Suppose the price S of a non-dividend-paying asset has a constant volatility . Assume the volatility at

Suppose the price S of a non-dividend-paying asset has a constant volatility σ. Assume the volatility at date t of a discount bond maturing at T > t is

φ

κ



1−e−κ(T−t)

 (17.17)

for constants κ > 0 and φ > 0. Assume the discount bond and stock have a constant correlation ρ. Let P denote the price of the discount bond.

(a) Calculate the volatility of S/P as a function σ (ˆ t).

(b) Define

σavg =

.

1 T

T 0

σ (ˆ t)2 dt .

Show that

σ2 avg = σ2 +

1

κ 2



φ2 − 2κρσ φ −(2φ2 −2κρσ φ)1 −e−κT

κT



+φ2

1 −e−2κT 2κT

.

(c) Apply results from Section 17.3 to derive a formula for the value of a European call option on the asset with price S that matures at T.

(d) Use l’Hôpital’s rule to show that σavg ≈ σ for small T.

(e) Show that σavg > σ for large T if ρ is sufficiently small.

Note: The bond volatility (17.17) arises in the Vasicek term structure model, with κ being the rate of mean reversion of the short rate process and φ being the

(absolute) volatility of the short rate process (Section 18.3).