If e is the base of natural logarithms, and (σ) is the standard deviation of the continuously compounded annual returns on the asset, and h is the interval as a fraction of a year, then the quantity (1 + upside change) is equal to:
A) e^[(σ) * SQRT(h)]
B) e^[h * SQRT(σ)]
C) (σ) * e^[SQRT(h)]
D) none of the above.

Question

Quizplus · Accepted Answer

According to the Black-Scholes model, the upside change in the stock price can be calculated using the formula:
1 + upside change = e^( (μ + 0.5σ^2 )h + σ√h*z ), where μ is the expected return, σ is the standard deviation of returns, h is the time horizon, and z is a standard normal random variable.
Assuming a continuously compounded annual return, we have:
μ = ln(1+r), where r is the continuously compounded annual return.
So, the formula can be rewritten as:
1 + upside change = e^( (ln(1+r) + 0.5σ^2 )h + σ√h*z )
Since we want the upper limit of the stock price, we need to consider the value of z that gives us the highest possible result. This occurs when z = 1, so we get:
1 + upside change = e^( (ln(1+r) + 0.5σ^2 )h + σ√h )
Simplifying this equation, we get:
1 + upside change = e^(ln(1+r)h) * e^(σ√h)
1 + upside change = (1+r)^h * e^(σ√h)
Dividing both sides by (1+r)^h, we get:
(1 + upside change)/(1+r)^h = e^(σ√h)
Therefore, the answer is A) e^[(σ) * SQRT(h)]

If E Is the Base of Natural Logarithms, and (σ)