Eric Wepsic, D. E. Shaw & Co., Vice President

Problem

This is a management-oriented example. We have trading parameters in our system. Sometimes, we wonder whether we could improve these parameters, and, if so, by how much. Suppose our current setting of a parameter is 120, and our system makes $5 million / year. If we find out that the ideal value of our parameter could be as small as 115 and as much as 125, what would determining the parameter more accurately be worth? Is it worth doing, if it would take a year?

Solution

Let f(p) be profitability as a function of the parameter setting p. Suppose that a parameter of 0 leads to 0 profitability, and that f is always positive. Suppose also that the best value is as far as possible from 120. Near a maximum, profitability is well-approximated by

f(120) ~= f(115) + f'(115) * 5 + f''(115) * 5^2 / 2.
          ~= f(115) + f''(115) * 5^2 / 2.

Note now that f''(115) is probably no more than 500, assuming no great concavity changes, since
0 < f(0) ~= f(120) + f''(120) * 120^2 / 2
so f(120) ~= $5M,
f''(120) ~<= $10M / 120^2 = 1000.

This means that

f(120) - f(115) <= 1000 * 12.5.

So the difference in profitability is around $12500. Therefore, this is probably not worth working on.

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