Valuation of bonds and bond options in continuous time.

Summary This research proposes to define the theoretical framework for determining the term structure of interest rates and the value of interest rate contingent assets, taking into account, in particular, the randomness of rate volatility. This study examines both the arbitrage and general equilibrium approaches. We present option pricing models based on the price dynamics of the underlying asset, the bond, following the example of the Black-Scholes model. The limitations of these models lead us to highlight the fact that the functional relationship between the bond price and the interest rate is the major relationship in the arbitrage approach. Models based on the term structure of rates are well suited for valuing bonds and bond options. The Martingale approach provides the conceptual and mathematical framework for removing risk premiums from the formulas for valuing interest rate contingent assets and incorporating the original term structure of rates. Finally, we abandon the assumption of a constant or deterministic volatility of rates, to highlight its stochastic character. Two directions are favored. On the one hand, the evaluation of bonds is carried out in the framework of an economy with imperfect information, by considering that volatility is an unobservable variable and by using the mathematical tool of filtering. On the other hand, we evaluate the bonds in the framework of a model.