Digital Options: Payoffs, Pricing, and Put–Call Relationships
Summary
The document explains cash-or-nothing digital options, whose call and put payoffs are complementary: one pays a fixed cash amount when its condition is met, while the other pays in the opposite outcome. Their combined payoff is therefore fixed at expiry, so its present value is discounted. This differs from an asset-or-nothing payoff, where the holder receives the underlying asset and its value varies with the spot price.
A coin-flip analogy illustrates why a digital option’s theoretical value relates to the probability of its payout event, with discounting for the time until payment. The discussion corrects the proposed parity equation: ordinary call–put parity does not apply by substituting digital payoffs into its formula. It also notes that real digital options are often valued through call spreads. The probability interpretation is theoretical and depends on pricing assumptions; actual market prices and conventions can differ.
Key ideas
- A cash-or-nothing digital call pays a fixed amount above its strike, while its put counterpart pays at or below the strike.
- Combining complementary digital call and put payoffs produces a fixed cash amount at expiry.
- An asset-or-nothing digital payoff delivers the underlying, so its value depends on the spot price.
- The theoretical digital price is related to the probability of the payout event and discounted for time value.
- Market participants often approximate or price digital options using call spreads.
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Full text
# Digital and binary put/call options
# Digital and binary put/call options
I'm looking for put-call parity for the call and put digital options, but I don't really know what is `digital options` and it's difference between `binary options`.
I found that payoff of the digital call option is: $$ C^b(T) = \begin{cases}0, \; S(T) \leq K \\ 1, \: S(T) > K \end{cases},$$ and payoff for digital put option is: $$ P^b(T) = \begin{cases} 1, \; S(T) \leq K \\ 0, \: S(T) > K \end{cases}.$$ Are they the same for binary options? Does "$^b$" in $C^b$ mean it binary ( = digital) option?
Next I found put-call parity is as follows: $$ C - P = S(t) - Ke^{-r(T-t)},$$ so put-call parity for the call and put digital options would be
$$ \begin{cases} -1 = S - Ke^{-r(T-t)},\; if \; S(T)\leq K \\ 1 = S - Ke^{-r(T-t)},\; if \; S(T)> K \end{cases} \;?$$
## Answer by AKdemy (score 3)
https://quant.stackexchange.com/a/64008
Generally, I would say it is a bit difficult to look for put-call parity of something you do not know what it is in the first place.
To give more intuition to what fesman wrote, look at your C and P equation. If you have a C+P, you get 1 no matter what. Why +P? That is just because it is the most natural way to look at it.
Combined, call plus put, will give you 1 no matter the outcome. That payoff, will be in the future though, so you discount to today. Furthermore, it can be any monetary value. This is the cash or nothing variant.
The other variant is not important here but intuitively, if you get the asset, further increases or decreases in Spot above or below the strike have an impact on the actual value. Assume the asset has a strike of 100 and you receive either 100 USD or 1 asset. 100 USD will always be 100 USD, no matter the value of the underlying. However, 1 asset will only be worth 100 if spot is equal to strike. If spot ends up 10% above strike, you actually gain 110 USD.
Simplified, it is like flipping a coin (binary means one of two outcomes). Heads you win (1 or any agreed cash payment), tails you lose (get nothing). In a fair coin example, the price will be 0.5. Why? Because your expected outcome is that it falls 50% of the time on heads and 50% on tails. Hence, you will get 1 in 50% of the time, and zero in the other outcome. Assume you agree to play in 1 year today. You can still expect to get 0.5 but the time value of money tells you it needs to be discounted to todays value.
The fair price of a digital options is also the probability of the event happening (S>K for call). At least theoretically. In reality they are usually priced as call spreads.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.