Binary payout on gold under normal dynamics

Question

Gold trades at $400 per ounce and follows arithmetic Brownian motion with volatility $60 per year; interest rates are zero. What is the value of a contract paying $1 million in six months if gold is then at $430 or higher?

Short answer

About $240{,}000. In six months gold is normal with mean $400$ and standard deviation $60\sqrt{0.5} \approx 42.43$ . The payout probability is $1 - \Phi(30/42.43) = 1 - \Phi(0.707) \approx 0.2398$ , giving about $239{,}750 with no discounting.

Adapted from Crack, Heard on the Street, Q2.37, p.30; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Barrier Options
The Four Faces of Brownian Motion

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