Pricing a Cash-or-Nothing Range Option with Black–Scholes

Summary

The document considers a European-style option that pays a fixed cash amount when the terminal stock price falls within a specified interval, and pays nothing otherwise. It shows how to express that range payoff as the difference between two cash-or-nothing digital payoffs: one triggered above the lower strike and one triggered above the upper strike. This decomposition reduces the valuation to pricing two digital claims.

Under the Black–Scholes assumptions, the risk-neutral probability that the stock finishes above a strike is given by the cumulative normal distribution evaluated at the model’s d₂ term. The range option’s value is therefore the cash amount multiplied by the difference between the corresponding probabilities, with discounting for payment at maturity. The response outlines this route rather than carrying through a numerical calculation. Its result depends on the standard model inputs and assumptions, including volatility, interest rates, and the stock’s modeled price dynamics; real-world frictions or alternative payoff boundary conventions may require adjustments.

Key ideas

• A fixed-payout range option can be decomposed into two digital options at its lower and upper boundaries.
• In Black–Scholes, a digital option’s risk-neutral value is based on the probability of finishing above its strike.
• The range payoff value follows from subtracting the upper-strike digital value from the lower-strike value.
• The formula relies on Black–Scholes assumptions and requires model inputs such as volatility and interest rates.

Tags

# How to price this option using the Black Scholes model?


# How to price this option using the Black Scholes model?












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 by Matt Wolf (score 2)

https://quant.stackexchange.com/a/11037

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 by Gordon (score 2)

https://quant.stackexchange.com/a/21759

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.