Past exam of the mathematical models of financial derivatives course of the University of Oxford / 2015 / 2 / c / Solution 2026-10-25
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Take the terminal financial payoff . The function satisfies the Black-Scholes equation: its time derivative and second stock derivative vanish, while . It also has the correct terminal value. Part a therefore gives
(1)
Thus the discounted stock is a martingale. The explicit lognormal distribution in part b supplies the necessary finite expectation, so here this is a true martingale identity.
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