Question

Section 26.9 gives two formulas for a down-and-out call. The first applies to the situation where the barrier, $H$, is less than or equal to the strike price, $K$. The second applies to the situation where $H \geqslant K$. Show that the two formulas are the same when $H=K$.

   Section 26.9 gives two formulas for a down-and-out call. The first applies to the situation where the barrier, $H$, is less than or equal to the strike price, $K$. The second applies to the situation where $H \geqslant K$. Show that the two formulas are the same when $H=K$.
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Options, Futures, and Other Derivatives
Options, Futures, and Other Derivatives
John C. Hull 10th Edition
Chapter 26, Problem 6 ↓

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For the first formula, when $H \leq K$, we have: $C = S_0 \left( \frac{e^{(r-q)T}}{S_0}N(d_1) - \frac{e^{-rT}}{S_0}N(d_2) \right) - K e^{-rT} \left( N(d_1 - \sigma \sqrt{T}) - \left( \frac{H}{S_0} \right)^{2 \lambda} N(d_1 - \sigma \sqrt{T} - 2 \lambda \sigma  Show more…

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Section 26.9 gives two formulas for a down-and-out call. The first applies to the situation where the barrier, $H$, is less than or equal to the strike price, $K$. The second applies to the situation where $H \geqslant K$. Show that the two formulas are the same when $H=K$.
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