Bounds on the Joint Probability of Two Digital Options

Summary

The document explains intuitive bounds for a dual digital payoff that pays when two digital option conditions are both met. If the individual event probabilities are 0.7 and 0.8, the joint probability cannot exceed the smaller probability: the less likely event can be entirely contained within the more likely one. This gives a maximum joint price of 0.7 when outcomes are perfectly positively aligned.

For the minimum, the reply arranges outcomes to overlap as little as possible. With event probabilities of 0.7 and 0.8, the events must still overlap by 0.5, yielding the minimum price under perfect negative dependence. The explanation uses conditional probabilities to illustrate the calculation. It assumes the digital prices can be read as event probabilities under the relevant pricing measure and abstracts from discounting, payoff amounts, and other pricing details.

Key ideas

• A dual digital’s value depends on the joint probability of its two underlying digital events.
• The maximum joint probability is bounded by the smaller of the two individual probabilities.
• The minimum joint probability occurs when the events overlap as little as their marginal probabilities allow.
• The example illustrates the bounds through conditional probabilities and assumes digital prices represent event probabilities.

Tags

# The Intuition Behind The Max and Min of a Dual Digital


# The Intuition Behind The Max and Min of a Dual Digital












How do we intuitively figure out what the max and min price of a dual is? We know that the factors to price a dual is the price of digital option 1 and price of digital option 2 as well as correlation.

For example, for a dual digi (both calls) with price of 1 being 0.7 and price of 2 being 0.8, we know that the min dual price (when the correlation is -1) is 0.5. But why? Why is the max 0.7 (when the correlation is 1)?

## Answer by Arshdeep (score 1)

https://quant.stackexchange.com/a/80200

Best case, if 1 happens, then 2 also always happens. So the max is $0.7$ (if 2 happens, 1 can't always happen because the mass at 1 is lower).

Worst case, 2 happens, then 1 happens with a chance of $x$. So $0.8*x+0.2*y=0.7$ so x is minimum if $y=1$, so $x=5/8$.

So the price is $2 happens * Pr(1 happens|2 happens)$=$0.8*5/8=0.5$

If you like you can also get the max price with maximizing $x$, which happens at $x=7/8$.

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