This is given by the natural logarithm of the new price divided by the old price; ln(80 / 90) = −0.1178.

Explanation in markdown format:

The continuously compounded rate of return is calculated using the formula:

$r = \ln\left(\frac{P_t}{P_{t-1}}\right)$

Where:

• $P_t$ = ending price = $80
• $P_{t-1}$ = beginning price = $90

Substituting the values:

$r = \ln\left(\frac{80}{90}\right) = \ln(0.888888...)$

$r = \ln(0.888888...) ≈ -0.1178$

Why other options are incorrect:

• Option B (-0.1000): This is the simple rate of return: (80-90)/90 = -10/90 = -0.1111, not the continuously compounded return.
• Option C (-0.1250): This would be (80-90)/80 = -10/80 = -0.1250, which is incorrect for either simple or continuously compounded returns.

Key Concept: Continuously compounded returns use natural logarithms, which provide the instantaneous rate of return and have useful mathematical properties for financial modeling, particularly in options pricing and time series analysis.