Summary

In this chapter, we briefly discussed the persistence of nonlinearity in economics and finance. We looked at some nonlinear models that are commonly used in finance to explain certain aspects of data left unexplained by linear models: the Black-Scholes implied volatility model, Markov switching model, threshold model, and smooth transition models.

In Black-Scholes implied volatility modeling, we discussed the volatility smile that was made up of implied volatilities derived via the Black-Scholes model from the market prices of call or put options for a particular maturity. You may be interested enough to seek the lowest implied volatility value possible, which can be useful for inferring theoretical prices and comparing against the market prices for potential opportunities. However, since the curve is nonlinear, linear algebra cannot adequately solve for the optimal point. To do so, we will require the use of root-finding methods.

Root-finding methods attempt to find the root of a...