contained in the drift and can be treated perturbatively using Feynman diagrams. Logarithmic Libor variables ␾共t , x兲 yield strictly positive coupon bonds and Libors. A nonlinear quantum field theory for flat quantum field ␾共t , x兲 is the most appropriate formalism for simultaneously analyzing coupon bond and Libor instruments. The Hamiltonian realization of the martingale evolution of an instrument entails that the instrument be annihilated by the Hamiltonian. The Hamiltonian is the appropriate framework for imposing the martingale condition for nonlinear interest rates and, in particular, yields the exact expression for the Libor market model’s nonlinear drift. A new feature of the interest rate dynamics is that, unlike the case for equity, the interest rates’ state space and Hamiltonian are time dependent; this leads to a number of new features for the Hamiltonian and, in particular, that the evolution operator is defined by a time ordered product. Choosing the forward bond numeraire leads to a major simplification: the interest rates’ state space becomes equivalent to a fixed state space and one can dispense with time ordering, leading to a time independent evolution operator. ACKNOWLEDGMENTS

I thank Cui Liang for many earlier discussions and for emphasizing the importance of the Libor market model. I thank Tang Pan and Cao Yang for useful discussions. APPENDIX A: A(t , x)’S GENERATING FUNCTIONAL

One is free to choose the dynamics of the quantum field A共t , x兲. Following Baaquie and Bouchaud 关16兴, the Lagrangian that describes the evolution of instantaneous forward rates is defined by three parameters ␮ , ␭ , ␩ and is given by

046119-15


BELAL E. BAAQUIE

再

冉

1 1 ⳵ A共t,z兲 L共A兲 = − A2共t,z兲 + 2 2 ␮ ⳵z ⫻

冏

⳵ A共t,z兲 ⳵z

冏

冊冉 2

1 ⳵2A共t,z兲 + 4 ␭ ⳵ 2z

冊冎 2

t

共A1兲

= 0,

冕冕 ⬁

S关A兴 =

⬁

dt

t0

dzdz⬘L共A兲.

共A2兲

The market value of all interest rates’ financial instruments is obtained by performing a path integral over the 共fluctuating兲 two-dimensional quantum field A共t , z兲. The expectation value of an instrument F关A兴, denoted by E兵F关A兴其, is defined by the functional average over all values of A共t , z兲, weighted by the probability measure eS / Z. Hence

冕

DAF关A兴eS关A兴,

0

冋再冕

⬅

1 Z

=exp

冕

t0

dt

冕

再

Z=

⬁

dzh共t,z兲A共t,z兲

0

再冕冕 ⬁

⬁

dt

t0

冕冕 ⬁

DA exp S关A兴 +

1 2

T

t*

t 0 + TFR

x

冕

Consider for simplicity the bond forward interest rates given by

⳵ f B共t,x兲 = ␣共t,x兲 + ␴共t,x兲A共t,x兲, ⳵t

DAeS关A兴 .

The quantum field theory of the forward interest rates is defined by the generating function given by Z关h兴 = E exp

t0

( t 0,T)

FIG. 12. The domain of the forward interest rates f共t , x兲, in calendar and future time, required for evaluating the time evolution of a zero coupon bond.

共A3兲

⬁

( t 0 , t 0)

0

t0

1 Z

( t* ,T)

z=0

where market 共psychological兲 future time is defined by z = 共x − t兲␯. The action S关A兴 of the Lagrangian is defined as

E兵F关A兴其 ⬅

( t* ,t* )

t*

⬁

dt

t0

冎册

f B共t,x兲 = f B共t0,x兲 +

t

dt␣共t,x兲 +

t0

dzh共t,z兲A共t,z兲

0

冎

冕

t

dt␴共t,x兲A共t,x兲,

t0

共B2兲

冎

共A4兲

where the drift ␣共t , x兲 and volatility ␴共t , x兲 are both deterministic functions. From Eq. 共1兲, a zero coupon and its forward bond are given by

再冕冎再冕冎再冕冕冎再冕冕冎

dzdz⬘h共t,z兲D共z,z⬘ ;t兲h共t,z⬘兲 . 共A5兲

0

冕

共B1兲

T

dxf B共tⴱ,x兲 ,

B共tⴱ,T兲 = exp −

Hence the correlator of the A共t , x兲 quantum field is given by

tⴱ

E关A共t,z兲兴 = 0,

T

E关A共t,z兲A共t⬘,z⬘兲兴 = ␦共t − t⬘兲D共z,z⬘ ;t兲.

共A6兲

tⴱ

For notational simplicity only the case of ␯ = 1 will be considered; all integrations over z are replaced with those over future time x. For ␯ = 1 from Eq. 共A2兲 the dimension of the quantum field A共t , x兲 is 1/time and the volatility ␴共t , x兲 of the forward interest rates also has dimension of 1/time. The expression for D共x , x⬘ ; t兲 has been studied in 关16,17兴 and provides a very accurate description of the correlation of the forward interest rates. In the present paper the explicit value of the propagator D共x , x⬘ ; t兲 is not required. APPENDIX B: TIME EVOLUTION OF A BOND

To illustrate the content of the singularity in the equal time quadratic product of the quantum field, namely, A共t , x兲A共t , x⬘兲 given in Eq. 共17兲, a concrete analysis is carried out of the evolution of zero coupon bond—with and without discounting.

dxf B共t0,x兲 ,

F共t0,tⴱ,T兲 = exp −

T

tⴱ

B共tⴱ,T兲 = F共t0,tⴱ,T兲exp −

t0

⫻ exp −

tⴱ

T

tⴱ

dx␴共t,x兲A共t,x兲 .

dt

t0

dx␣共t,x兲

dt

共B3兲

tⴱ

The domain of integration is a rectangle R, equal to 关t0 , tⴱ兴 ⫻ 关tⴱ , T兴 and shown in Fig. 12. To calculate the time evolution of the bond one needs to compute the time derivative of exp兵−兰ttⴱdt兰tT dx␴共t , x兲A共t , x兲其; for doing this computation ⴱ 0 one needs to take account of the fact that the quadratic power of A共t , x兲 is singular. Consider 关24兴

046119-16



再冕

tⴱ

exp +

dt

冕

T

dx␴共t,x兲A共t,x兲

再冕 tⴱ

t0

⳵ ⫻ exp − ⳵ tⴱ

冋再

冕冕

1 = exp − ⑀ ⑀

再

⫻ exp + ⑀

tⴱ

dt

冕

冎

T


tⴱ

t0

T

dx␴共tⴱ,x兲A共tⴱ,x兲

冎

1 ⳵ B共tⴱ,T兲 = f共tⴱ,tⴱ兲 − B共tⴱ,T兲 ⳵ tⴱ

+

Expanding the exponential to second order yields all the nontrivial terms as follows 关25兴:

冋再

1 exp − ⑀ ⑀

冕

再

T


tⴱ

⫻ exp + ⑀

冕

=−

冕

tⴱ

T

dx␴共tⴱ,x兲A共tⴱ,x兲 +

⑀ 2

冉冕

T


tⴱ

冕冕冕


tⴱ

T

1 2

+

2

冕冕

冊

2

dt

dx⬘M␴共x,x⬘ ;tⴱ兲M␴共x,x⬘ ;tⴱ兲

T


再冕 tⴱ

t0

⫻

冕

⳵ exp − ⳵ tⴱ

冕冕冕

tⴱ

t0

dt

冕


tⴱ


tⴱ

+

1 2

T

册

T

dx␴共tⴱ,x兲A共tⴱ,x兲 B共tⴱ,T兲.

tⴱ

tⴱ

dtf共t,t兲 B共tⴱ,T兲

冕

tⴱ

冋冕

册

T

dx␴共tⴱ,x兲A共tⴱ,x兲 D共tⴱ,T兲.

tⴱ

APPENDIX C: MARTINGALE

A martingale refers to a special category of stochastic processes. An arbitrary discrete stochastic process is a collection of random variables Xi, i = 1 , 2 , . . . , N that is described by a joint probability distribution function p共x1 , x2 , . . . , xN兲. The stochastic process is a martingale if it satisfies

冎

E关Xn+1兩x1,x2, . . . ,xn兴 = xn

共martingale兲.

共C1兲

In other words, the expected value of the random variable Xn+1—conditioned on the occurrence of xi for random variables Xi, i = 1 , 2 , . . . , n—is simply xn itself. Consider a general stochastic functional differential equation for a two-dimensional function ␹共t , x兲,

dt␴共t,tⴱ兲A共t,tⴱ兲

t0

T

dx

tⴱ

冎

T

T

=−

冕

From Eq. 共C9兲, it is seen that the discounted bond follows a martingale process.

tⴱ

tⴱ

冋

⳵ D共tⴱ,T兲 =− ⳵ tⴱ

Collecting all the results yields

再冕

dx⬘D共x,x⬘ ;t兲␴共t,x⬘兲.

and a derivation similar to the result obtained in Eq. 共B4兲 yields

⬅ ␴共tⴱ,x兲␴共tⴱ,x⬘兲D共x,x⬘ ;tⴱ兲.

exp +

x

D共tⴱ,T兲 ⬅ exp −

T

dx

tⴱ

dx⬘M共x,x⬘ ;tⴱ兲. 共B4兲

tⴱ

tⴱ

再冕冎


dx⬘M␴共x,x⬘ ;tⴱ兲,

T

T

dx

Similarly, the time derivative of a discounted bond obeys a martingale evolution. The discounted zero coupon bond is given by

+ O共⑀兲

tⴱ

dx␴共tⴱ,x兲A共tⴱ,x兲 1 ⑀

冕

T

1 2

冕

⳵ B共tⴱ,T兲 = f共tⴱ,tⴱ兲 − ⳵ tⴱ


tⴱ


t0

T

=

This yields

T

dx

tⴱ

tⴱ

冎册

t0

where, from Eq. 共17兲,

冉冕

冊

冕冕冕

t

t0

T

=−

冕

tⴱ

dx␣共tⴱ,x兲

tⴱ

The martingale condition on the drift ␣共t , x兲 for a numeraire using discounting by the money market account, namely, by exp兵−兰ttⴱdtr共t兲其, where r共t兲 = f共t , t兲 is the spot interest rate, is 0 given by

␣共t,x兲 = ␴共t,x兲

dt␴共t,tⴱ兲A共t,tⴱ兲 − 1

t0

tⴱ

+

冎

T

tⴱ

dt␴共t,tⴱ兲A共t,tⴱ兲 − 1 .

t0

冕

T

−

冎册

tⴱ

tⴱ

冎

⳵F共t0 , tⴱ , T兲 / ⳵tⴱ = +f共t0 , tⴱ兲F共t0 , tⴱ , T兲 and using Eq. 共B1兲, one obtains from Eq. 共B3兲

dx⬘M␴共x,x⬘ ;tⴱ兲.

⳵␹共t,x兲 = d共t,x兲 + ⳵t

tⴱ

The only term in the zero coupon bond that needs to be examined carefully is the one involving A共t , x兲; the other terms all obey the usual rules of calculus. Hence, since

冕

dx⬘G共x,x⬘ ;t兲v共t,x⬘兲A共t,x⬘兲, 共C2兲

where A共t , x兲 is the Gaussian two-dimensional quantum field defined by the action given in Eq. 共12兲. G共x , x⬘ ; t兲 is a deter-

046119-17


BELAL E. BAAQUIE

ministic function and the quantities d共t , x兲 and v共t , x兲 can, in general, depend on ␹共t , x兲. Equation 共C2兲 is a stochastic differential equation that is encountered in the study of Libor. An initial 共or final兲 condition needs to be specified to obtain a solution for Eq. 共C2兲; for applications in finance, the initial condition is specified as follows: boundary condition:␹共t0,x兲 = fixed.

共ii兲 the BGM-Jamshidian limit, and 共iii兲 the HJM limit. 1. Tenor 艎 \ 0

Consider the limit of the tenor ᐉ → 0. Let x = Tn, x⬘ = Tm, and x , x⬘ ⬎ TI. From Eq. 共28兲, v共t , x兲 ⯝ ᐉf共t , x兲␥共t , x兲; hence, from Eq. 共69兲

Discretizing time in Eq. 共C2兲 yields for infinitesimal ⑀

x 苸关Tn,Tn+1兴,

␹共t + ⑀,x兲 = ␹共t,x兲 + ⑀d共t,x兲 +⑀

冕

e␾共t兲 ⬃ ᐉf共t,x兲,

dx⬘G共x,x⬘ ;t兲v共t,x⬘兲A共t,x⬘兲.

共C3兲

冋

= ␹共t,x兲 + ⑀d共t,x兲.

共C4兲

The martingale condition given in Eq. 共C1兲 requires E关␹共t + ⑀ , x兲兩 ␹共t , x兲兴 = ␹共t , x兲; hence, from Eqs. 共C1兲 and 共C4兲共martingale condition兲.

d共t,x兲 = 0

n

冋

册

⳵␹共t,x兲 =0 ⳵t

⳵␹共t,x兲 = ⳵t

冕

dx⬘G共x,x⬘ ;t兲v共t,x⬘兲A共t,x⬘兲.

共C7兲

The martingale condition can be further generalized. The stochastic evolution equation 1 ⳵␰共t,x兲 = ␰共t,x兲 ⳵ t

冕

dx⬘G共x,x⬘ ;t兲v共t,x⬘兲A共t,x⬘兲

共C8兲

yields the following conditional expectation value: E关␰共t + ⑀,x兲兴 = ␰共t,x兲

共martingale兲,

n

+ ᐉf共t,x兲ᐉ2 ⯝ v共t,x兲

APPENDIX D: LIMITS OF THE LIBOR MARKET MODEL

冕

兺关f共t,Tm兲␥共t,x兲D共x,Tm ;t兲␥共t,Tm兲兴 m=I+1

x

dx⬘D共x,x⬘ ;t兲v共t,x⬘兲 + O共ᐉ兲.

共D1兲

TI

Hence, from Eq. 共9兲

⳵ f共t,x兲 ⯝ ᐉ2 f共t,x兲 ⳵t

冕

x

dx⬘D共x,x⬘ ;t兲f共t,x⬘兲 + ᐉ2 f共 2t,x兲A共t,x兲.

TI

共D2兲 The limit of tenor ᐉ → 0 yields an evolution equation for the Libor forward interest rates f共t , x兲 that looks similar to the HJM model, except v共t , x兲 = ᐉf共t , x兲␥共t , x兲 is stochastic. Recall the drift results from requiring a martingale evolution of the Libor forward interest rates f共t , x兲 with the forward bond numeraire being the zero coupon bond B共t , TI兲 with TI fixed.

共C9兲

which is the martingale condition.

.

ᐉ ␮共t,x兲 ⯝ − 关ᐉf共t,x兲␥共t,x兲兴D共x,x;t兲␥共t,x兲 2

共martingale兲.

In summary, for ␹共t , x兲 to be a martingale, it is sufficient that

册

The sum 兺 j H j共x兲 has collapsed to one term since x 苸关Tn , Tn+1兲. The limit ᐉ → 0 is taken holding v共t , x兲 = ᐉf共t , x兲␥共t , x兲 fixed; hence, in this limit

共C5兲

共C6兲

ᐉf共t,T 兲

m ᐉ␥共t,x兲D共x,Tm ;t兲␥共t,Tm兲兺 1 + ᐉf共t,x兲 m=I+1

Hence, the unconditional expectation value is given by E关␹共t + ⑀,x兲兴 = E关␹共t,x兲兴 ⇒ E

册

ᐉ ⯝ 关ᐉf共t,x兲兴 − ␥共t,x兲D共x,x;t兲␥共t,x兲 2 +

dx⬘G共x,x⬘ ;t兲v共t,x⬘兲E关A共t,x⬘兲兴

冋

⬁

1 ␮共t,x兲 = 兺 H j共x兲关1 − e−f j共t兲兴 − ⌳ j共t,x兲 + ␳ j共t,x兲 2 j=0

E关␹共t + ⑀,x兲兩␹共t,x兲兴 = ␹共t,x兲 + ⑀d共t,x兲

冕

1 − e−f n共t兲 ⯝ ᐉf共t,x兲,

⌳n ⯝ ᐉM ␥共x,x;t兲 = ᐉ␥共t,x兲D共x,x;t兲␥共t,x兲,

The martingale condition given in Eq. 共C1兲 requires that the expectation value of ␹共t + ⑀ , x兲 is taken conditioned on ␹共t , x兲 having a fixed value. In taking the expectation value of Eq. 共C3兲, the functions d共t , x兲 and v共t , x兲 are deterministic since they depend only on ␹共t , x兲 and hence can be taken outside the expectation value. Since E关A共t , x⬘兲兴 = 0, taking the conditional expectation value of both sides of Eq. 共C3兲 yields the following:

+⑀

L共t,Tn兲 ⬃ f共t,x兲 + O共ᐉ兲,

2. BGM-Jamshidian limit

The BGM-Jamshidian limit of the quantum finance formulation of Libor market model can be obtained using the following prescription:

The following three different limits of the Libor market model are taken: 共i兲 the limit of zero tenor, namely, ᐉ → 0, 046119-18

D共x,x⬘ ;t兲兩BGM → 1, A共t,x兲兩BGM → R共t兲,



E关R共t兲R共t⬘兲兴 = ␦共t − t⬘兲,

共D3兲

where R共t兲 is Gaussian white noise. The Libor evolution for TI ⬍ Tn, given in Eq. 共26兲, yields the BGM-Jamshidian limit. From Eq. 共39兲 ⌳mn共t兲 =

冕

Tm+1

dx␥共t,x兲

Tm

冕

Tn+1

dx⬘D共x,x⬘ ;t兲␥共t,x⬘兲

Tn

→ ␥n共t兲␥n共t兲,

where ␥n共t兲 =

冕

Tn+1

dx␥共t,x兲.

Tn

Hence, the BGM-Jamshidian limit of the Libor market model evolution equation, from Eq. 共26兲, 共53兲, and 共D3兲, is given by 1 ⳵ L共t,Tn兲 = ␨n共t兲 + ␥n共t兲R共t兲, L共t,Tn兲 ⳵ t n

␨n共t兲 = ␥n共t兲

ᐉL共t,T 兲

m ␥m共t兲. 兺 m=I+1 1 + ᐉL共t,Tm兲

FIG. 13. Time evolution of f共t , x兲. The singularity at time ts is spurious since nonlinear effects take over at time ts − ᐉ leading to a finite f共t , x兲 for all time.

v共t,x兲 =

共D4兲

␥共t,x兲,

␮共t,x兲 = v共t,x兲

The HJM limit requires the following three conditions: 共i兲 the zero tenor limit ᐉ → 0 is taken, 共ii兲 it assumed that v共t , x兲 = ᐉf共t , x兲␥共t , x兲 is a deterministic function equal to the HJM-volatility function, that is, v共t , x兲 → ␴共t , x兲, the HJM bond forward interest rate volatility is independent of f共t , x兲, and 共iii兲 D共x , x⬘ ; t兲兩HJM → 1. With these assumptions Eq. 共D2兲 yields

冕

ᐉ␥共t,x兲f共t,x兲, ᐉf共t,x兲 Ⰶ 1

x

APPENDIX E: NONSINGULAR LIBOR FORWARD INTEREST RATES f(t , x)

The underlying Libor forward interest rates driving all the Libors, from Eq. 共9兲, are given by the following:

⳵ f共t,x兲 = ␮共t,x兲 + v共t,x兲A共t,x兲. ⳵t The theory is nonlinear due to the dependence of the volatility v共t , x兲 and drift ␮共t , x兲 on the underlying Libor forward interest rates f共t , x兲. A nonlinear drift that renders the time evolution of Libor to be a martingale apparently implies that the underlying Libor forward interest rates are singular 关19兴. This aspect of the Libor forward interest rates is analyzed. An approximation of Eq. 共70兲 that is adequate for the analysis of this appendix is v共t,x兲 ⯝ 关1 − e−ᐉf共t,x兲兴␥共t,x兲.

Recall from Eq. 共54兲 that volatility v共t , x兲 has the following two limiting cases:

共E1兲

冕

x

dx⬘D共x,x⬘ ;t兲v共t,x⬘兲 + O共ᐉ兲.

The limiting cases for ␮共t , x兲, from Eq. 共E1兲, are the following 关26兴: For case 共i兲 ᐉf共t , x兲 Ⰶ 1,

␮共t,x兲 ⯝ 关ᐉf共t,x兲␥共t,x兲兴2 + O共ᐉ兲. For case 共ii兲 ᐉf共t , x兲 Ⰷ 1,

TI

which is the expected HJM equation; note that the drift is the one appropriate for the forward bond numeraire B共t , TI兲.

冎

TI+1

␮共t,x兲 ⯝ −

dx⬘␴共t,x⬘兲 + ␴共t,x兲R共t兲,

ᐉf共t,x兲 Ⰷ 1.

Here v共t , x兲 = ᐉf共t , x兲␥共t , x兲 ⬃ 0, ignoring the integration that does not qualitatively change the results. Recall from Eq. 共22兲 that the drift is

3. HJM limit

⳵ f共t,x兲 = ␴共t,x兲 ⳵t

再

1 2

冕

Tn+1

dx⬘M ␥共x,x⬘ ;t兲 +

Tn

冕

Tn+1

共E2兲

dx⬘M ␥共x,x⬘ ;t兲.

TI+1

In the limit ᐉf共t , x兲 Ⰷ 1, volatility v共t , x兲 → ␥共t , x兲. The results obtained in Eqs. 共E2兲 and 共E1兲 for this limit are consistent with the earlier result given in Eq. 共22兲; the two results agree only in the limit of ᐉ → 0. For the case when ᐉf共t , x兲␥共t , x兲 is independent of ᐉ, the extra term 兰TTn+1M ␥共x , x⬘ ; t兲 / 2 in Eq. 共E2兲 is of O共ᐉ兲 and goes to zero. n For small values of ᐉf共t , x兲, ␮共t , x兲 ⯝ 关ᐉ␥共t , x兲f共t , x兲兴2 follows from Eq. 共E2兲. The stochastic term can be ignored as these do not qualitatively change the impact of the quadratic term on the evolution of f共t , x兲. The simplified dynamics for the Libor forward interest rates, from Eq. 共9兲, is the following 关27兴:

⳵ f共t,x兲 ⯝ ᐉ2␥2 f 2共t,x兲 + random terms ⇒ f共t,x兲 ⳵t ⯝ ᐉ␥

f共0,x兲 + random terms ⇒ f共t,x兲 1 − 关ᐉ␥ f共0,x兲兴t

→ ⬁ for ts =

1 , ᐉ␥ f共0,x兲

f共0,x兲 ⬎ 0.

共E3兲

From Eq. 共E3兲, all the Libor forward interest rates seem to become infinite as t → ts = 1 / 关ᐉ␥ f共0 , x兲兴 ⬎ 0 and are shown as the dashed line in Fig. 13. If f共t , x兲, in fact, becomes singular

046119-19


BELAL E. BAAQUIE

冕 ⬘冋 tⴱ

dt

␦共t⬘ − t兲

t0

册

⳵ ␦˜␳共t,x兲 d␾共t⬘,x兲 = ␥共t,x兲dA共t,x兲 − ⳵ t⬘ ␦␾共t⬘,x兲

⇒ det关J兴D␾ = const ⫻ DA. The change of variables factorizes for the x variable; hence, for notational simplicity, the x coordinate is suppressed and only the time variable t is displayed. The Jacobian is equal to det关J兴, where matrix of transformation J is given by

FIG. 14. The behavior of volatility v共t , x兲 and drift ␮共t , x兲 as a function of f共t , x兲.

without the stochastic term, then, on including the stochastic term, the singularity is even more severe with the Libor forward interest rates becoming singular almost instantaneously 关19兴. However, the result that the forward interest rates become singular is not correct. When ᐉf共t , x兲 ⬃ 1, the approximation leading to Eq. 共E3兲 is no longer valid and the solution breaks down for t f given by f共t,x兲 ⯝

J共t⬘,t兲 = ␦共t⬘ − t兲 =U

U共t⬘,t兲 =

t ⬎ tf .

The different domains for f共t , x兲 are shown in Fig. 13; f共t , x兲 grows slowly for t ⬎ ts since the coefficient ␮0 ⬃ ␥2 Ⰶ 1. Recall that the drift ␮共t , x兲 and volatility v共t , x兲 are both functions of only exp兵−f n共t兲其 ⯝ exp兵−ᐉf共t , Tn兲其. As f共t , x兲 grows large, both ␮共t , x兲 and v共t , x兲 rapidly become deterministic and independent of f共t , x兲, as shown in Fig. 14. This in turn means that f共t , x兲 can be described by a linear Gaussian quantum field. Gaussian fields have configurations where f共t , x兲 takes small values and can, hence, revert back to the regime for its nonlinear evolution. In this manner, the Libor forward interest rates are driven by its exact evolution equation between the nonlinear and linear domains. APPENDIX F: JACOBIAN OF TRANSFORMATION A(t , x) \ ␾(t , x)

The change of variables from quantum field A共t , x兲 to ␾共t , x兲 is, from Eq. 共56兲, given by 共F1兲

Equation 共F1兲 is a nonlinear change of variables since drift ˜␳共t , x兲 depends on ␾共t , x兲 and, in principle, can have a nontrivial Jacobian. Taking the differential of Eq. 共F1兲 yields

␦˜␳共t⬘,x兲 . ␦␾共t,x兲

冋兿冕册兿

共F2兲

M−1

tⴱ

dtn

n=1

Instead, as follows from Eqs. 共E2兲 and 共E1兲, for ᐉf共t , x兲 ⬃ 1 the volatility and drift both become deterministic, leading to a finite evolution of f共t , x兲. Ignoring the stochastic term yields

t 苸关t0,tⴱ兴.

J共t⬘,t兲 =

M−1

exp兵⑀J共tn,tn+1兲其,

n=0

t0

boundary conditions:t1 = t⬘,

共E4兲

⳵ ␾共t,x兲 = ˜␳共t,x兲 + ␥共t,x兲A共t,x兲, ⳵t

⳵ −1 U , ⳵t

In Eq. 共F2兲 matrix multiplication is an integration over t given by 兰ttⴱdt. Time is discretized t → tn = t⬘ + 共n − 1兲⑀, n 0 = 1 , 2 , 3 , . . . , M = 共t − t⬘兲 / ⑀ to explicitly write out the matrix elements of U as follows:

f共0,x兲 1 = ᐉ␥ ⇒ t f = ts − ᐉ ⬍ ts . 1 − 关ᐉ␥ f共0,x兲兴t f ᐉ

f共t,x兲 ⬃ f共0,x兲e␮0t,

⳵ ␦˜␳共t⬘,x兲 ⳵ − ⬅ ␦共t⬘ − t兲 − J共t⬘,t兲 ⇒ J ⳵ t ␦␾共t,x兲 ⳵t

t M = t.

From Eq. 共F2兲 the Jacobian is given by

冋

det关J兴 = det U

册冋

册

⳵ ⳵ −1 U = det ␦共t⬘ − t兲 = const. ⳵t ⳵t

Hence, the Jacobian of the transformation in going from A共t , x兲 → ␾共t , x兲 is a constant; all constants involved in going from A共t , x兲 → ␾共t , x兲 cancel due to division by the partition function Z. Henceforth, the path integration measure will taken to be invariant, namely,

冕冕 D␾ =

DA.

The Jacobian being a constant is essential for the derivation of the Libor Hamiltonian given in Sec. XV. APPENDIX G: INTEREST RATE STATE SPACE Vt

The Hamiltonian and the state space are two independent ingredients of a quantum system; taken together they reproduce the Lagrangian and the path integral. The essential features of the interest rates’ Hamiltonian and state space are reviewed; a detailed discussion is given in 关2兴. The state space of a quantum field theory at time t, similar to all quantum systems, is a linear vector space—denoted by Vt. The dual space of Vt—denoted by Vt,Dual—consists of all linear mappings from Vt to the complex numbers and is also a linear vector space. The Hamiltonian Ht is an operator— the quantum generalization of energy—that is an element of the tensor product space Vt 丢 Vt,dual and maps the state space to itself, that is, Ht : Vt → Vt. For each time slice, the state space is defined for interest rates with x ⬎ t, as shown in Fig. 15. The state space has a nontrivial structure due to the underlying trapezoidal domain

046119-20



t

t1

t2

t2 +TFR t1 +TFR

t0

x

t0 +TFR

FIG. 15. The domain of the state space of the interest rates. In the figure the state space Vt is indicated for two distinct calendar times t1 and t2.

T of the xt space. On composing the state space for each time slice, the trapezoidal structure for finite time, as shown in Fig. 16, is seen to emerge from the state space defined for each time slice. The Hamiltonian for log Libor ␾共t , x兲 is nonlinear and defined on a nontrivial domain. Since the quantum field ␾共t , x兲 exists only for future time, that is, for x ⬎ t and hence x 苸关t , t + TFR兴. In particular, the interest rates’ quantum field has a distinct state space Vt for every instant t. The state space at time t is labeled by Vt, and the coordinate eigenvector of Vt is denoted by 兩␾t典. For fixed time t, the state space Vt consists of all possible functions of the interest rates, with future time x 苸关t , t + TFR兴. The elements of the state space of the forward interest rates Vt include all possible debt instruments that are traded in the market at time t. In continuum notation, the coordinate basis eigenstates of Vt are tensor products over the future time x, 兩 ␾ t典 =

兿

兩␾共t,x兲典,

tⱕxⱕt+TFR

and satisfy the following completeness equation:

兿

It =

tⱕxⱕt+TFR

冕

+⬁

−⬁

d␾共t,x兲兩␾t典具␾t兩 ⬅

冕

D␾t兩␾t典具␾t兩. 共G1兲

The time-dependent interest rate Hamiltonian H共t兲 is the backward Fokker-Planck Hamiltonian and propagates the interest rates backward in time—taking the final state 兩␾final典 given at future calendar time T f backward to an initial state 具␾initial兩 at the earlier time Ti. The transition amplitude Z for a time interval 关Ti , T f 兴 can be constructed from the Hamiltonian and state space by applying the time slicing method; since both the state space and Hamiltonian are time dependent one has to use the timeordering operator T to keep track of the time dependence: H共t兲 for earlier time t is placed to the left of H共t兲 that refers to later time. The transition amplitude between a final 共coordinate basis兲 state 兩␾final典 at time T f to an arbitrary initial 共coordinate basis兲 state 具␾initial兩 at time Ti is given by the following 关2兴:

再

Z = 具␾initial兩T exp −

Tf

冎

H共t兲dt 兩␾final典.

Ti

共G2兲

Due to the time dependence of the state spaces Vt the forward interest rates that determine Z form a trapezoidal domain shown in Fig. 16. APPENDIX H: INTEREST RATE HAMILTONIAN

Consider the Lagrangian density L共t , x兲 for logarithmic Libor field ␾共t , x兲 given by Eq. 共60兲,

冋

册册

1 ⳵ ␾共t,x兲/⳵ t − ˜␳共t,x兲 ⫻ D−1共t,x,x⬘兲 2 ␥共t,x兲

L共t,x兲 = − ⫻

冋

⳵ ␾共t,x⬘兲/⳵ t − ˜␳共t,x⬘兲 − ⬁ ⱕ ␾共t,x兲 ⱕ + ⬁. ␥共t,x⬘兲

The volatility ␥共t , x兲 is deterministic and ˜␳共t , x兲 is a nonlinear drift term defined in Eq. 共59兲. Neumann boundary condition, given in Eq. 共61兲, has been incorporated into the expression for the Lagrangian. The derivation for the Hamiltonian is done for an arbitrary propagator D−1共t , x , x⬘兲, although for most applications a specific choice, such as the stiff propagator, is made. Discretizing time into a lattice of spacing ⑀, with t → tn = n⑀, yields the Lagrangian L共tn兲, L共tn兲 ⬅

Figure 15 shows the domain of the state space as a function of time, shown for typical times t1 and t2.

冕

tn+TFR

dxL共tn,x兲,

tn

−

t

1 2⑀2

冕

冕冕 ⬅

x

tn+TFR

dx =

tn

A共tn,x兲D−1共t,x,x⬘兲A共tn,x兲,

共H1兲

x

A共tn,x兲 =

共␾tn+⑀ − ␾tn − ⑀˜␳tn兲共x兲

␥共tn,x兲

.

共H2兲

Note ␾共tn , x兲 ⬅ ␾tn共x兲 has been written to emphasize that time tn is a parameter for the interest rate Hamiltonian. The Dirac-Feynman formula relates the Lagrangian L共tn兲 to the Hamiltonian operator and yields

x

FIG. 16. The trapezoidal domain T of the forward interest rates required for computing the transition amplitude 具␾initial兩T兵exp − 兰TT f H共t兲dt其兩␾final典. i

冕

具␾tn兩e−⑀H f 兩␾tn+⑀典 = Ne⑀L共tn兲 ,

共H3兲

where N is a normalization. Equation H共H3兲 is rewritten using Gaussian integration and 共ignoring henceforth irrel-

046119-21


BELAL E. BAAQUIE

evant constants兲, using notation 兿x 兰dp共x兲 ⬅ 兰Dp yields e⑀L共tn兲 =

冕

+i

冋

Dp exp −

冕

⑀ 2

冕

册

x,x⬘

p共x兲D共t,x,x⬘兲p共x⬘兲共H4兲

p共x兲A共x兲 .

x

time at which it acts on the state space. The degrees of freedom ␾t共x兲 refer to time t only through the domain on which the Hamiltonian is defined. This is the reason that in the Hamiltonian the time index t can be dropped for the variables ␾t共x兲 and replaced by ␾共x兲 with t ⱕ x ⱕ t + TFR. The Hamiltonian for logarithmic Libor interest rates, from Eqs. 共H5兲–共H7兲, is given by

The propagator D共t , x , x⬘兲 is the inverse of D−1共t , x , x⬘兲. Rescaling the variable p共x兲 → ␥共t , x兲p共x兲, Eqs. 共H1兲 and 共H2兲 yield 共up to an irrelevant constant兲关28兴 exp兵⑀L共t兲其 =

冕

再冕

p共x兲共␾t+⑀ − ␾t − ⑀˜␳t兲共x兲

Dp exp i

再

⫻exp −

⑀ 2

冕

x

冎

H共t兲 = −

冕冕 1 2

t

冎

dxdx⬘M ␥共x,x⬘ ;t兲

t

t+TFR

−

␥共t,x兲p共x兲D共x,x⬘;t兲␥共t,x⬘兲p共x⬘兲 .

t+TFR

dx˜␳共t,x兲

␦2 ␦␾共x兲␦␾共x⬘兲

␦ , ␦␾共x兲

M ␥共x,x⬘ ;t兲 = ␥共t,x兲D共x,x⬘ ;t兲␥共t,x⬘兲.

共H8兲

For each instant of time, there are infinitely many independent interest rates 共degrees of freedom兲, represented by the collection of variables ␾t共x兲 , x 苸关t , t + TFR兴. Hence, one needs to use functional derivatives to represent the Hamiltonian as a functional differential operator and the Hamiltonian is written in terms of functional derivatives in the coordinates of the dual state space variables ␾t. Unlike the action S关␾兴 that spans all instants of time— from the initial to the final time—the Hamiltonian is an infinitesimal generator in time and refers to only the instant of

The derivation only assumed that the volatility ␥共t , x兲 is deterministic, which is a key feature of the Libor market model. The drift term ˜␳共t , x兲 in Hamiltonian is completely general and can be any 共nonlinear兲 functions of the interest rates 关29兴. General considerations related to the existence of a martingale measure rule out any potential terms for the interest rate Hamiltonian 关2,30兴. Nontrivial dynamics is contained in the kinetic term with the function M ␥共x , x⬘ ; t兲 encoding the model chosen for the interest rates; a wide variety of such models has been discussed in 关2兴. The drift term is completely fixed by the martingale condition and, in particular, by M ␥共x , x⬘兲. The quantum fields ␾共t , x兲 is more fundamental that the velocity quantum field A共t , x兲; the Hamiltonian cannot be written in terms of the A共t , x兲 degrees of freedom. The reason being that the dynamics of the forward interest rates are contained in the time derivative terms in the Lagrangian, namely, terms containing ⳵␾共t , x兲 / ⳵t; in going to the Hamiltonian representation, these time derivatives essentially become differential operators ␦ / ␦␾共t , x兲关31兴.

关1兴 B. E. Baaquie, Interest Rates and Coupon Bonds in Quantum Finance 共Cambridge University Press, Cambridge, 2009兲. 关2兴 B. E. Baaquie, Quantum Finance 共Cambridge University Press, Cambridge, 2004兲. 关3兴 The Euclidean quantum field, namely, A共t , x兲, driving L共t , T兲 is discussed in Sec. IV. 关4兴 D. Heath, R. Jarrow, and A. Morton, Econometrica 60, 1 共1992兲. 关5兴 A. Brace, D. Gatarek, and M. Musiela, Math. Finance 7, 127 共1997兲. 关6兴 F. Jamshidian, Finance Stoch. 1, 293 共1997兲. 关7兴 R. Rebonata, Modern Pricing of Interest-Rate Derivatives 共Princeton University Press, Princeton, 2002兲. 关8兴 R. Rebonata, Interest-Rate Option Models 共Wiley, New York, 1996兲. 关9兴 L. Andersen and J. Andreasen, Appl. Math. Finance 7, 1 共2000兲. 关10兴 M. Joshi and R. Rebonata, Quant. Finance 3, 458 共2003兲.

关11兴 Henry Labordére, Analysis, Geometry, and Modeling in Finance 共Chapman and Hall, Boca Raton, Fl, 2008兲. 关12兴 P. S. Hagan, D. Kumar, A. S. Lesniewski, and D. E. Woodward, Wilmott Magazine 84, 108 共2004兲. 关13兴 D. Brigo and F. Mercurio, Interest Rate Models—Theory and Practice 共Springer, New York, 2007兲. 关14兴 K. G. Wilson, Phys. Rev. 179, 1499 共1969兲. 关15兴 K. G. Wilson and W. Zimmermann, Commun. Math. Phys. 24, 87 共1972兲. 关16兴 B. E. Baaquie and J. P. Bouchaud, Wilmott Magazine 64, 67 共2004兲. 关17兴 B. E. Baaquie and Cao Yang, Physica A 388, 2666 共2009兲. 关18兴 I. Grubsic, Ph.D. thesis, Leiden University, 2002. 关19兴 S. E. Shreve, Stochastic Calculus for Finance II: ContinuousTime Models 共Springer, New York, 2008兲. 关20兴 P. Hunt and J. Kennedy, Financial Derivatives in Theory and Practice 共Wiley, New York, 2000兲. 关21兴 Recall that L共t , T兲 is the simple interest rate from T to T + ᐉ.

x,x⬘

共H5兲

Hence, the Dirac-Feynman formula given in Eq. 共H3兲 yields the Hamiltonian as follows: Ne⑀L共t兲 = 具␾t兩e−⑀H兩␾t+⑀典 =e−⑀Ht共␦/␦␾t兲

冕

冋冕

Dp exp i

x

共H6兲

册

p共␾t − ␾t+⑀兲 .

共H7兲

046119-22



关22兴 L. D. Landau and L. M. Lifshitz, Quantum Mechanics: NonRelativistic Theory, 3rd ed. 共Elsevier Science, Holland, 2003兲, Vol. 3. 关23兴 Henceforth, for notational convenience, the limit of TFR → +⬁ is taken. 关24兴 Note, on the right-hand side of the equation, the positive sign of the second term in the exponent. 关25兴 The second term in the exponential, namely, 兰ttⴱdt␴共t , tⴱ兲A共t , tⴱ兲, does not contribute to the singular piece; 0 when this term is squared the singular term coming from the quadratic product of A共t , x兲 at equal time is canceled by the vanishing integration measure. n → 共Tn 关26兴 Note that, for ᐉ → 0, one has the limit ᐉ兺m=I+1 − TI兲 : const. 关27兴 The dependence of ␥ and ␮ on t , x is henceforth ignored since

it simplifies the calculation and does not change the main conclusions. Since only two time slices are henceforth considered, the subscript n on tn is dropped as it is unnecessary. Drift is fixed by the choice of the numeraire. The logarithmic Libor rate ␾共t , x兲 has a nonlinear drift. A potential term is a function only of ␾共t , x兲 or f共t , x兲; the interest rate Hamiltonian can only depend on the ␦ / ␦␾共t , x兲 or ␦ / ␦ f共t , x兲. If one wants to use the velocity degrees of freedom A共t , x兲 in the state space representation, one needs to use the formalism of phase space quantization 关32兴. J. Zinn-Justin, Quantum Field Theory and Critical Phenomenon 共Oxford University Press, New York, 1993兲.

关28兴关29兴关30兴

关31兴

关32兴

046119-23