The below exercise and solution was found in "Models for Financial Economics" by Abraham Weishaus. My issues are:
- In this problem, $S(t)$ does not satisfy the Black-Scholes framework because volatility is time-varying. According to this paper on page 15, it would appear that we would need to modify the volatility of $S(t)$ for use in the Black-Scholes formula such that:
$$\text{Var}\left(\ln S(t) | S(0)\right) = \frac{1}{1 - 0}\int_0^1 (\sigma(t))^2 dt = \frac{1}{1 - 0}\int_0^1 (0.16t)^2 dt = 0.00853333.$$
- The second issue is that this appears, based on the wording, to be a gap option where the trigger is $S(1) > 60$ and not $S(1)^{0.9} > 60$.
The author mentions in the errata that the $r$ in the fourth expression from the top of the solution should be changed in the exponent to $\alpha$, but does not address these other concerns.
Are these concerns justified, or is the author's solution correct?