# Introduction to Visual Quantitative Finance

This chapter is from the book

### This chapter is from the book 

Because the adjustment (1/2 variance) has important implications, I want to show you what the spreadsheet looked like before the adjustment. In the next chapter, I will correct the mistake and walk through the spreadsheet components step by step.

To get a reference point, I priced an option with the Black-Scholes formula. I assumed a one-year term, 30% volatility, and \$100 for the current stock price and the option strike. I also assumed 0% interest and no dividends. The Black-Scholes price was \$11.92.

Then I started building the spreadsheet.

At the top, I entered the pricing assumptions. Then I started filling in the body of the sheet, following the rule that “continuously compounded stock returns are normally distributed.”

When dealing with a normal distribution, the usual place to start is with the standard normal distribution. This is just a special case in which the mean or average value is 0 and the standard deviation is 1. Excel has a built-in function, so I filled in the first two columns with an approximate version that fit on two pages. (I divided it into 81 points, ranging from –4 standard deviation to +4 standard deviations in 0.1 increments. To handle the tails, I put everything outside 4 standard deviations in the two endpoints.) I knew it would not be exact, but it would work for a first try.

Next, I used a common rule of statistics to transform the standard normal distribution into a normal distribution with a standard deviation of 30% and a mean of 0. I remembered that “continuous compounding” meant using the EXP function. That gave me the stock prices. Knowing the stock price makes it easy to calculate the option payoff. The option payoff is just the difference between the stock price and the strike price, not less than zero.

The only thing left to do was weight each option payoff by its probability and add the numbers. The answer was \$14.63, shown in Cell F95. I am intentionally showing you the wrong version so that I can focus on the correction in the next chapter. What is important here is the basic format.

When I was finished, the spreadsheet looked like Figures 1a and 1b.

This spreadsheet describes a simple world. In this world, stock returns, stock prices, and option payoffs are linked to each other, and each of them can be only one of 81 different values.

Columns A and B are constants representing the approximated standard normal distribution. Here are the other column formulas:

• Column C = Column A × 0.30
• Column D = EXP[Column C]
• Column E = MAX[0, Column D − Strike price]
• Column F = Column B × Column E

The option price is the total of Column F.

As an example in reading the spreadsheet, look at Row 53. It is one of the 81 possible outcomes. In this outcome, the stock return is 0%, the stock price is \$100, and the option payoff is \$0. The probability that this particular outcome will occur is 3.98776%.

Similarly, in Row 63 at the one standard deviation point, the stock return is 30%, the corresponding stock price is \$134.99 (the fact this differs from \$130 is explained later), and the option payoff (the difference between the stock price and the option strike price of \$100) is \$34.99. The probability that this particular outcome will occur is 2.41971%.

Notice that the only positive values for the option payoff are in Rows 54–93. These 40 outcomes are the only numbers factored into the option price. The last column shows the weighted values of the option payoffs.

Looking at Column F, the value of the option is concentrated between 0 and 3 standard deviations. The highest contributions occur at around 1 standard deviation, with a weighted value of \$0.85. In the tail of the distribution, the payoffs are very high but the probabilities are very low. For instance, even though the payoff goes as high as \$232.01 in Row 93, the effect on the value of the option is only 1 cent. The probability at this point is so low that a high payoff has almost no effect.

### InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

## Overview

Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

## Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

### Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

### Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

### Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

### Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

### Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

### Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

## Other Collection and Use of Information

### Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

### Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

### Do Not Track

This site currently does not respond to Do Not Track signals.

## Security

Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

## Children

This site is not directed to children under the age of 13.

## Marketing

Pearson may send or direct marketing communications to users, provided that

• Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
• Such marketing is consistent with applicable law and Pearson's legal obligations.
• Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
• Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

## Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

## Choice/Opt-out

Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

## Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

## Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

## Sharing and Disclosure

Pearson may disclose personal information, as follows:

• As required by law.
• With the consent of the individual (or their parent, if the individual is a minor)
• In response to a subpoena, court order or legal process, to the extent permitted or required by law
• To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
• In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
• To investigate or address actual or suspected fraud or other illegal activities
• To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
• To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
• To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.