3. Solving Nonlinear Equations

Calibration of financial models corresponds to the problem of finding the roots of a function. These might appear directly, as in the example of finding implied volatility in Sect. 3.1 or indirectly. Some examples of indirect problems are maximizing the likelihood (see Sect. 13.1) or minimizing a loss function, as in method of least squares in Chap. 17. Nonlinear equation solving is a central component in many numerical procedures. Suppose the price of one share of a particular stock at time t is denoted by \(P_t\). Then the ratio \(P_t/P_{s}\), for \(s<t\) is the return over the period (s, t). One of the central problems of mathematical finance is to find the volatility, since it is not an observed quantity and has to be inferred from observed market prices of stocks or derivatives. Volatility is a measure of dispersion of prices of a particular asset over a time period. Assuming that the distribution of returns does not change with time, volatility is the standard deviation of log returns for unit time. The estimate of volatility that is derived from the option price is called implied volatility.