Advanced Modelling in Finance Using Excel and VBA

Chapter 11. The Black-Scholes Formula

Although the binomial tree provides an easier way to understand option pricing, the analytic Black–Scholes formula remains the central ingredient for European options. Its strength is in providing the option value through a formula and also in determining the hedge ratio for the replicating portfolio. In this chapter, the Black–Scholes formula is derived formally and extended to cover financial assets with a continuous dividend, thus allowing options on currencies and futures to be valued. Hedging parameters can also be derived, and these allow us to create portfolios that are invariant in value to modest changes in share price.

Hull's (2000) textbook on Options is the best reference for most of the topics in this chapter (Chapter 11 for the derivation and explanation of Black–Scholes, Chapter 12 for the adaptation to include continuous dividends and hence how the Black–Scholes formula can be modified to price options on currencies and futures, and Chapter 13 for the 'greeks' and delta hedging). Models illustrating the various Black–Scholes pricing formulas are in the OPTION2.xls workbook, together with a range of useful valuation functions.

THE BLACK–SCHOLES FORMULA

The Black–Scholes pricing formula for a call option was introduced in section 9.2 and the inclusion of dividends in valuing options was briefly introduced in section 9.7. In this section, the formula of section 9.2 is extended using Merton's approach to allow for continuous dividends. ...