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

Factor the square root of the stochastic exponential asThe stochastic exponential in this expression is a positive local martingale and hence a supermartingale, so its expectation is at most one. If $[M{]}_T\ge a$ almost surely, then

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If $[M{]}_T\le b$, the Novikov condition holds for $(M-M_0)/2$ becauseIts stochastic exponential is therefore a true martingale with expectation one. Using the factorization from part (i),

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With zero interest rate, the bank account is constant. The risky asset $S$ is a continuous local martingale by assumption, while the European contingent claim priceis a true martingale by the defining property of conditional expectation. Thus the original probability measure is an equivalent local martingale measure for all traded discounted prices. The fundamental theorem of asset pricing then excludes arbitrage for admissible self-financing strategies.

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In the Black-Scholes model,Conditioning on ${\mathcal{F}}_t$ and using the moment-generating function of the independent Gaussian increment $W_T-W_t$ givesThe function $c$ satisfies the zero-rate Black-Scholes equation, so Itô formula leaves only its stochastic term:Consequently the required delta hedge is

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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.

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