Summary

In this chapter we review the basic financial mathematics behind option pricing. First, we discuss the no arbitrage assumption, which is the foundation for all modern financial mathematics. We present the binomial model of option pricing, and illustrate the principles of the risk neutral and real world measures, and of pricing by replication.

We discuss the Black–Scholes–Merton option pricing formula, and, in particular, demonstrate how it may be used both for pricing and risk management.

Introduction

In Section 12.4 we discussed the problem of non-diversifiable risk in connection with equity-linked insurance policies. A methodology for managing this risk, stochastic pricing and reserving, was set out in Sections 12.5 and 12.6. However, as we explained there, this methodology is not entirely satisfactory since it often requires the insurer to set aside large amounts of capital as reserves to provide some protection against adverse experience. At the end of the contract, the capital may not be needed, but having to maintain large reserves is expensive for the insurer. If experience is adverse, there is no assurance that reserves will be sufficient.

Since the non-diversifiable risks in equity-linked contracts and some pension plans typically arise from financial guarantees on maturity or death, and since these guarantees are very similar to the guarantees in exchange traded financial options, we can use the Black–Scholes–Merton theory of option pricing to price and actively manage these risks. When a financial guarantee is a part of the benefits under an insurance policy, we call it an embedded option.