This chapter will introduce some basic concepts of modern mathematical finance. One goal is to cover the fundamentals of option pricing. This has become an important tool in actuarial mathematics, since many insurance and annuity contracts today contain the so-called ‘embedded options’, which we discussed in Section 13.2. For the most part, we carry this out in a discrete setting, but we do move into the continuous-time approach briefly in order to introduce the Black–Scholes–Merton formula. Another major objective in this chapter is to revisit the basic quantity of a discount function which we introduced early on. In the first part of the book, we treated this as a deterministic function, but a more realistic approach would be to consider v(s, t) as a random variable, reflecting the stochastic nature of investment returns that we discussed in Chapter 14. In particular, we seek a version of the key identity, Formula (2.1) in this stochastic setting.
A prerequisite for this chapter is the starred Section 2.12. We assume familiarity with concepts discussed in that section such as as short selling, forward contracts, and arbitrage.
20.2 Modelling prices in financial markets
A financial market is an institution designed to facilitate the trading of financial assets, such as stocks or bonds. Certain individuals wish to buy such assets, while others wish to sell them, and the financial market provides a forum ...