Black-Scholes Formula: A Walkthrough

TABLE OF CONTENTS

LIST OF FORMULAS

1. INTRODUCTION

2. FROM THE BASICS TO THE PARITY
2.1 PUT AND CALL OPTIONS
2.2 PUT-CALL PARITY

3. BLACK-SCHOLES - AN OPTION PRICING MODEL
3.1 ASSUMPTIONS OF THE MODEL AND ITS INFLUENCING FACTORS
3.2 THE BLACK-SCHOLES FORMULA
3.3 THE BLACK-SCHOLES FORMULA IN PRACTICE
3.3.1 Fictional Example
3.3.2 General Electric Example
3.4 EXCURSION TO THE GREEKS

4. CONCLUSION

REFERENCE LIST

LIST OF FORMULAS

Formula 1 Put-Call Parity v1

Formula 2 Put-Call Parity v2

Formula 3 Black-Scholes Formula

Formula 4 Black-Scholes - d1

Formula 5 Black-Scholes - d2

1. INTRODUCTION

“ The most influential development in terms of impact on finance practice was the Black-Scholes model for option pricing. ” - Robert Merton Financial derivates found their way into the financial markets already hundreds of years ago. With the opening of the Chicago Board Options Exchange (CBOE) in 1973, they successfully established themselves in the market. Ever since, it is not possible to imagine a financial world without derivates (Balmer, 2002). Options build up one major type of financial derivates. With the opening of the CBOE, option trading has exponentially become popular in American securities markets. However, option pricing theory was still far off from being any precise. In fact, the question has always been on how to value options in a world where the underlying share was fluctuating (Cox, Ross & Rubinstein, 1979). The academics have also started to participate in this issue, believing that if they would be able to mathematically describe the emotional confidence of investors or traders, they would be able to solve the problem of how to value options. Hence, early econometric models kept on adding various parameters and factor to a mathematical formula in order to find the key to success - a problem to which its solution was later worth the Nobel Prize (French, 1983). However, it was Fischer Black and Myron Scholes who have published their groundbreaking findings in 1973 that revolutionized many aspects of option pricing today. According to Dan French, professor of Finance at Texas A&M University, “the Black-Scholes model, which was the first to define option pricing in equilibrium, is probably the most widely publicized and extensively tested of the probability-based models”. The formula, which brought Scholes and Merton the Nobel Prize in 1997, was also valid and used in the real world by traders who soon imagined having the ultimate tool for making money (Balmer, 2002).

2. FROM THE BASICS TO THE PARITY

“ The financial markets generally are unpredictable. So that one has to have different scenarios... ”

- George Soros

2.1 PUT AND CALL OPTIONS

When speaking of options, all economical text books describe them as granting the buyer of a call option the right to buy the underlying financial asset at a fixed price, and on the other hand, to give the buyer of a put option the right to sell the underlying financial asset at a given price. In both cases, parties can either gain or lose money through a fluctuation in the price of the underlying (Damodaran, 2001). Option prices reflect the intrinsic value of the option, plus any additional amount over its intrinsic value, where the intrinsic value is considered to be the difference in the strike price and the stock price. This addition is known as the time premium. Further, when dealing with options, it is important to understand that both put and call options have a fixed expiration date. The owner of an option has the right to execute the option up until the expiration date. However, there are differences between an American option and a European option. The American option is exercised at any time prior to -and at the expiration date. With a European option, the option can only be exercised on the expiration date itself (Fabozzi & Modigliani, 2009).

Next to the ordinary buying or selling of call and put options, certain combinations exist, which in their nature fall back on the put-call parity, a theory which also finds its relevance in the Black-Scholes formula. However, looking at two wide-spread examples of option combinations, the author wants to start with the so-called covered call strategy. This rather conservative strategy is simultaneously buying equity and writing a call on the same share. Here, if the share price increases, the investor makes profit out of his share. If, on the opposite, the share price drops, the loss in the equity can be absorbed by the written call. A second combination is called the protective-put strategy. In this example, the investor buys the share in combination with buying a put option on the same underlying. Here, the investor may want to protect the value of the underlying against the risk of a decline in market value. As a result of the two approaches, the investor gets the same returns from the two possible strategies. Hence, a relationship exists, in showing that both options yield the same result (Jaffe, Ross & Westerfield, 2002).

2.2 PUT-CALL PARITY

In fact, this relationship is called the put-call parity.

Formula 1

Abbildung in dieser Leseprobe nicht enthalten

Looking at the formula ( Formula 1 ), there are now two ways to buy a protective put. The

investor may either buy a put and buy the underlying share, in which the total cost would be the share price and the price of the put, or the investor can buy a call and buy a zero-coupon bond simultaneously. In the latter case, the costs would be the zero-coupon bond, which is equal to the present value of the exercise price, and the price of the call. There are a few assumptions that are important to this relationship, being that the put and call must have the same exercise price and expiration date and so does the maturity of the zero-coupon bond.

Formula 2

Abbildung in dieser Leseprobe nicht enthalten

Rearranging the formula, one can understand that buying the underlying share can be exactly replicated in buying a call, selling a put and buying a zero-coupon bond. The author will refer back to this fundamental relationship, when dealing with the results of the Black-Scholes formula (Jaffe, Ross & Westerfield, 2002).

3. BLACK-SCHOLES - AN OPTION PRICING MODEL

3.1 ASSUMPTIONS OF THE MODEL AND ITS INFLUENCING FACTORS

The previous sections aimed to provide an overview of the basics when dealing with options. The Black-Scholes model requires being equipped with those fundamentals. The Black-Scholes model is one which is based on probabilities, as it determines the value of an option upon the probability that the share will be situated above exercise price on the specific expiration date. As with many models, also the Black-Scholes falls back on given conditions, being (Black & Scholes, 1973):

1. A constant risk-free interest rate
2. Share price changes are log-normally distributed
3. No dividends occur
4. Options can only be exercised at maturity date
5. No transactions costs occur
6. Borrowings are unlimited

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