/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 Consider a commodity with consta... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 13

Consider a commodity with constant volatility, \(\sigma\). Assuming that the risk-free interest rate is constant, show that in a risk-neutral world: $$\ln S_{T} \sim \phi\left[\ln F-\frac{\sigma^{2}}{2}(T-t), \sigma \sqrt{T-t}\right]$$ where \(S_{T}\) is the value of the commodity at time \(T\) and \(F\) is the futures price for a contract maturing at time \(T\).

Short Answer

Expert verified
In a risk-neutral world, \(\ln S_T\) is normally distributed with mean \(\ln F - \frac{\sigma^2}{2}(T-t)\) and variance \(\sigma^2 (T-t)\).

Step by step solution

01

Understanding the Assumptions

We're considering a commodity with constant volatility \(\sigma\) and a constant risk-free interest rate. Under the risk-neutral measure, futures prices follow a specific distribution.
02

Using Log-normal Distribution Property

Recall that under a risk-neutral world, the future price of any asset follows a log-normal distribution. This means that \(\ln S_T\), which is the natural logarithm of the spot price at time \(T\), will be normally distributed.
03

Defining the Mean of the Normal Distribution

In a risk-neutral world, the expected spot price growth is equal to the risk-free interest rate. Therefore, the present value of the expected future spot price is the current futures price, \(F\). The expected value of \(\ln S_T\) thus becomes \( \ln F - \frac{\sigma^2}{2}(T-t) \).
04

Determining the Variance of the Normal Distribution

The variance of \(\ln S_T\) is determined by the volatility \(\sigma\) and the time \(T-t\) until maturity. It is expressed as \(\sigma^2 (T-t)\). Hence, the standard deviation is \(\sigma \sqrt{T-t}\).
05

Conclusion

Putting together the mean and the variance, we conclude that \(\ln S_T \sim \phi \left[\ln F - \frac{\sigma^2}{2}(T-t), \sigma \sqrt{T-t} \right]\), confirming that \(\ln S_T\) is normally distributed with the specified parameters.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Commodity Pricing
Commodity pricing refers to the process of determining the market value of basic goods or raw materials like oil, gold, or agricultural products. Unlike traditional assets, commodities have prices that are influenced by global supply and demand, geopolitical factors, and seasonal variations. This dynamic nature makes it complex and fascinating.
**Factors Influencing Commodity Pricing:**
  • **Supply and Demand:** Basic economics dictate that if demand surpasses supply, prices rise, and vice versa.
  • **Geopolitical Events:** Conflicts, sanctions, and trade policies can disrupt supply chains and affect prices.
  • **Weather Conditions:** Especially relevant for agricultural commodities, where crops depend heavily on climate.
  • **Macroeconomic Indicators:** Interest rates, inflation, and currency fluctuations also play a critical role.
In the context of risk-neutral valuation, commodity pricing assumes that all investors are indifferent to risk, which implies a streamlined pricing process that solely factors the risk-free rate and intrinsic market dynamics.
Log-normal Distribution
A log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed. This means that if you take the natural logarithm of a log-normal distribution, it transforms into a normal distribution. This is particularly useful in finance as returns on investments and commodity prices often follow a log-normal distribution.
**Key Characteristics:**
  • **Positive Values:** The distribution is skewed to the right, meaning it only produces positive values, which aligns well with pricing since prices can't be negative.
  • **Modeling Growth:** Simulates the multiplicative processes of growth and decay, mimicking real-world financial behaviors.
In a risk-neutral world, the future price of commodities is modeled as a log-normal distribution, as shown in the exercise. The natural logarithm of the future spot price becomes normally distributed, helping forecast future price trends effectively under risk-neutral assumptions.
Volatility in Finance
Volatility, represented by \( \sigma \,\), is a statistical measure of the dispersion of returns for a given security or market index. In simpler terms, it indicates how much the price of a commodity can vary over time. In finance, a higher volatility means more risk as the price can significantly differ above or below the expected value.
**Significance of Volatility:**
  • **Risk Assessment:** Higher volatility suggests higher risk, potentially leading to higher returns or losses.
  • **Pricing Model Inputs:** Essential for pricing derivatives and other financial instruments like options or futures.
  • **Impact on Asset Allocation:** Influences investors' decision-making process regarding asset distribution in portfolios.
In the context of risk-neutral valuation, constant volatility is an assumption that simplifies calculations. This helps in establishing models where future price expectations are easier to compute and predict.
Risk-Free Rate
The risk-free rate represents the return on an investment with zero risks of financial loss. It is the hypothetical rate of return expected from an absolutely risk-free investment over a specific period. Normally, government bonds of stable countries are used as a proxy for the risk-free rate due to their reliability.
**Role in Risk-Neutral Valuation:**
  • **Baseline for Expected Returns:** Sets the foundation for comparing other investment options and assessing risk premiums.
  • **Impact on Pricing Models:** Integral to calculating fair values of options and futures, ensuring that expected returns can be built upon a risk-free base benchmark.
  • **Influence on Economic Decisions:** Plays a pivotal role in monetary policy and economic forecasting.
In risk-neutral world assumptions used in financial modeling, the risk-free rate helps in determining future pricing predictions without accounting for risk, allowing for simpler evaluation of potential future scenarios.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that an interest rate, \(x,\) follows the process $$d x=a\left(x_{0}-x\right) d t+c \sqrt{x} d z$$ where \(a, x_{0},\) and \(c\) are positive constants. Suppose further that the market price of risk for \(x\) is \(\lambda .\) How should the drift rate in \(x\) be adjusted when the extension of the risk-neutral valuation argument is used to value a derivative security?

Suppose that the market price of risk for gold is zero. If the storage costs are \(1 \%\) per annum and the risk-free rate of interest is \(6 \%\) per annum, what is the expected growth rate in the price of gold?

An oil company is set up solely for the purpose of exploring for oil in a certain small area of Texas. Its value depends primarily on two stochastic variables: the price of oil and the quantity of proven oil reserves, Discuss whether the market price of risk for the second of these two variables is likely to be positive, negative, or zero.

The convenience yield for soybean oil is \(5 \%\) per annum, the storage costs are \(1 \%\) per annum, the risk-free interest rate is \(6 \%\) per annum, and the expected growth in the price of soybean oil is zero. What is the relationship between the 6 -month futures price and the expected price in 6 months?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.