期貨理論與實務英文版課件：Chapter 11 Stochastic Interest Rates

1、chapter 11Stochastic Interest Ratesmain content11.1 Binomial Tree Model11.2 Arbitrage Pricing of Bonds11.2.1 Risk-Neutral Probabilities11.3 Interest Rate Derivative Securities11.3.1 options11.3.2 Swaps11.3.3 Caps and FloorsThe differences between the model of bond prices and the model of stock:1.Bon

2、d prices is described by the running time and the maturity time, whereas stock prices are functions of the running time.2.There are many ways of describing the term structure: bond prices, implied yields, forward rates, short rates.3.The model needs to match the initial data. For a stock this is jus

3、t the current price. In the case of bonds the whole initial term structure is given.4.Bonds become non-random at maturity. This is in sharp contrast with stock prices.5.The dependence of yields on maturity must be quite special. Bonds with similar maturities will typically behave in a similar manner

4、.11.1 Binomial Tree ModelThe probabilities will be allowed to depend on particular states. We write p(sn) to denote the probability of going up at time n + 1, having started at state sn at time n.definitionAt time 0,we are given the initial bond prices for all maturities up to N, that is, a sequence

5、 of N numbersB(0, 1),B(0, 2),B(0, 3), . . . ,B(0,N . 1),B(0,N).At time 1,two possibilities distinguished by the states u and dB(1, 2; u),B(1, 3, u), . . . ,B(1,N 1; u),B(1,N; u),B(1, 2; d),B(1, 3; d), . . . ,B(1,N 1; d),B(1,N; d).The evolution of bond prices can be described be means of returns. Sup

6、pose we have reached state sn1 and the bond price B(n 1,N; sn1) becomes known. Then we can writeB(n,N; sn1u) = B(n 1,N; sn1) expk(n,N; sn1u);B(n,N; sn1d) = B(n 1,N; sn1) expk(n,N; sn1d).examplethe length of each step is one monthThe evolution of bond prices is in perfect correspondence with the evol

7、ution of implied yields to maturity. Namely,with the same tree structure as for bond prices. In particular, the final yields are non-random given that the state sn1 at the penultimate step is known.the random evolution of forward rates:At time 1 we have two possible sequences of N -1 forward rates o

8、btained from two sequences of bond pricesf(1, 1; u), f(1, 2; u), . . . , f(1,N 1; u),f(1, 1; d), f(1, 2; d), . . . , f(1,N 1; d).11.2 Arbitrage Pricing of BondsExample： A(0)=1 A(1)=1.01 portfolio (x, y), with x being the number of bonds of maturity 3 and y the position in the money market, such that

9、 the value of this portfolio matches the time 1 prices of the bond maturing at time 2.Are the prices provide an arbitrage opportunity?How can we adjust it?P(0) = C1 expy(0, 1) + C2 exp2y(0, 2) + + (CN + F) expNy(0,N).where p:The coupon bond price P at a particular time will not include the coupon du

10、e (the socalled ex-coupon price) y(0,N):spot yields Ck + P(k; sk)= Ck + Ck+1 exp.y(k, k + 1; sk) + + (Cn + F) exp. (n . k)y(k, n; sk).PropositionA coupon bond maturing at time N with random couponsCk(sk1) = (expr(k 1; sk1) 1)Ffor 0 k N is trading at par. (That is, the price P(0) is equal to the face

11、 value F.)11.2.1 Risk-Neutral Probabilitiesbond price B(n,N; sn) and two possible values at the next step, B(n+1,N; snu) and B(n+1,N; snd). These values represent a random variable, which will be denoted by B(n + 1,N; sn). If n = N -1,We are looking for a probability p* such thatB(n,N; sn) = p.B(n +

12、 1,N; snu) + (1-p*)B(n + 1,N; snd)*expr(n; sn).PropositionThe lack of arbitrage implies that the risk-neutral probabilities are independent of maturity.exerciseSuppose that the risk-neutral probabilities are equal to 1/2 in every state. Given the following short rates, find the prices of a bond matu

13、ring at time 3 (with a one-month time step, = 1/2 ):11.3 Interest Rate Derivative SecuritiesOptionsWith the bond prices as in Example 11.5 (Figure 11.10), consider a call option with exercise time 2 and strike price X = 0.99 on a zero-coupon bond maturing at time 3.As a result, the price of the opti

14、on is 0.000 SwapsWhy we need swaps?How can we get benifites from swaps?example:Suppose that company A wishes to borrow at a variable rate, whereas B prefers a fixed rate. Banks offer the following effective rates (that is, rates referring to annual compounding):In this case we say that A ha

15、s comparative advantage over B in the fixed rate, with B having comparative advantage over A in the variable rate. (Notwithstanding the fact that the overall credit rating of A is better, as reflected by the lower interest rates offered.) In these circumstances A should borrow at the fixed rate, B s

16、hould borrow at the variable rate, and they can swap their interest payments.11.3.3 Caps and FloorsA cap is a provision attached to a variable-rate bond which specifies the maximum coupon rate paid in each period over the lifetime of a loan.A caplet is a European option on the level of interest paid

17、 or received. A cap can be thought of as a series of caplets.exampleWe take a loan by selling a par floating-coupon bond maturing at time 3.(a bond which always has the par value)The cash flow shown below includes the initial amount received for selling the bond together with the coupons and face value to be paid:Consider a caplet that applies at time 1 (one month) with strike interest rate of 8% (corresponding to 0.67% for a one-month period). The coupon determined by the caplet rate is 0