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Questions and Answers
Which of the following is a key characteristic of temperate deciduous forests?
Which of the following is a key characteristic of temperate deciduous forests?
- Trees that lose their leaves annually. (correct)
- Sparse vegetation due to extreme cold.
- Dominated by coniferous, evergreen trees.
- High rainfall and consistently warm temperatures.
Which of the following best describes the climate of temperate deciduous forests?
Which of the following best describes the climate of temperate deciduous forests?
- Hot and dry summers with cold and wet winters
- Mild and wet weather, categorized as temperate maritime (correct)
- Extremely cold winters and short, cool summers.
- Consistently warm temperatures with high humidity year-round.
In which of the following regions are coniferous forests primarily found?
In which of the following regions are coniferous forests primarily found?
- Scandinavia, Russia, and Canada (correct)
- The Mediterranean and coastal California
- The Australian Outback and the Sahara Desert
- Central Africa and the Amazon basin
What is a defining characteristic of mountain areas regarding vegetation growth?
What is a defining characteristic of mountain areas regarding vegetation growth?
Which of the following biomes is characterized by extremely cold climates and limited plant and animal life?
Which of the following biomes is characterized by extremely cold climates and limited plant and animal life?
How does the distribution of large-scale ecosystems (biomes) primarily depend on?
How does the distribution of large-scale ecosystems (biomes) primarily depend on?
In what regions are tropical forests commonly located?
In what regions are tropical forests commonly located?
What is a distinctive climatic feature of savannas or tropical grasslands?
What is a distinctive climatic feature of savannas or tropical grasslands?
Which of the following is a characteristic of Mediterranean climates?
Which of the following is a characteristic of Mediterranean climates?
Which of the following options accurately describes an ecosystem?
Which of the following options accurately describes an ecosystem?
Flashcards
Biosphere
Biosphere
The biosphere is a global ecosystem composed of living organisms (biota) and the abiotic (non-living) factors from which they derive energy and nutrients.
Ecosystem
Ecosystem
An ecosystem is a small-scale community of living things that interact with each other and their non-living environment; it may be as large as a desert or as small as a puddle.
Abiotic Factors
Abiotic Factors
Abiotic factors refer to non-living physical and chemical elements in the ecosystem such as climate, soil, water, air, sunlight, and relief (geology).
Biotic Factors
Biotic Factors
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Desert
Desert
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Savanna
Savanna
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Temperate Grassland
Temperate Grassland
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Tropical Forest
Tropical Forest
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Study Notes
Setting up a Risk-Free Portfolio
- Option price is represented by $C$.
- Stock price is represented by $S$.
- $\Delta$ represents the number of shares to short.
- The portfolio is represented as $-\Pi = C - \Delta S$.
- The change in portfolio value is $-\Delta \Pi = \Delta C - \Delta \mathrm{S} \Delta$.
- $\Delta C$ can be expressed as $\frac{\partial C}{\partial S} \Delta S + \frac{1}{2} \frac{\partial^2 C}{\partial S^2} (\Delta S)^2 + \frac{\partial C}{\partial t} \Delta t + \dots$.
- Substituting $\Delta C$, the change in portfolio value becomes $-\Delta \Pi = \frac{\partial C}{\partial S} \Delta S + \frac{1}{2} \frac{\partial^2 C}{\partial S^2} (\Delta S)^2 + \frac{\partial C}{\partial t} \Delta t - \Delta S \Delta$.
- To eliminate risk, $\Delta$ is chosen such that $\Delta = \frac{\partial C}{\partial S}$.
- This simplifies the change in portfolio value to $-\Delta \Pi = \frac{1}{2} \frac{\partial^2 C}{\partial S^2} (\Delta S)^2 + \frac{\partial C}{\partial t} \Delta t$.
Black-Scholes PDE
- Change in stock price, $\Delta S$, is given by $\mu S \Delta t + \sigma S \Delta W$.
- $(\Delta S)^2$ can be expressed as $\sigma^2 S^2 (\Delta W)^2$.
- The expectation of $(\Delta W)^2$ is $\Delta t$, i.e., $E[(\Delta W)^2] = \Delta t$.
- Therefore, $(\Delta S)^2 = \sigma^2 S^2 \Delta t$.
- Substituting $(\Delta S)^2$ the change in portfolio value becomes $-\Delta \Pi = \frac{1}{2} \frac{\partial^2 C}{\partial S^2} \sigma^2 S^2 \Delta t + \frac{\partial C}{\partial t} \Delta t$.
- Risk-free return is $-\Delta \Pi = r \Pi \Delta t = r(C - S \frac{\partial C}{\partial S}) \Delta t$.
- Equating the two expressions for $-\Delta \Pi$ yields $\frac{1}{2} \frac{\partial^2 C}{\partial S^2} \sigma^2 S^2 \Delta t + \frac{\partial C}{\partial t} \Delta t = r(C - S \frac{\partial C}{\partial S}) \Delta t$.
- Dividing by $\Delta t$ and rearranging, the Black-Scholes PDE is obtained: $\frac{\partial C}{\partial t} + rS \frac{\partial C}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} = rC$.
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