Closed Formula for Options with Discrete Dividends and Its Derivatives

Abstract

We present a closed pricing formula for European options under the Black–Scholes model as well as formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and a proposition that relates expectations of partial derivatives with partial derivatives themselves. The closed formulas are attained assuming the dividends are paid in any state of the world. The results are readily extensible to time-dependent volatility models. For completeness, we reproduce the numerical results in Vellekoop and Nieuwenhuis, covering calls and puts, together with results on their partial derivatives. The closed formulas presented here allow a fast calculation of prices or implied volatilities when compared with other valuation procedures that rely on numerical methods.

Key Words:

• Equity option
• discrete dividend
• hedging
• analytic formula
• closed formula

Acknowledgements

The authors thank an anonymous referee for valuable comments. Carlos Veiga wishes to thank Millennium bcp investimento, S.A. for the financial support being provided during the course of his PhD studies. Uwe Wystup thanks the Fulbright commission for supporting this research and Carnegie Mellon University for providing a working environment during the sabbatical in Fall 2008.

Notes

¹We omit the model parameters and the option-specific quantities, like maturity or strike, from the function C to preserve clarity.

²Note that is known at time t _n − 1 and thus below is only a derivative with respect to an argument of the function and not a derivative with respect to a stochastic variable.

³The derivatives of the option price are usually called Greeks because Greek alphabet letters are commonly used to denote them.

⁴Using a C++ ‘.xll’ added to MS Excel03 running on an Intel Core2 4400@2 GHz.

⁵Please note that each element of the vector is just the previous plus an extra term composed by the Hermite polynomial (that should be computed by the well known recursive relation) multiplied by a quantity already available in memory.

⁶And even derivatives of I₁ with respect to r and to σ if the first routine is ready to return them.