A variational formula for private sector bonds.

Summary A new methodology for the valuation of optional products is introduced based on a variational approach. This approach facilitates: (i) the study of the comparative statistics of the price of option products with respect to the model parameters, (ii) the numerical determination of these prices. The methodology introduced is more particularly applied to private sector coupon bonds that can be valued according to Geske (1977) as option compositions. We show that the valuation formula of Geske (1977) for private sector coupon bonds can be written as a solution to a minimization problem. This allows us to study the behavior of the price of a private sector bond as a function of the volatility of the issuing firm, the amount of the promised coupons, and the maturity of the bond. A study of the spread of a private sector bond is then proposed. It includes, more particularly, an analysis of the variations of the spread according to the distribution of the debt over the maturity of the bond considered. These results are established under the assumption of a flat rate structure. The last part of the work concerns an extension of our methodology to a random curve.