Trade Fast, Trade Slow

Question

Minimize f(x) = x + frac1x over x > 0 without calculus. Then solve the desk version: execution cost is C(v) = a v + fracbv in trading speed v (impact grows with speed, risk grows with slowness), find the optimal speed and the minimal cost.

Accepted Answer

AM-GM does it in one line. For x > 0, x + frac1x ge 2sqrtx cdot frac1x = mathbf2, with equality iff x = frac1x, i.e. x = 1. The trick works because the product of the two terms is constant, so the sum is minimized when they're equal. (Calculus agrees: 1 - frac1x^2 = 0 Rightarrow x = 1, second derivative frac2x^3 > 0.) The general form. av + fracbv ge 2sqrtab, equality at v^ = sqrtfracba, at the optimum the two cost terms are exactly equal, each sqrtab. Why this is the execution-scheduling problem. Trade a block too fast and you pay market impact (roughly increasing in participation speed: the av term); trade too slowly and you sit exposed to price risk and adverse drift while unfilled (the fracbv term). The AM-GM answer encodes two portable lessons: 1. Optimal speed balances marginal costs: at v^, impact cost equals risk cost. If your impact bill is running well above your risk estimate, you're trading too fast, no derivative required to see it. 2. Square-root economics: the minimal total cost 2sqrtab scales like the geometric mean of the two coefficients. Halving impact (a to fraca2) cuts total optimal cost only by a factor frac1sqrt2 approx 0.71, because you partially re-spend th

Trade Fast, Trade Slow

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