# Basic black scholes option pricing and trading pdf

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Statistics of Financial Markets pp Cite as. Simple generally accepted economic assumptions are insufficient to develop a rational option pricing theory. Assuming a perfect financial market in Section 2. While these relations can be used as a verification tool for sophisticated mathematical models, they do not provide an explicit option pricing function depending on parameters such as time and the stock price as well as the options underlying parameters K, T. To obtain such a pricing function the value of the underlying financial instrument stock, currency,

## Basic Black-Scholes: Option Pricing and Trading (Revised Fourth)

Black-Scholes Option Pricing Model Nathan Coelen June 6, 1 Introduction 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 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 , or he would have also received the award [Hull, ]. 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 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. A person who buys a financial asset in hopes that it will increase in value has taken a long position.

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.

In this case, the buyer would lose the purchase price of the option. The theory states that two otherwise identical assets cannot sell at different prices. Here we are assuming the risk-free rate to be that of a bank account or a government bond, such as a Treasury bill. To illustrate the concept of arbitrage, consider a simple example of a stock that is traded in the U.

In the U. A person could make an instantaneous profit by simultaneously buying shares of stock in New York and selling them in London. Speculators take long or short positions in derivatives to increase their exposure to the market. They are betting that the under- lying asset will go up or go down. Arbitrageurs find mispriced securities and instantaneously lock in a profit by adopting certain trading strategies like those discussed above.

The last group is hedgers who take positions in derivative securities opposite those taken in the underlying asset in order to help manage risk.

The person is worried that the stock might decline sharply in the next two months. One very important hedging strategy is delta hedging. The call option gives the buyer the right to buy shares, since each option contract is for shares.

The gain loss on the option position would then tend to be offset by the loss gain on the stock position. There are different forms of this hypothesis, but all say the same two things.

First, the history of the stock is fully reflected in the present price. Second, markets respond immediately to new information about the stock. With the previous two assumptions, changes in a stock price follow a Markov process. A Markov process is a stochastic process where only the present value of the variable is relevant for predicting the future. So, our stock model states that our predictions for the future price of the stock should be unaffected by the price one week, one month, or one year ago.

As stated above, a Markov process is a stochastic process. In the real world, stock prices are restricted to discrete values, and changes in the stock price can only be realized during specified trading hours. Nevertheless, the continuous-variable, continuous-time model proves to more useful than a dis- crete model. Another important observation is to note that the absolute change in the price of a stock is by itself, not a useful quality. The relative change of the price of a stock is information that is more valuable.

The relative change will be defined as the change in the price divided by the original price. Now consider the price of a stock S at time t. Consider a small time interval dt during which the the price of the underlying asset S changes by an amount dS. The first part is completely deterministic, and it is usually the risk free interest rate on a Treasury bill issued by the government. The second part of the model accounts for the random changes in the stock price due to external effects, such as unanticipated news.

The B in dB denotes Brownian motion, which will be described in the next section. This is not of interest for our model. Brow- nian motion, which was originally used as a model for stock price movements in by L. Bachelier[Klebaner, ], is a stochastic process B t charac- terized by the following three properties: 1.

These three properties alone define Brownian motion, but they also show why Brownian motion is used to model stock prices. Property 2 shows stock price changes will be independent of past price movements. This was an important assumption we made in our stock price model. An occurence of Brownian motion from time 0 to T is called a path of the process on the interval [0,T]. There are five important properties of Brownian motion paths.

Properties 4 and 5 show the distinction between functions of Brownian motion and normal, smooth functions. Quadratic variation plays a very important role with Brownian motion and stochastic calculus. The integral over 0, T ] should be the sum of integrals over subintervals [0, a1 , a1 , a2 , a2 , a3 , So if X t takes values ci on each subinterval then the integral of X with respect to B is easily defined.

First we consider the integrals of simple processes e t which depend on t and not on B t. Zero mean property. Isometry Property.

Also the integral that arises this way still satisfies properties above. It can be shown that if a general predictable process satifies certain conditions, the eneral process is a limit in probability of siple predictable processes we discussed earlier.

For example, we find the 0T B t dB t. Notice that from property 5 of Brownian RT n motion patht the second sum converges to the limit T. The quadratic variation of continuous functions, x t , of finite variation we work withR in standard calculus is 0. Recall that Brownian motion has quadratic variation on [0,t] equal to t, for any t. One last very important case for us to consider is for functions of the form f X t , t.

Therefore, we let the function V S, t be twice differentiable in S and differentiable in t. Note from above that this portfolio is hedged. Since this portfolio contains no risk it must earn the same as other short-term risk-free securities. If it earned more than this, arbitrageurs could make a profit by shorting the risk- free securities and using the proceeds to buy this portfolio.

If the portfolio earned less arbitrageurs could make a riskless profit by shorting the portfolio and buying the risk-free securities. For this project we will concern ourselves with a European call, C S, t with exercise price E and expiry date T. The first step is to get rid of the S and S 2 terms in equation The first integral can be solved by completing the square in the exponent. This means that a person can use the Black-Scholes differ- ential equation to solve for the price of any type of option only by changing the boundary conditions.

The Black-Scholes model truly revolutionized the world of finance.

## Black-Scholes Option Pricing Model

Basic Black Scholes. Book Contents The explanations do not go far beyond basic Black-Scholes. There are three reasons for this: First, a novice need not go far beyond Black-Scholes to make money in the options markets; Second, all high-level option pricing theory is simply an extension of Black-Scholes theory; and Third, there already exist many books that look far beyond Black-Scholes without first laying the firm foundation given here. The author studied PhD-level option pricing at MIT and Harvard, taught undergraduate and MBA option pricing at Indiana University winning many teaching awards in the process , and has traded options for over ten years. This special mixture of learning, teaching, and trading is reflected in every page. What is in this book that makes it special or unique: Basic intuition you need if you are trading options for the first time, or interviewing for an options job.