| The classical approach to modeling prices of financial instruments is to identify a certain (small) family of "underlying" processes, whose dynamics are then described explicitly, and compute the prices of all other financial derivatives by taking expectations under the risk-neutral measure or maximizing the utility function. Such is the famous Black-Scholes model, where the underlying stock price is assumed to be given by geometric Brownian motion. On contrary, the present paper is concerned with the construction of so-called market models, which describe the simultaneous dynamics of the prices of all liquidly traded derivative instruments. This framework was originally advocated by Heath, Jarrow and Morton for the Treasury bond markets, and, therefore, is sometimes referred to as the HJM approach.;We discuss the arbitrage free dynamic stochastic models for the markets with European call options of infinitely many strikes and maturities as the liquid derivatives. It turns out that, in order to prescribe the dynamics of all the call prices simultaneously, it is important to choose the right code-book for the option prices. More precisely, we need to find a deterministic mapping (the code-book) of the call price surface (as a function of strike and maturity) into some convenient state space and then use a system of stochastic differential equations (SDE's) to prescribe the dynamics of the call prices in this state space. One popular example of the code-book is the implied volatility surface, which is widely used by the practitioners. Unfortunately, although the implied volatility code-book provides a good representation of the call price surface at a given moment of time, it is not well suited for describing the dynamics of the option prices. Therefore, we have to introduce different code-books. In fact, it turns out that the choice of the appropriate code-book depends heavily upon the assumptions we make on the paths of the underlying. In particular, we choose to represent the option prices through the local volatility in case when the underlying has continuous paths, and we use the tangent Levy density in case when the underlying is a pure jump process.;After the dynamics of call prices are specified, it is, of course, important to make sure that these "prices" are, indeed, the prices of the call options, namely, that they coincide with the conditional expectations of the respective payoffs under some pricing measure. Thus, the main thrust of our work is to characterize the consistency between the explicit dynamics of option prices and their definition as conditional expectations. We then address the issue of construction of the consistent models, and provide some examples that help better understand the scope of the proposed families of market models. |
