(Henry-Labordre (2009), 3.5). a) Using the gamma probability density function and integration by parts or Laplace...

(Henry-Labordère (2009), § 3.5).

a) Using the gamma probability density function and integration by parts or Laplace transform inversion, prove the formula

$$
\int_{0}^{\infty} \frac{\mathrm{e}^{-u x}-\mathrm{e}^{-\mu x}}{x^{ho+1}} d x=\frac{\mu^{ho}-u^{ho}}{ho} \Gamma(1-ho)
$$

for all $ho \in(0,1)$ and $\mu, u>0$, see Relation 3.434.1 in Gradshteyn and Ryzhik (2007).

b) By the result of Question (a), generalize the volatility swap pricing expression (8.19).

c) By Lemma 8.2 and the result of Question (b), find an expression of the volatility swap price using call and put functions.