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6.1 The Black-Scholes Option Pricing Model

The Black-Scholes 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 Black-Scholes 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 Black-Scholes 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 risk-free 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 Black-Scholes Option Pricing Model. The option price will be obtained from the value of the risk-free portfolio, as in the binomial model.

Finally, we discuss how the model is applied in the topic Interpretation of the Black-Scholes 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.

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