By Dr Hans-Peter Deutsch (auth.)
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2. 5 Spot Rates Spot rates are the current yields on securities which generate only one single payment (cash ow) upon maturity. A zero coupon bond is an example of such a security as are coupon bonds whose last coupon payment prior to maturity has already been made. The spot rates as a function of time to maturity W is called the spot rate curve or the term structure. These spot rate curves can be represented by the discount factors EU (w> W ). 6 Forward Rates Forward rates are the future spot rates from today’s point of view which are consistent with the current spot rates in the sense of the following arbitrage argument: One monetary unit is to be invested today (time w) until a specied maturity date W 0 .
They are actually unnecessary but the markets are inconceivable without them because of strong historical ties. This is particularly true of holiday calendars, some of which even have religious roots. 3 Discount Factors In order to concentrate on the essentials, a general notation will be observed in this book which holds for all compounding, day count, business day and other market conventions. To accomplish this, discount factors rather than interest rates will be employed throughout. The discount factor is the value by which a cash ow to be paid at a time W is multiplied in order to obtain the value of the cash ow at an earlier time w (for example, today).
As a result, only statistical statements are available to describe the properties of this vector. For example, the length of this vector cannot be determined with certainty but we could calculate its mean length. This requires a large number of random walks with the same number of steps. For each of these random walks, the square of the Euclidean ® norm of the end-to-end vector R is determined and used to calculate the mean R2 = The mean end-to-end distance is than dened as the square root of this value.
Derivatives and Internal Models by Dr Hans-Peter Deutsch (auth.)