feynman perturbation expansion for coupon bond option price in a field theory of interest rates by...
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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
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.
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
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
;
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.
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.