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I don't believe you will necessarily find a cite-able source as, I believe, this comes from a practical rather than theoretical motivation. As you know option prices are a function of: future prices, discount rates and implied volatility, volatility surface skew and other supple/demand factors. So when you are trading these instruments, you need to ...

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Ftse100 would not have a smooth dividend yield, as your formula has, it would be discrete, being much higher on certain days of year than others. In pricing options on ftse, u need to take into account implied dividends (dividends that are implied by put call parity)

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Sorry to disagree but if interest rates is 0, the binary is still not worth $1 now. Suppose spot$S(0) = 100$, assume$x = 110\$ and upon touch (whenever it happens as the option has no maturity) you receive one dollar. Suppose I buy 1 stock. If the barrier hits, i sell the stock and receive 110 USD. What if I buy N stocks at t=0? upon hit of barrier i ...

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