Investment Theory and Risk Management, + Website

Summary

This chapter had three objectives. The first was to tie together our work on the binomial lattice to the unifying underlying mathematics. That required us to develop a simple, yet appealing, model of stock price dynamics based on a stochastic extension of the basic model of continuous compounding. We introduced Wiener processes and generalized these with Ito's lemma to produce the basic lognormal property of stock prices and Brownian motion. The second objective tied the lattice parameters to the properties of stock price dynamics, in particular, their volatility and units of time Δt. From that discussion, we could develop the intuition of Black-Scholes-Merton and use this result to price options analytically. We learned that the BSM solution is the limit as Δt → 0 in the lattice (that is, as the number of nodes in the lattice goes to infinity). Finally, we extended the Brownian motion model to accommodate correlated stock prices and used this basic model in a Monte Carlo framework to check the accuracy of BSM as well as to offer a few thoughts on the properties of option portfolios as they relate to the correlation of the underlying stock price returns.