For the two factor CIR model proposed by Longstaff and Schwartz (1992), the short rate r(t) is defined by where and are positive constants, and Under the risk neutral measure, the risk factors x and y are governed by where Z 1 and Z 2 are uncorrelated standard Brownian processes Let V denote the in...

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For the two-factor CIR model proposed by Longstaff and Schwartz (1992), the short rate r(t) is defined

Question:

For the two-factor CIR model proposed by Longstaff and Schwartz (1992), the short rate r(t) is defined by

r = ax + By,

where α and β are positive constants, and α = β. Under the risk neutral measure, the risk factors x and y are governed by

dx = (y - 8x) dt + xdZ dy (ny) dt + ydZ2, = where Z₁ and Z₂ are uncorrelated standard Brownian processes. Let V denote the instantaneous variance of changes in the short rate.

(a) Show that

V = ax + By.

(b) Using Ito’s lemma, show that the dynamics of r and V are governed by

dr = ( + dV = 20 + - r V V ( ) =artin- + ta? - + dZ1 + , ( ) Br - V ( ) dZ1 + 2, - V dt

Note that r and V together form a joint Markov process.

and variance E[r] var(r) = + + 282 22 Similarly, show that V also has a stationary distribution with mean

and variance ELVI-/+ E[V] = var(V) = 'n + 282 22

(d) Let B(r,V,τ) denote the price function of a unit discount bond with τ periods until maturity. Show that

where and B(r, V, t) = AY (T) Bn (T) exp(KT + C(t)r + D(t)V), A(T) = B(T) = C(T) = D(T) = 20 (8 + 0) (exp(pt) (e) Suppose a < , show that V (t) is limited to the range (ar (t), Br(t)) at any time t.