Option Pricing: Black-Scholes for Actuaries

The Black-Scholes Framework

The Black-Scholes model provides a framework for pricing options on assets that follow geometric Brownian motion. The key insight is that a riskless portfolio can be constructed by continuously hedging the option with the underlying asset, leading to risk-neutral pricing. The Black-Scholes formula for a European call option depends on five inputs: the current stock price, strike price, time to expiration, risk-free rate, and volatility. The model assumes constant volatility, continuous trading, no transaction costs, and log-normal stock prices. While these assumptions are violated in practice, the framework provides the foundation for option pricing theory.

Actuarial Applications

Many insurance products contain embedded options that require option pricing techniques. Variable annuity guaranteed minimum death benefits (GMDB) and guaranteed minimum income benefits (GMIB) are essentially put options on fund performance. Equity-indexed annuity crediting formulas involve call options on equity indices. Surrender options in life insurance policies can be analyzed as interest rate options. Actuaries use Black-Scholes and its extensions (stochastic volatility models, jump-diffusion models) to value these embedded guarantees. Understanding Greeks (delta, gamma, vega, rho) helps actuaries design hedging strategies for the market risk created by these embedded options.