Derivatives.

7 The Cox-Ingersoll-Ross (CIR) model is a one-factor interest-rate model of the form: dr = a(p-r)di +ourd= which leads to the solution at time t for the prices B(1, 7) of zero-coupon bonds of maturity 7 as follows: But, T)= exp[a(v)-b(o).r()] where a(7) = c In G exp(c, ) and b(t) = = exp(G () -1 [c[exp(cT)-1]+q ] cy[exp(q c) containing terms in o, u and q. (0) (a) List the key features that a good model of the entire interest rate yield curve should have. (b) Discuss how well the CIR model matches your criteria. (c) Describe in particular how the CIR model copes with two typical problems facing interest rate models: too wide a "dispersion" of rates over time due to future uncertainty, and allowing interest rates to become negative. [8] By considering the behaviour of the logarithm of the bond price, prove that for large T: B(t, T) = exp(-RT) where R= (q -c )c; is a constant long-term rate. [5] (iii) The CIR model is a "no-arbitrage" model. Outline what is meant by this term, and the significance of such a property for an interest-rate model. (3]Let $, be a stock price process which follows a geometric Brownian motion with parameters . of, and with stochastic differential equation: dS, = 05,dll, + (u+ to' )5,di where W, is a Brownian motion process. Let B, be a risk free asset whose price grows deterministically according to B, =2", and let Z, = B, 'S, be the discounted stock price process. Consider a dynamic portfolio (6, w,) consisting of o, units of S, and v, units of B, and let X =/(5,) be a path-independent claim on ST. Derive the stochastic differential equation for Z [5] Explain what is meant by a self-financing and replicating strategy for X. [5] (wi) (a) Explain how the Cameron-Martin-Girsanov (sometimes referred to as Girsanov's) theorem and the Martingale Representation theorem can be used to construct a replication strategy for X. (b) Derive an expression for the stochastic differential equation for the value of the claim. [10]