How to price this option using the Black Scholes model?

Question

I have a question regarding regular option pricing.

In the standard Black-Scholes model, with interest r and volatility $\sigma$, I have to eetermine the arbitrage free price at time $t$ of an option which at $T>t$ pays the holder the amount of 100 USD dollar if the stock price is between 50 and 100 USD.

I.e. an option with payoff function:

$$\phi(S) = 100 ~ \text{if} ~ 50<S_T<100 ~ \text{else} ~ 0$$

A thorough walk through in how to calculate this price would be highly appreciated.

Answer 1

I am too lazy to write up a longer answer and I do not know how to write LateX, so here you go

Pricing formulas for Double Knock Out and Binary Range Options

Answer 2

The payoff can be decomposed as \begin{align*} \phi(S) &= 100 \, I_{50 \le S_T < 100}\\ &= 100 \, \big(I_{S_T \ge 50} - I_{S_T \geq 100}\big). \end{align*} Note that, under the risk-neutral measure $P$, \begin{align*} E(I_{S_T \ge K} \mid \mathcal{F}_t) &= P(S_T \ge K \mid \mathcal{F}_t)\\ &= N(d_2), \end{align*} where \begin{align*} d_2 = \frac{\ln \frac{S_t}{K} + \big(r-\frac{1}{2}\sigma^2\big) (T-t)}{\sigma\sqrt{T-t}}. \end{align*} The valuation of the above option payoff is now straightforward.