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Chapter 13: Problem 25

A company's stock price is $$\$ 50$$ and 10 million shares are outstanding. The company is considering giving its employees 3 million at-the-money 5 -year call options. Option exercises will be handled by issuing more shares. The stock price volatility is \(25 \%\), the 5-year risk-free rate is \(5 \%\), and the company does not pay dividends. Estimate the cost to the company of the employee stock option issue.

Short Answer

Expert verified
The estimated cost is $37.59 million.

Step by step solution

01

Understanding the Black-Scholes Model

To estimate the cost of the employee stock options, we will use the Black-Scholes Option Pricing Model, which calculates the price of a European call option based on specific parameters. The key inputs are the stock price, strike price, volatility, time to expiration, risk-free rate, and dividend yield.
02

Define the Given Parameters

We have the following parameters: current stock price (\(S_0\)) is \(50; the option's strike price (\(K\)) is also \)50 since it is at-the-money; the volatility (\(\sigma\)) is 25% or 0.25; the time to expiration (\(T\)) is 5 years; the risk-free rate (\(r\)) is 5% or 0.05; and the dividend yield (\(q\)) is 0% since no dividends are paid.
03

Calculate Parameters d1 and d2 using Black-Scholes Formulas

Using the formulas, \[d_1 = \frac{\ln\left(\frac{S_0}{K}\right) + (r - q + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}}\] and \[d_2 = d_1 - \sigma\sqrt{T}\], we substitute the values: \[d_1 = \frac{\ln\left(\frac{50}{50}\right) + \left(0.05 - 0 + \frac{0.25^2}{2}\right)5}{0.25\sqrt{5}}\] and \[d_2 = d_1 - 0.25\sqrt{5}\].
04

Calculate d1 and d2

Calculate \(d_1\) and \(d_2\). First, compute: \[d_1 = \frac{0 + (0.05 + 0.03125)5}{0.559016994}\] which simplifies to approximately \[d_1 \approx 0.559\]. Then calculate \[d_2 = 0.559 - 0.559\approx 0\].
05

Use Standard Normal Distribution Functions

The next step involves calculating \(N(d_1)\) and \(N(d_2)\), where \(N\) is the cumulative distribution function of the standard normal distribution. Look up \(N(0.559)\approx 0.712\) and \(N(0) = 0.5\).
06

Calculate the Option Price

Using the Black-Scholes formula for the call option price: \[C = S_0e^{-qT}N(d_1) - Ke^{-rT}N(d_2)\] we calculate \[C = 50\cdot e^{0}(0.712) - 50\cdot e^{-0.25}(0.5)\], which results in \[C \approx 12.53\].
07

Estimate Total Cost for All Options

Now that we know each option is worth approximately \(12.53, multiply by the total 3 million options issued: \[3,000,000\times 12.53 = 37,590,000\]. The estimated cost is approximately \)37.59 million.

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Key Concepts

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

European Call Option in the Black-Scholes Model
In financial markets, a European call option gives the owner the right, but not the obligation, to purchase an asset, like a stock, at a specified price known as the strike price, on a specified future date. Unlike American options, which can be exercised any time before expiration, European options can only be executed at expiration. This characteristic makes them simpler to model, and they're specifically relevant when applying mathematical models like the Black-Scholes Model.
  • The price of a European call option is influenced by several factors: the current stock price, the volatility of the stock price, the time until expiration, interest rates, and any dividends paid by the stock.
  • The Black-Scholes Model is a widely-used formula that estimates the theoretical value of a European call option using these inputs.
Understanding Stock Price Volatility
Stock price volatility measures how much the price of a stock fluctuates over a period of time. In the context of the Black-Scholes Model, volatility is a crucial input because it directly affects the option's price. High volatility means the stock price can move significantly, increasing the chance of the option being profitable at expiration.
  • Volatility is often expressed as a percentage and is typically calculated based on historical stock price movements.
  • A higher volatility increases the price of the option, as there's a greater chance the stock price will exceed the strike price at expiration, making the option valuable to exercise.
The Role of Risk-Free Rate
The risk-free rate represents the theoretical return on an investment with no risk, typically associated with government treasury bonds. In option pricing models, it's used to discount the expected future payoffs to their present value. By doing so, the model accounts for the time value of money, which is the principle that receiving money in the future is less valuable than receiving the same amount today.
  • This rate helps in evaluating the present value of the expected money flow from exercising the option, affecting the option鈥檚 current price in the model.
  • In the Black-Scholes formula, the risk-free rate is involved in calculating the present value of the strike price adjusted for the expiration period, denoted as \( Ke^{-rT} \), where \( r \) is the risk-free rate and \( T \) is the time to expiration.
Cumulative Distribution Function in the Black-Scholes Model
The cumulative distribution function, or CDF, is a statistical tool used to describe the probability that a random variable takes on a value less than or equal to a specific number. In the context of the Black-Scholes Model, the CDF is applied to the standard normal distribution to find the probabilities \(N(d_1)\) and \(N(d_2)\).
  • These probabilities are used to account for the likelihood that the option will be in-the-money, meaning that it will be profitable at expiration.
  • The CDF provides a way to quantify the dynamics of stock prices under the assumed normal distribution of returns. This helps in determining how much an option is likely to be worth at expiration.

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Most popular questions from this chapter

Consider a derivative that pays off \(S_{T}^{n}\) at time \(T\), where \(S_{T}\) is the stock price at that time. When the stock price follows geometric Brownian motion, it can be shown that its price at time \(t(t \leqslant T)\) has the form $$ h(t, T) S^{n} $$ where \(S\) is the stock price at time \(t\) and \(h\) is a function only of \(t\) and \(T\). (a) By substituting into the Black-Scholes-Merton partial differential equation, derive an ordinary differential equation satisfied by \(h(t, T)\). (b) What is the boundary condition for the differential equation for \(h(t, T) ?\) (c) Show that $$ h(t, T)=e^{\left[0.5 \sigma^{2} n(n-1)+n(s-1)(T-\theta)\right.} $$ where \(r\) is the risk-free interest rate and \(\sigma\) is the stock price volatility.

A portfolio manager announces that the average of the returns realized in each year of the last 10 years is \(20 \%\) per annum. In what respect is this statement misleading?

Consider an American call option when the stock price is $$\$ 18$$, the exercise price is $$\$ 20$$, the time to maturity is 6 months, the volatility is \(30 \%\) per annum, and the risk-free interest rate is \(10 \%\) per annum. Two equal dividends are expected during the lifc of the option with ex- dividend dates at the end of 2 months and 5 months. Assume the dividends are 40 cents. Use Black's approximation and the DerivaGem software to value the option. How high can the dividends be without the American option being worth more than the corresponding European option?

Explain carefully why Black's approach to evaluating an American call option on a dividend-paying stock may give an approximate answer even when only one dividend is anticipated. Does the answer given by Black's approach understate or overstate the true option value? Explain your answer.

What is the price of a European call option on a non-dividend-paying stock when the stock price is $$\$ 52$$ the strike price is $$\$ 50$$ the risk-free interest rate is \(12 \%\) per annum, the volatility is \(30 \%\) per annum, and the time to maturity is 3 months?
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