Feynman Perturbation Expansion for Coupon Bond Option Price in a Field Theory of Interest Rates

byBelal E. Baaquie

Department of PhysicsNUS

Outline of Talk

• Field theory of forward interest rates• Empirical tests of the model• Coupon Bond Options• Feynman Perturbation Expansion• Empirical Analysis of Swaptions Reference: ‘Quantum Finance’ by B E Baaquie Cambridge University Press (2004)

http://www.cambridge.org/catalogue/catalogue.asp?isbn=0521840457

Forward Interest Rates

where the forward price of the bond at time t* is defined by

Domain of Forward Interest Rates f(t,x)

Note that x>t for f(t,x), with f(t,t)=r(t): spot interest rate.

The maximum future time TFR for which the interest rates is about 30 years, and is usually taken to be infinite.

Forward Interest Rates: A Quantum Field

Both the forward interest rates f(t,x) and its derived velocity field A(t,x) are considered to be two dimensional quantum fields; for each t and each x f(t,x) (and A(t,x)) is an independent (random) integration variable.

Quantum Theory of Forward Rates

‘Stiff’ Action for the Forward Interest Rates

Empirical Test of the Field Theory Model

• Discretize time so that t=nwhere 1day. Define

Then, for ’x’-t, the propagator is

Hence

Empirical PropagatorMarket data for Libor (London Interbank Offer Rates) for Eurodollar deposits yields the following market normalized correlator

Libor Data Fitted by the ‘Stiff’ Propagator

Inset line is the slope orthogonal to the diagonal with dashed line for

Recall z=(x-t

Treasury Coupon Bonds

Coupon Bond Option

Note C(t0, t*, K) is the price of the option and K is it’s strike price.

The Payoff FunctionThe fundamental idea in evaluating the price of the coupon bond option is to perturb the price about the Forward Coupon Bond price.

where the forward price of the bond F and the perturbation term V (later seen to be a ‘potential’ term in the action S[A]) are given by

;

The Partition Function

The Dirac delta function is given by

Nonlinear Quantum Field Theory

Perturbation Expansion

Forward Price CorrelatorThe expansion coefficients are given in terms of the correlator Gij, which is the correlation between two Forward Bond Prices Fi=F(t0,t*,Ti) with Fj=F(t0,t*,Tj). Given below is a graph of correlator Gij

and its diagramatic representation.

Feynman Diagrams for the Perturbation Expansion

Coefficient ACoefficient B

The value of the coefficient D=0 due to the martingale condition.

Convergent Expansion for Z()

Coupon Bond Option Price

Note in graph B=C=0.

Swaps

Swaptions

Swapti on Pri ce for Li bor of 2by10 Swapti on

0

20000

40000

60000

80000

100000

120000

1 32 63 94 125 156 187 218 249

T

Pric

e (P

rinc

ipal

US$

1mi

llio

n)

Market pri ceModel pri ce

Market and Model’s Price for Libor Swaptions (At the Money)

Conclusions• Historical data for forward interest rates is described to an

accuracy of over 99% by the quantum field theory model.• Coupon bond option yields a highly nonlinear theory

quantum field theory. The regularity of the effective propagator M(x,x;t)=2(t,x) is reason that divergences characteristic of field theories are absent in the Feynman diagrams for the swaption price.

• Pricing of coupon bond option is possible as a perturbation expansion because the volatility of the forward rates 2(t,x) is a small parameter.

• Field theory model predicts swaption prices for the market quite accurately and also matches the trends of the market.

• All the correlators of the two or more swaptions can be computed in the field theory model.