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Chapter 24: Problem 18

What is the value of a derivative that pays off \(\$ 100\) in 6 months if the S\&P 500 index is greater than 1,000 and zero otherwise? Assume that the current level of the index is 960 , the risk-free rate is \(8 \%\) per annum, the dividend yield on the index is \(3 \%\) per annum, and the volatility of the index is \(20 \%\).

Short Answer

Expert verified
The derivative value is approximately $4.59.

Step by step solution

01

Identify the Derivative Type

This derivative is a European call option with a binary (or digital) payoff structure. It pays $100 if the S&P 500 index is greater than 1,000 in 6 months.
02

Define the Parameters

Set the parameters needed for the Black-Scholes Model. These include the current index level \(S_0 = 960\), strike price \(K = 1000\), risk-free rate \(r = 0.08\), dividend yield \(q = 0.03\), volatility \(\sigma = 0.2\), and time to expiration \(T = 0.5\; \text{years}\).
03

Calculate the Black-Scholes d2

Use the formula for \(d_2\):\[d_2 = \frac{\ln\left(\frac{S_0}{K}\right) + \left(r - q - \frac{\sigma^2}{2}\right)T}{\sigma\sqrt{T}}\].Substituting the values, \[d_2 = \frac{\ln\left(\frac{960}{1000}\right) + \left(0.08 - 0.03 - \frac{0.2^2}{2}\right) \times 0.5}{0.2 \sqrt{0.5}}\].
04

Compute the Probability

Calculate \(d_2\) from the previous step, then use the standard normal cumulative distribution function (CDF), \(N(d_2)\), to find the probability that the index will be above the strike price at expiration.
05

Calculate Present Value of Payoff

The value of the binary call option is the present value of the payoff times the probability, \(100 \cdot e^{-rT} \cdot N(d_2)\). Substitute the values into this formula to calculate the option value.
06

Execute the Calculation

After computing all necessary values and substituting them into the equation, determine the option's present value by calculating \(N(d_2)\), then multiply by \(100 \times e^{-rT}\).

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Key Concepts

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

Black-Scholes Model
The Black-Scholes Model is a mathematical model used to calculate the theoretical price of European call and put options. Market conditions and various factors are taken into account.

Essential elements include:
  • The current price of the underlying asset, denoted as \(S_0\).
  • The strike price \(K\), which is the price at which the option can be exercised.
  • The time to expiration \(T\), which marks the duration until the option can be exercised.
  • The risk-free interest rate \(r\), representing the return on an investment with zero risk.
  • The volatility \(\sigma\), indicating the degree of variation of the asset's returns.
  • The dividend yield \(q\), relevant if the underlying is a stock paying dividends.
The model helps investors understand the potential value and risk associated with an option. By inputting these parameters into the Black-Scholes formula, expected pricing of the option can be estimated, providing a basis for decision-making.
Risk-Free Rate
The risk-free rate is one of the most significant components in the Black-Scholes Model. It represents the theoretical return of an investment considered to have zero risk of financial loss.

In practice, this rate is usually derived from government bonds, such as U.S. Treasury bills. Here鈥檚 why the risk-free rate is crucial:
  • It informs the present value calculation of future cash flows, providing a benchmark for evaluating investment returns.
  • Helps in discounting future option pay-offs, ensuring that calculations align with realistic financial expectations.
When calculating an option鈥檚 fair price, the risk-free rate allows us to accurately assess the time value of money, which is essential for correctly accounting for how much future cash flows are worth in today鈥檚 terms.
Volatility
Volatility in the context of options is a measure of how much the price of the underlying asset is expected to fluctuate over a given period. It's a critical component of the Black-Scholes Model.
  • Expressed as a percentage, volatility ultimately affects premium prices for options.
  • Higher volatility signals greater uncertainty, meaning potential for more significant price movements. This often increases the value of an option.
  • Conversely, low volatility might lead to a lower premium since price movements are expected to be minimal.
Understanding volatility helps investors gauge the degree of risk associated with an option. It's vital for predicting how price changes can impact the value of an option, influencing strategies and pricing models.
Cumulative Distribution Function
In the Black-Scholes Model, the cumulative distribution function (CDF) is used to determine the probability of certain events related to price change. Specifically, it estimates the probability that a random variable, such as asset price, will be less than or equal to a certain value.
  • The CDF reflected by \(N(d_2)\) is crucial in computing the likelihood that the option will finish in-the-money, meaning the asset price exceeds the strike price at expiration.
  • This function helps translate the \(d_2\) value calculated earlier into a probability measurement.
Employing the CDF allows investors to make informed predictions about future price movements, significantly aiding in making investment decisions based on probability.

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Most popular questions from this chapter

Suppose that \(c_{1}\) and \(p_{1}\) are the prices of a European average price call and a European average price put with strike price \(K\) and maturity \(T, c_{2}\) and \(p_{2}\) are the prices of a European average strike call and European average strike put with maturity \(T\), and \(c_{3}\) and \(p_{3}\) are the prices of a regular European call and a regular European put with strike price \(K\) and maturity \(T\). Show that \(c_{1}+c_{2}-c_{3}=p_{1}+p_{2}-p_{3}\).

Calculate the price of a 1 -year European option to give up 100 ounces of silver in exchange for 1 ounce of gold. The current prices of gold and silver are \(\$ 380\) and \(\$ 4\), respectively; the risk-free interest rate is \(10 \%\) per annum; the volatility of each commodity price is \(20 \%\); and the correlation between the two prices is \(0.7\). Ignore storage costs.

Explain why a down-and-out put is worth zero when the barrier is greater than the strike rice.

In a 3-month down-and-out call option on silver futures the strike price is \(\$ 20\) per ounce and the barrier is \(\$ 18\). The current futures price is \(\$ 19\), the risk-free interest rate is \(5 \%\), and the volatility of silver futures is \(40 \%\) per annum. Explain how the option works and calculate its value. What is the value of a regular call option on silver futures with the same terms? What is the value of a down-and-in call option on silver futures with the same terms?

Consider an up-and-out barrier call option on a non-dividend-paying stock when the stock price is 50 , the strike price is 50 , the volatility is \(30 \%\), the risk-free rate is \(5 \%\), the time to maturity is 1 year, and the barrier at \(\$ 80\). Use the software to value the option and graph the relationstup between (a) the option price and the stock price, (b) the delta and the stock price, (c) the option price and the time to maturity, and (d) the option price and the volatility. Provide an intuitive explanation for the results you get. Show that the delta, gamma, theta, and vega for an up-and-out barrier call option can be th
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