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Difficulty: Hard
Category: Stochastic Calculus
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Topics: digital-option, delta, stochastic-calculus, dirac-delta, option-pricing
Consider a digital call option that pays 1 if the underlying asset $S$ is above the strike price $K$ at expiry $T$, and 0 otherwise. The current time is $t$, and we are close to expiry ( $T - t$ is small). Assume the underlying asset follows a geometric Brownian motion: $dS = \mu S dt + \sigma S dW$, where $\mu$ is the drift, $\sigma$ is the volatility, and $dW$ is a Wiener process. What happens to the delta of this digital call option as $S$ approaches $K$ near expiry ( $T-t \rightarrow 0$ )
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