## Interest rate Hamiltonian and option theory

The Hamiltonian is a differential operator that acts on an underlying state space To .1 A Hamiltonian formulation of option theory is discussed and shown to be equivalent to the Black-Scholes approach. In particular, it is shown that the Black-Scholes equation is mathematically identical to the imaginary time Schrodinger equation of quantum mechanics 7o . The Hamiltonian formulation of quantum field theory is equivalent to, and independent of, the framework based on the Feynman path integral...

## D 1 Lf fLtoT

ql V 1 tK 2 2 q q to, t , Tn The domain of integration for evaluating q2 is given in Figure 4.6. Note that q is the effective volatility for the caplet linear pricing formula and that the propagator for forward interest rates is required for pricing the caplet. The pricing formulas for caplets and floorlets are fixed by the volatility function a t, x , the correlation parameters i, X, n contained in the Lagrangian for the forward interest rates, as well as the initial interest rates term...

## Libor Market Model of interest rates

Libor L t, T is one of the primary interest rate instruments in the capital markets, the other being Euribor. The term Libor will be used generically for all interest rates on fixed deposits. The Libor Market Model LMM is defined in the framework of quantum finance and leads to a key generalization the Libors, for different future times, are imperfectly correlated. A major difference between a forward interest rates' model and the LMM lies in the fact that the LMM is calibrated directly from...

## Coupon bond European option HJM limit

In Section 4.11, the limiting cases for the quantum finance formulation of coupon bonds and interest rates were discussed. The HJM and BGM-Jamshidian models are special cases of forward interest rates being exactly correlated and correspond to the propagator being a constant. The limit of D x, xt 1 for all x, x' in turn yields M x,x' t a x - t D x,x' t a x' - t a x - t a x' - t . The limit of D x, x' t 1 is studied in order to determine the importance of having a nontrivial correlation for the...

## Numeraires for bond forward interest rates

Various numeraires are defined in the framework of the bond forward interest rates f t, x discussed in Chapter 5. Eqs. 5.1 and 2.20 yield the following t,x a t,x a t,x A t,x -x lt f t,x lt x where L t, Tn is the three-month benchmark Libor with tenor denoted by i. The main result of this chapter is that a numeraire, called the forward numeraire, can be chosen for the bond forward interest rates, such that all forward bond prices for future Libor time Tn To in with tenor i have a martingale...

## Hedging

All financial instruments are subject to the random evolution of underlying instruments such as stock prices, interest rates, exchange rates, etc. There are many ways of defining risk as discussed in Bouchaud and Potters 28 . Hedging is the general procedure for reducing, if not completely eliminating, an investor's exposure to risk. Derivative instruments are essential for hedging. For example, to hedge a portfolio that contains a security one also needs to include in the portfolio another...

## Libor and Euribor

The two main international currencies are the US Dollar and the Euro, which is the currency of the European Union. As can be seen from Table 2.1, almost 90 of international cash reserves are in the form of US Dollars or Euros. Cash fixed deposits in these currencies account for almost 90 of simple interest rates that are traded in the capital markets. Cash deposits in US Dollars as well as British Pounds earn simple interest at the rate fixed by Libor and deposits in Euros earn interest rates...

## Zero Coupon Floorlet

4.5 Put-call empirical Libor caplet andfloorlet 4.5 Put-call empirical Libor caplet and floorlet Put-call parity for caps and floors is a model-independent result that market prices obey. The prices of interest rate caplets and floorlets for Eurodollar futures contracts - expiring on 13 December 2004 with a fixed interest rate strike price of 2 - are analyzed for empirically testing put-call parity. Daily prices, from 12 September 2003 to 7 May 2004, are quoted with the interest rate in basis...

## Caps Caplets Floors Forwards

With par fixed interest rate at time T0 given by Libor have derivatives written on them, such as caps and floors these instruments are important interest rate derivatives and have many applications in the financial markets 65 . Interest rate contracts, such as caps and floors, can cover many years and involve a sequence of quarterly payments ranging from one to ten years. Consequently, pricing and hedging such derivatives require modeling of Libor over a long interval of time. Caps, floors, and...

## Instantaneous forward interest rates

Forward interest rates play a central role in the study of interest rates and coupon bonds. The forward interest rates provide a representation of zero coupon term structure that is analytically and conceptually very useful in the study of the bond market. To derive the instantaneous forward interest rates from the term structure of the zero coupon bonds, consider two bonds that are mature at infinitesimally separated future times. More precisely, in Eq. 2.6 let T2 T1 e hence one obtains the...

## Zero coupon yield curve and interest rates

Both the ZCYC and the forward interest rates are descriptions of the same financial instrument, namely the zero coupon bonds in the case of Libor, the ZCYC does not correspond to any actual traded zero coupon bonds, but, rather, is a compact way of expressing market data on the term structure for all the Libors taken together. The two different descriptions, namely the ZCYC and the forward interest rates, are useful for representing different aspects of the interest rate and bond markets....

## Discrete discounting zero coupon yield curve

Recall that, from Section 2.5, the yield-to-maturity z of a zero coupon bond is the annual simple interest that is discretely compounded every year. Let T, t be the maturity and issue date of the bond as before, let T t T t year be an integer equal to the number of years. On maturing, the bond value of 1 will compound to 1 z T t . Since, on maturity, the payoff of the bond is 1, the relation of z to the price of the zero coupon bond at t is given by Note the yield-to-maturity varies for the...

## Libor Zero Coupon Curve Excel

2 Interest rates and coupon bonds 3 2.2 Expanding global money capital 4 2.3 New centers of global finance 8 2.5 Three definitions of interest rates 10 2.6 Coupon and zero coupon bonds 12 2.7 Continuous compounding forward interest rates 14 2.8 Instantaneous forward interest rates 16 2.10 Simple interest rate 20 2.11 Discrete discounting zero coupon yield curve 22 2.12 Zero coupon yield curve and interest rates 26 2.14 Appendix De-noising financial data 29 3 Options and option theory 32 3.5...