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Please explain the formula in the picture below for me 资产不同状态下的回报率 \[ r_{u}=\fra...

May 19, 2024

Please explain the formula in the picture below for me

Solution

资产不同状态下的回报率: The formula

r_{u}=\frac{X_{u}}{P}-1

and

r_{d}=\frac{X_{d}}{P}-1

represents the return rate of an asset in different states. Here,

r_{u}

is the return rate in the up state, and

r_{d}

is the return rate in the down state.

X_{u}

and

X_{d}

are the asset prices in the up and down states, respectively, and

P

is the initial price of the asset

资产期望回报率: The expected return rate of the asset is given by

E[r] = q r_{u} + (1-q) r_{d}

. This can be expanded as

E[r] = q\left(\frac{X_{u}}{P}-1\right) + (1-q)\left(\frac{X_{d}}{P}-1\right)

. Simplifying further, we get

E[r] = \frac{1}{P}\left[q X_{u} + (1-q) X_{d}\right] - 1

. This shows that the expected return rate is a weighted average of the returns in different states, adjusted by the initial price

Answer

The formulas describe the return rates of an asset in different states and the expected return rate, considering the probabilities of each state.

Key Concept

Return Rate Calculation

Explanation

The return rate formulas

r_{u}

and

r_{d}

calculate the returns in up and down states, while the expected return rate

E[r]

is a weighted average of these returns, adjusted by the initial price.