Math for Quantitative Finance
# Expected Value

You are trading a weather contract that pays out $1 each day that it rains in a given 7-day week. If the probability of rain on any given day is 20%, what is the fair value of this 7-day contract (in dollars)?

**Remark:** These types of contracts can be used by agricultural companies and energy companies as hedges against unfavorable weather.

A useful construct for exploiting linearity of expectation is the indicator variable. It is 1 when some event occurs, and 0 otherwise. We can reframe the previous problem in terms of an indicator variable.

Take \(\mathbb{1}_{n}\) to be an indicator variable on the event of rain on day \(n.\) Then, the expected number of rainy days is \[E(\mathbb{1}_{1} + \mathbb{1}_2 + \cdots + \mathbb{1}_7),\] which by linearity of expectation is \[E(\mathbb{1}_{1}) + E(\mathbb{1}_2) + \cdots+ E(\mathbb{1}_7).\] Each of these variables has expected value \((0.20)(1) +(0.80)(0) = 0.20,\) so the expected value was \(7 \cdot 0.20 = 1.40.\)

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