Actuarial Finance

18Applications and extensions of the Black-Scholes formula

In Chapter 16 we derived the so-called Black-Scholes formula, which gives compact and instructive expressions for the price of a call and of a put option in the Black-Scholes-Merton model. It turns out it is also possible to further apply and extend the Black-Scholes formula in various contexts, including financial products sold by insurance companies.

The aim of this chapter is mostly to analyze the pricing of options and other derivatives such as options on dividend-paying assets, currency options and futures options, but also insurance products such as investment guarantees, equity-indexed annuities and variable annuities, as well as exotic options (Asian, lookback and barrier options). In most of these cases, we can easily find a Black-Scholes-like formula for the no-arbitrage price of these contracts. The specific objectives of this chapter are to:

• use the Black-Scholes formula to price options on dividend-paying assets, currency options, futures options and exchange options;
• use financial engineering arguments and the Black-Scholes formula to obtain pricing formulas for investment guarantees, equity-indexed annuities and variable annuities;
• understand how to compute the break-even participation rate (annual fee) for common equity-linked insurance and annuities;
• derive the distribution of the discrete geometric mean of lognormally distributed random variables;
• understand why it might be difficult for various ...