Explore BrainMass

Explore BrainMass

    Black-Scholes Model

    We often use a discounted cash flow analysis to determine the value of a capital budgeting project or the value of a stock or bond. So, why can't we simply use the same present value formula to value put and call options? Early traders and academics based their original attempts at valuing options by using a NPV formula. But, what discount rate should they use? Options, one can imagine, are more risky than the underlying asset, but by how much?

    Black and Scholes set to work developing a formula to represent the value of European Options on non-dividend paying underlying stock. They key idea behind the development of the formula was that the value of the option could be found by finding the price of a trading strategy that would duplicate the payoff of the option. Black and Scholes pointed out the the hedging strategy known as "delta hedging" could be used. That is, the strategy of borrowing money and buying a stock could be used to duplicate the risk of a call option on that stock. As a result, if you know the cost of borrowing and the price of the stock, an appropriate price for the related call option can be found. 

    Delta Hedging 

    How much stock?

    What we need to do is use the strategy of borrowing and buying a stock to duplicate the risk of a call option. How do we make sure the risk is the same? Imagine a call with an exercise price of $50. If next year the underlying stock is worth $60, the call will have a value of $10. However, if next year the stock is worth only $40, the value of the call is $0. We see that the "swing of the call" (how much the price of the call moves with changes in the stock price) is not the same as the "swing of the stock" (how much the stock price actually changes). By comparing these to swings, we get what we call the delta of the call. We will purchase an amount of stock equal to the delta of the call in our hedging strategy. 

    *A one dollar swing in the price of the stock causes a 1/2 dollar swing in the price of the call. 

    How much to borrow?

    If we buy a call option with an exercise price of $50, we will end up with $10 next year if the stock price rises to $60. Therefore, in pricing our strategy, we need to hypothetically borrow an amount that will make the payoff of the borrow-and-buy strategy equal to the payoff of buying the call. If we buy one half of a stock, and the stock price rises to $60 next year, the value of our one half of a stock will be $30. This is $20 more than the value of the call. Therefore, we should borrow an amount today with a future value next year of $20. The present value of $20 (assuming a 10% discount rate) is $18.18 ($20/1.10). 

    Value of the call.

    We know that if two strategies have an equal payoff at expiry their initiatial costs must be the same. Therefore, if we calculate the cost of buying the stock and the amount borrowed, we can find the price of the call.


    The Black-Scholes Model

    The delta hedging strategy shows us how to value an option with a two-state (binomial strategy) over a one-year time period. However, in the real world, there are more than two possibilities for the price of the stock one year from now. As a result, Black and Scholes set to work to derive a formula that could be used to value an option in the real world. The Black-Scholes formula relies on the fact that, while there might be numerous possibilities for the price of the stock over a year, the two-state assumption can be used when the time period is shorter. By calculated the price of a dynamic duplicating strategy, that is, a borrowing-to-buy strategy that changes from one instant to the next, the Black-Scholes model is able to calculate the price of a call in the real world using the same intuition that we use for our two-state delta strategy. 

    The value of a call option is a function of five variables, which are incorporated into the formula above. 

    In addition to these five parameters, the formula also incorporates the notation N(d), which stands for cumulative distribution function of the standard normal distribution. This represents the probability that a standardized, normally distributed, random variable will be less than or equal to d. We call this probability the “cumulative probability” and can find N(d) by calculating statistically the cumulative probability or by looking at a cumulative normal probability table. 


    Photo by Maxwell Young on Unsplash

    © BrainMass Inc. brainmass.com August 17, 2022, 6:07 pm ad1c9bdddf

    BrainMass Solutions Available for Instant Download

    Binomial Option Pricing and Black - Scholes Model

    Attached is word doc describing requirements as well as the Excel template. Please let me know if you can help me with this request. Excel programming: option pricing with six-step binomial tree You need to have six input cells: S, X, rannual, σannual, T, and N=6. All other cells should be formulas and automatically c

    Finance: Black-Scholes Model

    Questions: 1) On 10/15/2012, identify two call options for Apple (AAPL); one that is slightly in-the-money (ITM) and a second that is slightly out-of-the-money (OTM). Both options should expire on 12/21/2012. a) Stock close price; b) slightly ITM call option close price; c) slightly OTM call close price. On 10/15/2012, in add

    Cabou Coffee and Fazer Group: Financial Statements Ratio Analysis

    We will continue to review the financial statements of Cabou Coffee and Fazer Group to learn additional information. The emphasis of this case is to review the income statement, balance sheet and computation of ratios. Review the financial statements for the companies and answer the following questions for the last reporting

    Black-Scholes Option Pricing Model: Call Option

    could you please help me with this problem. Can I find the probabilities using Excel? A firm wants one to show what they know about the Black-Scholes option pricing model through finding the call price of an U.S call option with the following characteristics: stock price = $60 exercise price = $60 risk-free rate is

    Models for Price Call Options

    Chose two different models used to price call options. Detail each model and focus on comparing and contrasting the models.

    Black-Scholes and Binomial Option Pricing Models

    Research the variables that impact the pricing of options. Focus your energy on comparing the attributes of the two widely accepted models used for option pricing: Black-Scholes and Binomial Models.

    Leveraging Stock Options to Offset Employee Compensation

    Use the Internet to research a company in which you are interested and review their financial statements, with a focus on how the company is currently using options. Then determine the best way the company you researched could leverage stock options to offset employee compensation. Explain your rationale.

    Preparing a Flexible budget: Home Depot

    Prepare a flexible budget for next year for the company of your choice. The budget needs to be realistic and based on corporate and economic trends. Set up the flexible budget at three levels for the income statement. Companies prepare budgets based on absorption and/or variable costing. Due to lack of information, we're lim

    Match the audit step below with its appropriate area.

    Match the audit step below with its appropriate area. Each audit step has two areas and you may use an area more than once. A. Cash B. Investments C. Current liabilities D. Deferred tax asset/liability E. Pension liability F. Bonds payable G. Dividend payable H. Taxes payable I. Common stock J. Pension expense

    Components of Financial Statements

    Select a company's financial statements available on the Internet, post the link to the statements, and assume the role of a specific type of reader (investor, potential creditor, management, etc). Describe the components of the different financial statements based on this perspective and explain which of the four different fina

    Applying the Black-Scholes Model

    The current stock price of Johnson and Johnson is $64 and the stock does not pay dividends. The instantaneous risk free rate of return is 5%. The instantaneous standard deviation of J&J's stock is 20%. You wish to purchase a call option on this stock with an exercise price of $55 and an expiration date 73 days from now. Using

    Derivative Security Analysis

    Consider the following questions on the pricing of options on the stock of Hatters Yachts. a. A share of Hatteras Yachts sells for $65 and has a standard deviation of 20 percent. The current risk-free rate is 4 percent and the stock pays two dividends: (1) $1.00 just prior to the option's expiration date, which is 91 days fro

    Long Term Debt, Contingencies and Leases

    See the attachment. Discuss trends, financial position, ratios, etc. For example, it may be helpful to discuss inventory turnover when assessing inventory valuation risk. Obtain the 2010 financial statements and notes (annual report or 10-K) for Monro Muffler Brake, Inc. (MNRO). Use Tenneco for a benchmark and comparison pu

    Corporate Strategy Challenges

    Using the concept from Chapter 2 and Chapter 4 of Contemporary Strategy Analysis, address Ford's (Case 4) challenge in meeting the needs of shareholders versus stakeholders. (Please refer to attached files) 1. What ethical challenges and dilemmas might this pose? 2. How should a corporate strategy be adjusted to meet those c

    Horizon value, value of operations, common stock price

    See attached Excel file. 1 Akyol Corporation is undergoing a restructuring, and its free cash flows are expected to be unstable during the next few years. However, FCF is expected to be $95 million in Year 5, i.e., FCF at t = 5 equals $95 million, and the FCF growth is expected to be constant at 8% bey

    Using the Black-Scholes-Merton Calculation

    Given the Black-Scholes-Merton Model: The Microsoft stock price is 25.80 and the risk free rate is 0.20%. The option expires on July 15th and is 17 days to maturity. a. Calculate the intrinsic value and the price of a call option with a strike of $25, assume 23% volatility. b. Calculate the intrinsic value and the price

    Black Scholes pricing model.

    Assume that you have been given the following information on Purcell Industries. Current Stock price=$15 Strike price of option =$15 Time to maturity of option =6 mos Risk-free rate =6% Variance of stock return=0.12 d1 = 0.24495 N(d1) =0.59675 d2 = 0.00000 N(d2)= 0.50000 According to

    Investment and Stock Questions

    1. Which of the following statements is CORRECT? a. Put options give investors the right to buy a stock at a certain strike price before a specified date. b. Call options give investors the right to sell a stock at a certain strike price before a specified date. c. Options typically sell for less than their exercise value.

    Derivative security analysis: Microsoft

    Microsoft plans on issuing a 10 year bond with European style equity warrants that expire in 6 years and have an exercise price of $260. Microsoft currently has 1500 million shares outstanding and wishes to raise an additional $300 million with these warrants. The last closing price of Microsoft shares was $250. The risk free 6,

    Questions based on the 2010 annual report of Danaher Corporation

    Answer the following questions based on the 2010 annual report of Danaher Corporation at the link below: http://www.sec.gov/Archives/edgar/data/313616/000119312511044412/d10k.htm The financial statements are on pp. 62-104 of the annual report, and all questions can be answered within those pages. For each answer, specify

    Call option/stock price/annual risk free rate

    1. Suppose you believe that Sherwin Williams Co.'s stock price is going to increase from its current level of $22.50 sometime during the next 5 months. For $435.75 you can buy a 5-month call option giving you the right to buy 100 shares at a price of $25 per share. If you buy this option for $435.75 and Sherwin Williams Co.'s

    Ricardo International: Black Scholes Option Pricing Model

    Ricardo International would like you to demonstrate your knowledge of the Black-Scholes option pricing model by finding the call price of an U.S. call option with the following characteristics: stock price = $60 exercise price = $60 risk-free rate is 12% volatility (variance of stock returns) = 9% per year time to mat

    Behavioral Finance model: Spinoza's Theory of Emotions as Derivatives

    On Models & Theories- including Spinoza's Theory of Emotions as Derivatives and the Black-Scholes Option Pricing Model The full article is here: http://www.quantnet.com/emanuel-derman-fischer-black Read the above short article and: 1. Given models are fundamentally metaphors and using the materials you have learne

    David Lyons, CEO of Lyons Solar Technologies, is concerned about his firm's level of debt financing. The company uses short-term debt to finance its temporary working capital needs, but it does not use any permanent (long-term) debt. Other solar technology companies average about 30% debt, and Mr. Lyons wonders why they use so much more debt and how it affects stock prices. To gain some insights into the matter, he poses the following questions to you, his recently hired assistant: a. BusinessWeek recently ran an article on companies' debt policies, and the names Modigliani and Miller (MM) were mentioned several times as leading researchers on the theory of capital structure. Briefly, who are MM, and what assumptions are embedded in the MM and Miller models? b. Assume that Firms U and L are in the same risk class, and that both have EBIT = $500,000. Firm U uses no debt financing, and its cost of equity is rsU = 14%. Firm L has $1 million of debt outstanding at a cost of rd = 8%. There are no taxes. Assume that the MM assumptions hold, and then: (1) Find V, S, rs, and WACC for Firms U and L. (2) Graph (a) the relationships between capital costs and leverage as measured by D/V, and (b) the relationship between value and D.

    David Lyons, CEO of Lyons Solar Technologies, is concerned about his firm's level of debt financing. The company uses short-term debt to finance its temporary working capital needs, but it does not use any permanent (long-term) debt. Other solar technology companies average about 30% debt, and Mr. Lyons wonders why they use so m

    Zero Coupon Bonds: Value of Option to Default on the Bonds

    2. Assume your firm has zero coupon bonds maturing in 10 years with a face value of $183.75 million. a) If the assets are worth $150 million, what is the value of the stocks and the bonds? b) If the assets are worth $400 million, what is the value of the stocks and the bonds? c) If the assets are worth $400 million, the

    Volatility Stock Prices Trading Days

    1. Suppose the closing stock prices for 10 consecutive trading days are $20.00, $20.10, $19.90, $20.00, $20.50, $20.25, $20.90, $20.90, $20.90, $20.75. Estimate the volatility of the stock per annum. 2. A stock price is currently $70. Over each of the next two three-month periods it is expected to go up by 8% or down by 6%

    Stock Prices Calculations for Non-Dividends

    1. The price of an American call on a non-dividend-paying stock is $4.67. The stock price is $41.26, strike price is $39, and the expiration date is in 3 months. The risk-free rate is 6%. What is the upper bound for the price of an American put on the same stock with the same strike price and expiration date? 2. Calculate th

    Interest Compounded Annually

    Unless stated otherwise, interest is compounded annually and payments occur at the end of the period. Numbers in parentheses are the points for each problem. FORMAT: You can use a Word document, an Excel spreadsheet or both. If you use Excel, submit the Excel file rather than embedding Excel into a Word document. Please u