Decomposing a Piecewise Binary Payoff into Cash-or-Nothing Calls

Summary

The document shows how to value a stock-contingent payoff that pays one cash amount when the terminal stock price lies between two thresholds, a larger amount above the upper threshold, and nothing below the lower threshold. The key construction is to express the payoff as the sum of two cash-or-nothing call options: one struck at the lower boundary and another at the upper boundary, each paying the lower cash amount when its strike condition is met.

For terminal prices between the strikes, only the lower-strike digital pays; above the upper strike, both pay, producing the larger amount. The question supplies the stock price, volatility, risk-free rate, maturity, and payoff levels, but the accepted answer gives the replication rather than a numerical valuation. Pricing still requires a suitable digital-option valuation framework and consistent assumptions, such as the risk-neutral model and treatment of the strict boundary conditions.

Key ideas

• A payoff that is constant over separate terminal-price regions can be represented by digital options.
• The lower-strike cash-or-nothing call creates the payment above the lower threshold.
• A second digital call at the upper threshold adds the extra payment above that threshold.
• The document explains payoff decomposition but does not calculate a numerical option value.

Tags

# Cash or nothing option question


# Cash or nothing option question












I got stuck in one exercise of binary options, it says that I have to consider a stock that does not pay dividends, the current price of the stock is 100, the volatility of it is 20%, the risk-free rate is 4%, consider that the option has a term of one year. Let S(1) be the price of the stock at the end of the contract. The option will pay 10 if 100 <S(1)<120. And it will pay 20 if S (1)> 120. It will pay 0 otherwise. Calculate the price of the option. Now I know that for the second part I will have to use a Cash or nothing call option for the price for I'm not sure what to use in the first part, I also asume that my strike price is 120, is it right? What can I do for the first part? Thanks

## Answer by Valometrics.com (score 1, accepted)

https://quant.stackexchange.com/a/60721

This is the sum of the folowing cash or nothing (also called digital or binary) options:

1-Call with strike 100 and paying 10.

2-Call with strike 120 and paying 10.

hope it helps!

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