Quadratic transaction costs are a widely-used tractable specification for trading frictions in continuous-time financial models. The cost is levied on the square of each agent’s trading rate (order flow), so that agent n’s cumulative cost over [0,T] is lambda * integral_0^T (phi-dot_t^n)^2 dt, where lambda > 0 is the cost parameter and phi-dot denotes the rate of change of the portfolio position.
This specification is popular because it preserves linearity of the first-order conditions, leading to linear FBSDEs that can be solved explicitly via matrix exponentials (Herdegen et al. 2018). The quadratic form can be interpreted as either a temporary price impact (proportional to trade size and speed) or as a progressive transaction tax levied by an exchange (Gonon et al. 2020).
The qualitative and quantitative properties of equilibria are robust across different convex cost specifications when costs are calibrated to match trading volume (Gonon et al. 2020, Section 5), justifying the use of quadratic costs as a proxy for more realistic cost structures. The quadratic specification G(x) = lambda x^2 / 2 is the q = 2 special case of the power cost family G_q(x) = lambda |x|^q / q.
Key Details
- Cost functional: lambda * integral (phi-dot)^2 dt, penalising the trading rate phi-dot
- Key advantage: preserves linearity of optimality conditions, enabling explicit FBSDE solutions
- Optimal trading: agents track a moving target portfolio at speed proportional to sqrt(gamma * sigma^2 / (2 * lambda)), where gamma is risk aversion
- Tracking function: G(t) = cosh(sqrt(gamma/lambda)(T-t)) determines the discount kernel for future target values
- Small-cost scaling: leading-order correction to frictionless equilibrium is O(sqrt(lambda)), i.e. the equilibrium price deviation from frictionless scales with the square root of the transaction cost
- Equivalence: for zero net supply, quadratic costs with parameter lambda are equivalent to holding costs with parameter gamma when their ratio gamma/lambda is matched