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Math for Quantitative Finance

Take a guided tour through the powerful mathematics and statistics used to model the chaos of the financial markets.

Expected Value in Quant Finance

                 

A stock is currently trading at $50. Tomorrow it will announce either good news (and rise to $55) or bad news (and fall to $40). If you believe the probability of good news is 80%, should you buy or sell the stock for $50?

Whether you answered the last question intuitively or rigorously, you were computing an expected value. For a discrete random variable \(X,\) this is \[\sum_{x \in S} x \cdot P(X=x)\] Or, the sum of values times the probability of achieving that value, where \(S\) is the set of values that \(X\) takes. For example, the expected value of the stock in the previous question was \[\sum_{x \in \{40,55\}} x \cdot P(X=x) = 40(0.2) + 55(0.8) = 52.\]

A stock is trading at $14. Tomorrow it will announce either good news (and rise to $20) or bad news (and fall to $10). What is the probability of good news, given that the stock is trading at its post-announcement expected value?

One of the most important properties of expected value is the linearity of expectation. This means that \[E(X+Y) = E(X)+E(Y)\] That is the expected value of the sum of two variables is equal to the sum of their individual expected values. Even when \(X\) and \(Y\) are dependent. The next two questions make use of this property.

The price of Stock A in one month is (approximately) normally distributed with mean $100 and standard deviation $25. The price of Stock B in one month is (approximately) normally distributed with mean $50 and standard deviation $5. They have a correlation of 90%. If you buy 3 shares of stock A and (short) sell two shares of stock B, what is the expected value of your portfolio (assuming these actions are all it contains) in one month (in dollars)?

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 \(7 \cdot 0.20 = 1.40.\)

You flip 10 coins and are paid $1 for each consecutive pair of heads. For example, if you flipped TTHHHTHHTT, you would be paid $3. What is the expected value of this game in dollars?

A work event gets very rowdy, and no one can remember which coat they brought. So each person takes a random coat as they leave. If each of the 100 employees brought exactly one coat, what is the expected number of people who get their correct coat?

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