CHAPTER 6: BLACKSCHOLES OPTION PRICING MODEL
6.1
Overview

he BlackScholes model is the cornerstone of modern option pricing theory, and has led to many insights into the valuation of derivative securities. Online, the Option Calculator lets you calculate option prices using this model and its extensions.
For
the BlackScholes analysis, we assume that stock prices evolve in
"continuous time," and that trades can take place at any point in
time. This means that, if
necessary, trades can take place at two "infinitely close" points of
time. This is like the binomial
model in which every period covers an extremely short span of time.
We
further assume that we have an ideal capital market, which means that there are
no liquidity problems, bid ask spreads, and so on.
The BlackScholes model assumes that the stock price follows geometric
Brownian motion which, is a formal way of saying that price changes over
extremely short periods of time are random.
In
this setting, Black and Scholes (1972) derive the prices of European call and
put options. In solving the
problem, they show how a riskless hedge argument (which you saw in the binomial
model) can be used to obtain the solution.
The riskless hedge allows the prices to be determined purely on the basis
of arbitrage. The price
is independent of the expected return of either the stock or the option
(again, as in the binomial model).
This
chapter first presents the main topics at an intuitive and graphical level.
In the technical topics that follow, we provide detailed derivations of
the results.
Topic
6.2, Option Valuation under Certainty introduces the idea of continuous
time where there is no uncertainty. Here, the valuation problem is easy.
Since there is no uncertainty, we simply discount the future cash flow
from the option at the riskfree interest rate.
You will see that when it is applied to actual options data, this model
underprices call options. One
reason for this is that we ignore the fact that stock prices are volatile.
We
then introduce price volatility into the model in topic 6.3, Stock
Price Dynamics. You now
have to worry about how to discount the future cash flow from an option because
the cash flow is uncertain. You
will see how the riskless hedge argument solves this problem in
topic 6.4, The BlackScholes Option
Pricing Model. The option price
will be obtained from the value of the riskfree portfolio, as in the binomial
model.
Finally,
we discuss how the model is applied in the topic Interpretation
of the BlackScholes Model. You
then should be able to calculate the prices of stock options using information
reported in the financial press. At
the end of the chapter, we offer several appendices with technical derivations.
Online, you can now click on the topic Option
Valuation under Certainty.