| This work addresses the problem of pricing interest-rate derivative securities and the use of quoted prices of traded instruments to calibrate the corresponding interest-rate dynamics. To this end, an arbitrage-free model of interest rate evolution is adopted, for which the local drift will depend on the history of volatility, thus leading to path-dependent pricing. This model is based on the Heath-Jarrow-Morton formulation but, in addition, presupposes that the volatility process is not defined a-priori. This leads to a path-dependent problem that becomes Markovian on by considering additional state-variables and hence increasing the dimensionality of the computation. Instead of solving the resulting 3-dimensional partial differential equation, an alternative approach, based on conditional expectations of the history of volatility, is taken. This pricing method is adapted to a non-linear (adverse volatility) setting, and used as the core of a non-parametric model calibration technique. The algorithm, by performing an optimization over volatility surfaces, finds a volatility surface that matches the market prices of a given set of securities. This method also finds a hedge for volatility risk, using derivative securities as hedging instruments. In particular, we present results obtained for the problem of hedging bond options using interest-rate swaptions.; The conditional expectation approach is explored further, and found to be of interest in its own right for the pricing of several kinds of path-dependent instruments, providing an alternative to increasing state-space dimension in order to satisfy a Markov property. In particular, we show how this method speeds up the computation of prices for some types of exotic options, while being general enough to apply to both linear and non-linear pricing of books of options. |
