June 6, 2002
Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change, modern ﬁnancial instruments have become extremely complex. New mathematical models are essential to implement and price these new ﬁnancial instruments. The world of corporate ﬁnance once managed by business students is now controlled by mathematicians and computer scientists.
In the early 1970’s, Myron Scholes, Robert Merton, and Fisher Black made an important breakthrough in the pricing of complex ﬁnancial instruments by developing what has become known as the Black-Scholes model. In 1997, the importance of their model was recognized world wide when Myron Scholes and Robert Merton received the Nobel Prize for Economics. Unfortunately, Fisher Black died in 1995, or he would have also received the award [Hull, 2000]. The Black-Scholes model displayed the importance that mathematics plays in the ﬁeld of ﬁnance. It also led to the growth and success of the new ﬁeld of mathematical ﬁnance or ﬁnancial engineering.
In this paper, we will derive the Black-Scholes partial diﬀerential equation and ultimately solve the equation for a European call option. First, we will discuss basic ﬁnancial terms, such as stock and option, and review the arbitrage pricing theory. We will then derive a model for the movement of a stock, which will include a random component, Brownian motion. Then, we will discuss some basic concepts of stochastic calculus that will be applied to our stock model. From this model, we will derive the Black-Scholes partial diﬀerential equation, and I will use boundary conditions for a European call option to solve the equation.
Financial assets are claims on some issuer, such as the federal government or a corporation, such as Microsoft. Financial assets also include real assets such as real estate, but we will be primarily concerned with common stock. Common stock represents an ownership in a corporation. Stocks provide a claim to the corporation’s income and assets. A person who buys a ﬁnancial asset in hopes that it will increase in value has taken a long position. A person who sells a stock before he/she owns it hoping that it decreases in value is said to be short an asset. People who take short positions borrow the asset from large ﬁnancial institutions, sell the asset, and buy the asset back at a later time.
A derivative is a ﬁnancial instrument whose value depends on the value of other basic assets, such as common stock. In recent years, derivatives have become increasingly complex and important in the world of ﬁnance. Many individuals and corporations use derivatives to hedge against risk. The derivative asset we will be most interested in is a European call option. A call option gives the owner the right to buy the underlying asset on a certain date for a certain price. The speciﬁed price is known as the exercise or strike price and will be denoted by E . The speciﬁed date is known as the expiration date or day until maturity. European options can be exercised only on the expiration date itself. Another common option is a put option, which gives the owner the right to sell the underlying asset on a certain date for a certain price.
For example, consider a July European call option contract on Microsoft with strike price $70. When the contract expires in July, if the price of Microsoft 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 immediately sell 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 expire worthless. In this case, the buyer would lose the purchase price of the option.
One of the most fundamental theories to the world of ﬁnance is the arbitrage...