In Chapters 3, 4, and 5, I discussed the valuation of options when the variability entered only through future price (value or rate) movements. As a consequence, the bulk of my discussion in this chapter will focus on the estimation of model parameters when there is only price stochasticity. As in Chapter 5, I will continue to assume that the underlying is an index that pays continuous dividends.
I showed in Chapter 3 how equations (3.7a) and (3.7b)—were used to value European-style options on a stock that pays a continuously compounded dividend. For the ease of reading, I have re-stated these equations again:
When illustrating the implementation of equations (3.7a) and (3.7b) in Table 3.4, I assumed that all the inputs going into the model (e.g., St, K, t, T, rt, T, qt, T, σt, T) are known—which is far from the truth in practice. In reality, the only inputs that are known with 100 percent objectivity are St, K, t, T. In fact, while St represents the value of the observable index at the time of valuation, the ...