The Proof Of Agliardi Elettra's Conjecture And Its Extension In Jump-diffusion Process

When a firm financed with a combination of debt and stock, the stockcan be viewed as an "European call option"on the value of the firm. Inthis setting, an option on the stock is an option on an option, that is a com-pound option. R.Geske (1979) derived an analytical formula for compoundoption when the firm value follows "Geometric Brownian Motion"and thevolatility of the firm value and the interest rate are constant. Agliardi Elet-tra and Agliardi Rossella (2003)derived a formula of compound option incase of time-dependent volatility and the interest rate. In his paper, AgliardiElettra has mentioned that Geman had derived an analytical formula forcompound option in case of constant volatility and stochastic interest rateby using change of numeraire and change of probability measure. And inhis paper, Agliardi Elettra conjectured an analytical formula for compoundoption in case of time-dependent volatility and stochastic interest rate.In this paper,by using change of numeraire and change of probabilitymeasure, I have derived the analytical formula for compound option in case oftime-dependent volatility and stochastic interest rate ,which proved AgliardiElettra's conjecture, and I have applied the theory of change of numeraireand change of probability measure to jump-di?usion process, and made someresearch on compound option when underlying asset follows jump-di?usionprocess and interest rate is stochastic.