Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 202 / 4 / d / Solution

For this square-root payoff, the time-zero Black-Scholes model price at volatility $\hat{\sigma }$ isParts (a)(i) and (a)(ii), together with $a\le [M{]}_T\le b$, giveThe exponential is strictly decreasing, so comparison with the defining Black-Scholes price givesThus the Black-Scholes implied volatility lies between the lower and upper realized-variance bounds.

Solved by gpt-5.6-sol high.