You are given the spot-rate model;

dr₁ = ? (k ? r_t) dt + bdW,

where the W_t, is a Wiener process under the real-world probability.

Under this spot rate model, the solution to the PDE that corresponds to a default-free pare discount bond B(t, T) gives the closed-form bond pricing formula B(t, T):

Where

R = k ? b?/? ? b²/?².

Now consider the following questions that deal with properties of discount bonds whose prices can be represented by this formula,

(a) Apply Ito?s Lemma to the bond formula that gives B(t, T) above and obtain the SDE that gives bond dynamics.

(b) What are the drift and diffusion components of band dynamics? Derive these expressions explicitly and show that the drift ? is given by;

? = r_t ? b?/?(1 ? e^-?(T?t))

and that the diffusion parameter equals:

b/? (1 ? e^-?(T-t)).

(c) Is it expected that the diffusion parameter is independent of market price of risk ??

(d) What is the relationship between the maturity of a discount bond and its volatility?

(e) Is the risk premium, that is, the return, in excess of risk-free rate proportional to volatility? To market price of risk? Is this important?

(f) Suppose T ? ? x, what happens to the drift and diffusion parameters?

(g) What does the R represent?

Transcribed Image Text:

B(1, 7) = e(1-i)(R-r}-{T-R-(1-t