Quantitative Finance

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

$d_1=\frac{\ln \left(\frac{S}{K}\right)+\left(r+\frac{1}{2}{s}^2\right)(T-t)}{s\sqrt{T-t}},d_2=d_1-s\sqrt{T-t}$

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 ...