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Chapter 18: Problem 4

Assume that you have been given the following information on Purcell Industries: Current stock price \(=\$ 15\) Time to maturity of option \(=6\) months Variance of stock price \(=0.12\) \(\mathrm{d}_{2}=0.08165\) \(\mathrm{N}\left(\mathrm{d}_{2}\right)=0.53252\) Exercise price of option \(=\$ 15\) Risk-free rate \(=10 \%\) \(d_{1}=0.32660\) \(\mathrm{N}\left(\mathrm{d}_{1}\right)=0.62795\) Using the Black-Scholes Option Pricing Model, what is the value of the option?

Short Answer

Expert verified
The value of the option is approximately $1.82.

Step by step solution

01

Identify the Black-Scholes Formula

The Black-Scholes Option Pricing Model is given by the formula: \[C = S_0 N(d_1) - X e^{-rT} N(d_2) \]where:- \( C \) is the call option price.- \( S_0 = \\( 15 \) is the current stock price.- \( X = \\) 15 \) is the exercise price of the option.- \( r = 10\% = 0.10 \) is the risk-free rate.- \( T \) is the time to maturity in years, which is 0.5 years for 6 months.- \( N(d_1) = 0.62795 \) and \( N(d_2) = 0.53252 \) are the cumulative distributions of the standard normal variables.
02

Substitute Known Values into the Formula

Substitute all the known values into the Black-Scholes formula:\[C = 15 \times 0.62795 - 15 \times e^{-0.10 \times 0.5} \times 0.53252\]
03

Calculate the Present Value of the Exercise Price

Compute the present value of the exercise price:\[PV(X) = 15 \times e^{-0.05}\]Using \( e^{-0.05} \approx 0.95123 \), then:\[PV(X) = 15 \times 0.95123 = 14.26845\]
04

Compute the Option Value

Now, compute the value of the option using the substituted values:\[C = 15 \times 0.62795 - 14.26845 \times 0.53252\]Calculate each part:- \( 15 \times 0.62795 = 9.41925 \)- \( 14.26845 \times 0.53252 = 7.59724 \)Subtract to find \( C \):\[C = 9.41925 - 7.59724 = 1.82201\]
05

Finalize the Value of the Option

The calculated option value is approximately \( C = 1.82 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Call Option Pricing
Call option pricing is a central aspect of financial markets, used to evaluate the potential worth of stock options. In simple terms, a call option is a contract that gives the investor the right, but not the obligation, to purchase a stock at a specified price鈥攌nown as the exercise price鈥攂efore a certain date. This potential to "call" the stock makes it valuable, especially if the stock鈥檚 market price rises above the exercise price.

Understanding the calculation of a call option鈥檚 price involves several factors:
  • Current Stock Price: What the stock is presently trading at matters as it serves as the baseline for potential profit.
  • Exercise Price: This is the locked-in price at which the stock can be purchased through the option.
  • Time to Maturity: The duration until the option expires. Longer times allow for more opportunities for the stock price to exceed the exercise price.
  • Risk-Free Rate: An assumed rate of return on a theoretically riskless investment, often government bonds.
  • Volatility: This represents the stock price's fluctuations; more variance often makes the option more valuable.
The Black-Scholes model uses these components to derive a theoretical call option price, helping investors make informed decisions.
Financial Derivatives
Financial derivatives include instruments such as options, futures, and swaps. They derive their value from an underlying asset, which could be stocks, bonds, currencies, or commodities. Options, specifically call options, are a popular derivative, allowing investors to speculate or hedge.

Why are derivatives important?
  • Risk Management: They help investors and companies manage risk by providing a tool to hedge other investments.
  • Leverage: Derivatives can significantly increase return possibilities while using relatively small capital amounts, allowing for greater exposure to asset movements.
  • Market Efficiency: Derivatives facilitate arbitrage opportunities and price discovery, contributing to market efficiency.
Options like call options particularly benefit investors looking for leveraged stock exposure or who plan to use options in conjunction with other investment strategies, allowing for a broader range of financial maneuvers.
Risk-Free Rate
The risk-free rate is a theoretical interest rate that assumes investors will not face any risk of financial loss. Typically represented by the return on government securities, such as Treasury bills, it's a fundamental component in option pricing models like Black-Scholes.

Why is the risk-free rate used?
  • Present Value Calculations: Used to determine the present value of an expected future cash flow, crucial for making accurate financial assessments.
  • Time Value of Money: Reflects the value attached to money over time, essential for determining fair option pricing by understanding opportunity costs.
  • Benchmark for Investments: Offers a baseline for evaluating other potential investments by providing a low-risk return reference point.
In the Black-Scholes formula, the risk-free rate adjusts the exercise price鈥檚 present value, contributing to the call option鈥檚 overall valuation. A higher risk-free rate generally decreases the call option's value, as it raises the present value of money.
Stock Options Valuation
Stock options valuation is an analysis process to determine the worth of stock options. Through this, investors can assess whether options are a good investment. One widely used method for valuation is the Black-Scholes Option Pricing Model.

Here's the process of evaluating a stock option using this model:
  • Input Values: Incorporate current stock price, exercise price, time to maturity, volatility of the stock, and the risk-free rate into the model.
  • Calculate Key Metrics: Compute values such as \(d_1\) and \(d_2\) that contribute to the option pricing through their respective cumulative normal distribution functions \(N(d_1)\) and \(N(d_2)\).
  • Formulate Option Price: Use the Black-Scholes formula to find the option's theoretical value, specifically focusing on the components like the present value of the exercise price to complete your calculations.
Careful consideration of each factor leads to a calculated option value, providing strategic insight for financial decision-makers and guiding potential stock option investments.

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Most popular questions from this chapter

A call option on Bedrock Boulders stock has a market price of \(\$ 7 .\) The stock sells for \(\$ 30\) a share, and the option has an exercise price of \(\$ 25\) a share. a. What is the exercise value of the call option? b. What is the premium on the option?

Audrey is considering an investment in Morgan Communications, whose stock currently sells for \(\$ 60\). A put option on Morgan's stock, with an exercise price of \(\$ 55,\) has a market value of \(\$ 3.06 .\) Meanwhile, a call option on the stock with the same exercise price and time to maturity has a market value of \(\$ 9.29 .\) The market believes that at the expiration of the options, the stock price will be \(\$ 70\) or \(\$ 50\) with equal probability. a. What is the premium associated with the put option? the call option? b. If Morgan's stock price increases to \(\$ 70,\) what would be the return to an investor who bought a share of the stock? if the investor bought a call option on the stock? if the investor bought a put option on the stock? c. If Morgan's stock price decreases to \(\$ 50,\) what would be the return to an investor who bought a share of the stock? if the investor bought a call option on the stock? if the investor bought a put option on the stock? d. If Audrey buys 0.6 share of Morgan Communications and sells one call option on the stock, has she created a riskless hedged investment? What is the total value of her portfolio under each scenario? e. If Audrey buys 0.75 share of Morgan Communications and sells one call option on the stock, has she created a riskless hedged investment? What is the total value of her portfolio under each scenario?

Stewart Enterprises' current stock price is \(\$ 60\) per share. Call options for this stock exist that permit the holder to purchase one share at an exercise price of \(\$ 50 .\) These options will expire at the end of 1 year, at which time Stewart's stock will be selling at one of two prices, \(\$ 45\) or \(\$ 70\). The risk-free rate is \(7 \%\). As an assistant to the firm's treasurer, you have been asked to perform the following tasks to arrive at the value of the firm's call options. a. Find the range of values for the ending stock price and the call option at the option's expiration in 1 year. b. Equalize the range of payoffs for the stock and the option. c. Create a riskless hedged investment. d. What is the cost of the stock in the riskless portfolio? e. What is the present value of the riskless portfolio? f. From your answers in Parts d and e, what is the value of the firm's call option?

Which of the following events are likely to increase the market value of a call option on a common stock? Explain. a. An increase in the stock's price b. An increase in the volatility of the stock price c. An increase in the risk-free rate d. A decrease in the time until the option expires
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