108156.pdf

230 Mathematics for Finance Remark 10.1 To determine the initial term structure we need the prices of zero-coupon bonds. However ...

Variable Interest Rates 231 Buy a bond maturing at timeNand write a fractionB(n, N)ofabond maturing atn. (Here we use the assum ...

232 Mathematics for Finance be ln(110, 517. 09 / 100 ,000)∼=10%. Financial intermediaries may simplify your task by offering a s ...

Variable Interest Rates 233 months from now, at a rate of 10.23%. Does this present an arbitrage op- portunity? All rates stat ...

234 Mathematics for Finance Proposition 10.3 The bond price is given by B(n, N)=exp{−τ(f(n, n)+f(n, n+1)+···+f(n, N−1))}. Proof ...

Variable Interest Rates 235 We can see that the yields also increase. (See Exercise 10.20 below for a gener- alisation of this ...

236 Mathematics for Finance as follows: B(0,1) = 0. 9901 , B(0,2) = 0. 9828 , B(0,3) = 0. 9726 , B(1,2) = 0. 9947 , B(1,3) = 0. ...

11. Stochastic Interest Rates................................... This chapter is devoted to modelling the time evolution of rand ...

238 Mathematics for Finance terms this means that they are strongly positively correlated. 11.1 Binomial Tree Model............. ...

Stochastic Interest Rates 239 Next, we shall describe the evolution of bond prices. At time 0 we are given the initial bond pr ...

240 Mathematics for Finance Figure 11.2 Evolution of bond prices in Example 11.1 implicitly defining the logarithmic returns k(n ...

Stochastic Interest Rates 241 describes the random evolution of a single bond purchased at time 0 for 0.9726. The returns are ...

242 Mathematics for Finance that the words ‘up’ and ‘down’ lose their meaning here because the yield goes down as the bond price ...

Stochastic Interest Rates 243 At time 2 we have four sequences ofN−2 forward rates, and so on. At timeN− 1 we have 2N−^1 singl ...

244 Mathematics for Finance Figure 11.8 Forward rates in Exercise 11.3 We are now ready to describe the money market account. It ...

Stochastic Interest Rates 245 11.2 Arbitrage Pricing of Bonds Suppose that we are given the binomial tree of bond pricesB(n, N ...

246 Mathematics for Finance The model in Example 11.1 turns out to beinconsistent with the No-Arbitrage Principleand has to be r ...

Stochastic Interest Rates 247 Exercise 11.5 Evaluate the prices of a bond maturing at time 2 given a tree of prices of a bond ...

248 Mathematics for Finance replaced by k(n, N;sn− 1 u)>τr(n−1;sn− 1 )>k(n, N;sn− 1 d). (11.2) Any future cash flow can be ...

Stochastic Interest Rates 249 These fractions, defining thecoupon rate, are obtained by converting the short rate to an equiva ...