Chapter 25

# Some Approximate Values of the Black Scholes Call Formula

## 25.1 INTRODUCTION

In this chapter we show how to approximate the value of European call options to make the Black Scholes formula easier to evaluate. We do this for at-the-money, near-the-money, and deep out-of-the-money options. Our traditional Taylor Series tool is well used. We also demonstrate a newer “asymptotic series” approach for the deep out-of-the-money options.

In Chapter 23 we solved the Black Scholes equation for a Call Option to yield:

$C(S,t)=SN({d}_{1})-K{e}^{-r\left(T-t\right)}N({d}_{2})$

where

${d}_{1}=\frac{\mathrm{ln}\left(\frac{S}{K}\right)+\left(r+\frac{1}{2}{s}^{2}\right)(T-t)}{s\sqrt{T-t}},\text{\hspace{1em}}{d}_{2}={d}_{1}-s\sqrt{T-t}$

and where N(x) denotes the standard cumulative normal distribution.

This is a fairly complicated formula, although of course it is easy to code it into ...