ch5 - Chapter 5 Interest Rate Models 5.1 Why is There a...

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Chapter 5 Interest Rate Models 5.1 Why is There a Bond Market? In our treatment of the Black-Scholes model, we have always dealt with a risky stock and a riskless investment which could be a bond. However bonds are traded in their own right and we should try to understand why this is so. All bonds have a “par” or “nominal” value e.g. £ 100 or £ 1000. This is the amount on which interest is calculated and it is repaid to the buyer on maturity. Now suppose that the UK government issues a 25 year 10% bond in 2008 at par value £ 100. Suppose that you buy £ 5000 worth in 2008. Two years later you need the money and decide to sell but no-one wants to buy your bonds at the asking price of £ 100. Why is this? Well consider the scenario whereby interest rates have risen so that 25 year government bonds now pay out £ 12 . 5% per annum. Who would want to pay £ 100 to get £ 10 per annum when they can get £ 12 50p? They might oFer you £ 80 since 12 . 5 × 80 = 10, so the market price of your bond has dropped from £ 100 to £ 80. However if another two years pass and interest rates drop to 8%, your bond is now more attractive and you should check that its market price is now £ 125. So if we are modelling the price of bonds, we must take into account the variation of interest rates. Another factor we need to consider is closeness to redemption date. Con- sider again the scenario we discussed above where interest rates have fallen to 8% but suppose this happens after 24 years and 10 months rather than after 2. Now if you try to sell your £ 100 bond for £ 125, you’ll ±nd no takers as there are only two monthly interest payments to go which are worth roughly £ 1 . 33. After the two months have passed the bond is redeemed for its par value of £ 100 and a loss of £ 23 . 67p has been incurred. 44
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These considerations might lead you to think that modelling bond prices is a complicated business. This is undoubtedly true and the bond market is an active area of current research. In this short section we’ll simply aim to survey some of the most accessible concepts and look at one of the simplest models. 5.2 Term Structure of Interest Rates When studying the bond market it is usual to assume that we have a zero- coupon bond i.e. one that doesn’t pay out any interest. Although this may seem a gross simpliFcation, we will see that the problem of understanding such markets generates a great deal of interesting mathematics. We will study bonds that are bought at time zero and which pay out 1 (pound/euro/dollar) at the redemption time T and we will treat T as a variable. The value of the bond at time t where 0 t T is denoted by P ( t, T ) and we treat this as a function of two variables. Of course we must have P ( T,T ) = 1. As we don’t know the market price of the bond at time t , we should introduce a probability space (Ω , F ,P ) and model P ( t, T ) as a random variable. ±or Fxed T we can then regard ( P ( t, T ) , 0 t T ) as a stochastic process while if we Fx t and ω Ω, the mapping T P ( t, T )( ω ) is a function from [ t, )to R and we will assume that it is di²erentiable. The family
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ch5 - Chapter 5 Interest Rate Models 5.1 Why is There a...

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