Most of the explicit pricing formulas presented in this book are centered around computations related to Brownian motion and to affine processes. In many practical situations these solutions are not sufficient since securities and derivatives may have American features or complicated payoff structures which preclude explicit pricing formulas. Examples of this include the pricing of risky coupon debt with compound option features due to asset-sale restrictions and swap pricing with two-sided default risk, which we met in Chapter 7. A quick way to obtain numerical solutions is through discrete-time tree models for security prices. This section gives an overview of how this is solved conceptually for our two main cases. For the explicit implementation the reader will have to consult the cited references. The goal of this section is to convey the basic idea behind building discrete-time arbitrage-free models for bond prices.
We have given a probability space (Ω, F, P) with Ω finite. Assume that there are T + 1 dates, starting at date 0. A security is defined in terms of two stochastic process—an ex-dividend price process S and a dividend process δ—with the following interpretation: St (ω) is the price of buying the security at time t if the state is ω. Buying the security at time t ensures the buyer (and obligates the seller to deliver) the remaining dividends δt+1(ω), δt+2(ω), …, δT(ω). We will follow the tradition of probability theory and almost always suppress the ω in the notation below. Of course, a buyer can sell the security at a later date, thereby transferring the right to future dividends to the new owner. Since there are no dividends after the final date T we should think of ST as 0 and δT as a final liquidating dividend if the security is still traded at this point.
In all models considered in this appendix (and in almost every model encountered in practice), we will assume that the model contains a money-market account which provides locally riskless borrowing and lending. The interest rate on this account is defined through the spot rate process: