Black-Scholes Option Pricing Model

1 Black-Scholes Option Pricing ModelNathan CoelenJune 6, 20021 IntroductionFinance is one of the most rapidly changing and fastest growing areas in thecorporate business world. Because of this rapid change, modern financialinstruments have become extremely complex. New mathematical models areessential to implement and price these new financial instruments. The worldof corporate finance once managed by business students is now controlled bymathematicians and computer the early 1970 s, Myron scholes , Robert Merton, and Fisher black madean important breakthrough in the Pricing of complex financial instruments bydeveloping what has become known as the Black-Scholes Model . In 1997, theimportance of their Model was recognized world wide when Myron Scholesand Robert Merton received the Nobel Prize for Economics.

2 Unfortunately,Fisher black died in 1995, or he would have also received the award [Hull,2000]. The Black-Scholes Model displayed the importance that mathematicsplays in the field of finance. It also led to the growth and success of the newfield of mathematical finance or financial this paper, we will derive the Black-Scholes partial differential equationand ultimately solve the equation for a European call Option . First, wewill discuss basic financial terms, such as stock and Option , and review thearbitrage Pricing theory. We will then derive a Model for the movement of astock, which will include a random component, Brownian motion. Then, wewill discuss some basic concepts of stochastic calculus that will be applied toour stock Model .

3 From this Model , we will derive the Black-Scholes partialdifferential equation, and I will use boundary conditions for a European calloption to solve the DefinitionsFinancial assets are claims on some issuer, such as the federal governmentor a corporation, such as Microsoft. Financial assets also include real assetssuch as real estate, but we will be primarily concerned with common stock represents an ownership in a corporation. Stocks provide aclaim to the corporation s income and assets. A person who buys a financialasset in hopes that it will increase in value has taken a long position. Aperson who sells a stock before he/she owns it hoping that it decreases invalue is said to be short an asset.

4 People who take short positions borrowthe asset from large financial institutions, sell the asset, and buy the assetback at a later derivative is a financial instrument whose value depends on the valueof other basic assets, such as common stock. In recent years, derivativeshave become increasingly complex and important in the world of individuals and corporations use derivatives to hedge against risk. Thederivative asset we will be most interested in is a European call Option . Acall Option gives the owner the right to buy the underlying asset on a certaindate for a certain price. The specified price is known as the exercise or strikeprice and will be denoted byE. The specified date is known as the expirationdate or day until maturity.

5 European options can be exercised only on theexpiration date itself. Another common Option is a put Option , which givesthe owner the right to sell the underlying asset on a certain date for a example, consider a July European call Option contract on Microsoftwith strike price $70. When the contract expires in July, if the price ofMicrosoft stock is $72 the owner will exercise the Option and realize a profit of$2. He will buy the stock for $70 from the seller of the Option and immediatelysell the stock for $72. On the other hand, if a share of Microsoft is worth$69 the owner of the Option will not exercise the Option and it will expireworthless. In this case, the buyer would lose the purchase price of the ArbitrageOne of the most fundamental theories to the world of finance is the arbitragepricing theory.

6 The theory states that two otherwise identical assets cannotsell at different prices. This also means that there are no opportunities to2make an instantaneous risk-free profit. Here we are assuming the risk-freerate to be that of a bank account or a government bond, such as a illustrate the concept of arbitrage, consider a simple example of a stockthat is traded in the and in London. In the the price of the stock is$150 and the asset sells for 100 in London, while the exchange rate is $ pound. A person could make an instantaneous profit by simultaneouslybuying 100 shares of stock in New York and selling them in London. Aninstantaneous profit of100 (($ 100) $150) = $1000is realized without HedgingThree types of traders are attracted to derivative securities: speculators, arbi-trageurs, and hedgers.

7 Speculators take long or short positions in derivativesto increase their exposure to the market. They are betting that the under-lying asset will go up or go down. Arbitrageurs find mispriced securitiesand instantaneously lock in a profit by adopting certain trading strategieslike those discussed above. The last group is hedgers who take positions inderivative securities opposite those taken in the underlying asset in order tohelp manage risk. For example, consider an investor who owns 100 shares ofMicrosoft which is currently priced $62. The person is worried that the stockmight decline sharply in the next two months. The person could buy putoptions on Microsoft to sell 100 shares at a price of $60.

8 The person wouldpay the price of the options, but this would ensure that he could sell thestock for $60 at expiration if the stock declines sharply. One very importanthedging strategy is delta hedging. The delta, ,of the Option is defined asthe change of the Option price with respect to the change in the price of theunderlying asset. In other words, delta is the first derivative of the optionprice with respect to the stock price: = V S For example, suppose that the delta of a call Option is .60, the price of astock is $100 and the price of a call Option is $10. Imagine an investor who3has sold 1 call Option . The call Option gives the buyer the right to buy 100shares, since each Option contract is for 100 shares.

9 The seller s positioncould be hedged by buying 100 = 60 shares. The gain (loss) on theoption position would then tend to be offset by the loss (gain) on the stockposition. If the stock price goes up by $1 (producing a gain of $60 on theshares purchased) the Option price would tend to go up by $1 = $ (producing a loss of $ * 100 = $60 on the call Option written)[Hull, 2000].5 Stock Price ModelMost people agree that stock prices move randomly because of the efficientmarket hypothesis. There are different forms of this hypothesis, but all saythe same two things. First, the history of the stock is fully reflected inthe present price. Second, markets respond immediately to new informationabout the stock.

10 With the previous two assumptions, changes in a stockprice follow a Markov process. A Markov process is a stochastic processwhere only the present value of the variable is relevant for predicting thefuture. So, our stock Model states that our predictions for the future priceof the stock should be unaffected by the price one week, one month, or oneyear stated above, a Markov process is a stochastic process. In the realworld, stock prices are restricted to discrete values, and changes in the stockprice can only be realized during specified trading hours. Nevertheless, thecontinuous-variable, continuous-time Model proves to more useful than a dis-crete important observation is to note that the absolute change in theprice of a stock is by itself, not a useful quality.