Suppose the dynamics of the short rate r(t) is governed by the governing differential equation for the

Question:

Suppose the dynamics of the short rate r(t) is governed by

the governing differential equation for the price of a zero coupon bond B(r,t) is given by [see (7.2.8)] the governing differential equation for the price of a zero coupon bond B(r,t) is given by [see (7.2.8)]

at + p2 22 B 2 ar2 ar + ( - 2p) - - r B = 0. =

For any noncoupon bearing paying claim whose payoff depends on r(T ), its price function U(r,t) is governed by the same differential equation as above. Now, suppose we relate the price of the claim to the bond price by defining

V(B(r, t), t) = U(r, t),

show that V (B,t) satisfies

av at + av B +rB-rV = 0, 202 a B2 2

where the volatility of bond returns σ_B is given by

OB (r, t) = p(r, t) B(r, t) B r (r, t). Suppose the claim’s payoff is f (BT ) at maturity T , by applying the Feynman– Kac Theorem, show that V (B,t) admits the following representation