Name: \underline{Jaime Frade}

Financial Homework 6.3 (Already turned in 6.1-6.2 in mailbox)

1. 6.3 Verify the following:

d_{t}P (t, T) = P (t, T)[Σ(t, T)d_{t}W̃_{t} + r_{t} d_{t}]  (1)

Given the following, will use to verify (1):

W̃_{t} = W_{t} + ∫_{t}^{T} γ_{s}ds  (2)
Σ(t, T) = − ∫_{t}^{T} σ(t, u)du  (3)
Z(t, T) = β_{t}^{−1}P (t, T)  (4)

Z(t, T)is the SDE of the discounted bond, where

β_{t} ⇒ cash bond and

P (t, T) ⇒ price of the T-maturity bond.

d_{t}Z(t, T) = Z(t, T)Σ(t, T)dW̃_{t}  (5)
for discounted bond under underℚ  (6)

Since β_{t} is a zero-vaildity process satisfying SDE,

dβ_{t} = r_{t}β_{t}dt,  (7)

Using above, will verify (1) by using the product rule and simplify using above equations.

d_{t}(P (t, T)) = d_{t}(β_{t} Z(t, T))  (8)
applying the product rule obtains  (9)
= β_{t} \underbrace{d_{t}Z(t, T)}_{(5)} +Z(t, T) \underbrace{d_{t}β_{t}}_{(7)}  (10)
= β_{t} Z(t, T)Σ(t, T)dW̃_{t} + Z(t, T)r_{t}β_{t}dt  (11)
= β_{t}Z(t, T)(Σ(t, T)dW̃_{t} + r_{t}dt)  (12)
= P (t, T)(Σ(t, T)dW̃_{t} + r_{t}dt)  (13)

∴