Pump and Dumb (part 2)

The two-asset case

We saw in a previous post how assuming a nonlinear market impact function that only depends on the speed of trading leads to statistical arbitrage. In the present post (loosely adapted from ¹²), we shall address the case of two assets experiencing cross-market impact. Does the same conclusion hold in this case?

We are assuming that we have two traded assets s^1 and s^2 who are impacted by trading in either asset, through speed of trading only:

\((1)\ \ \ \ \ \ \begin{equation*} \left\{ \begin{split} ds^1_t &= \left(f^{11}(\lambda^1_t) + f^{12}(\lambda^2_t)\right)dt + \sigma_1 dW^1_t\\ \\ ds^2_t &= \left(f^{21}(\lambda^1_t) + f^{22}(\lambda^2_t)\right)dt + \sigma_2 dW^2_t\\ \\ dq^i_t &= \lambda^i_t dt,\ \ \ \ \ \ i=1,2\\ \\ dx_t &= -s^1_t dq^1_t - s^2_t dq^2_t, \end{split} \right. \end{equation*}\)

where q^i is our inventory in asset i, x_t is our cash account, and W^i_t are (possibly correlated) Brownians. To shorten the proof, we will restrict ourselves to smooth, odd functions f^{ij} (the latter property could actually be shown to be necessary). We will consider deterministic trading strategies, which start and end at 0: q^i_0 = q^i_T = 0, and for which x_0 = 0. Under these assumptions, a statistical arbitrage occurs if E(x_T) > 0. Let’s show that, for any such arbitrage to be ruled out, we must impose linearity on all functions f^{ij}, i,j = 1,2.

For that, first note that choosing either q^1_t or q^2_t to be identically zero reduces the problem to the one-asset case, for which we showed that the market impact function must be linear for no statistical arbitrage to exist. This readily implies that f^{11} and f^{22} are linear.

As to cross terms, let’s first observe that

\((2)\ \ \ \ \ \ \ \begin{equation*} \begin{split} \mathbb{E}\left(x_T\right) &= -\int_0^T s^1_t dq^1_t + s^2_t dq^2_t\\ \\ &= \int_0^T q^1_t\left(c^{11}\frac{dq^1}{dt} + f^{12}\left(\frac{dq^2}{dt}\right) \right)dt\\ &\ \ +\int_0^Tq^2_t\left(c^{22}\frac{dq^2}{dt} + f^{21}\left(\frac{dq^1}{dt}\right) \right)dt\\ &= \int_0^Tq^1(t)f^{12}\left(\frac{ dq^2}{dt}\right)dt + q^2(t)f^{21}\left(\frac{ dq^1}{dt}\right)dt\\ &= \int_0^T q(t)g\left(\frac{dq}{dt}\right)dt, \end{split} \end{equation*}\)

(where we used q^i_0 = q^i_T = 0), with the special choice that q^2(t) = q(t) is the piecewise linear function starting at 0, climbing to Q at t=t1>0, and returning to 0 at t=T (see our previous post), and q^1(t) = ξq^2(t) = ξq(t), for some nonzero ξ, and with the notation

\((3)\ \ \ \ \ \ g(v) = \xi f^{12}(v) + f^{21}\left(\xi v\right).\)

Repeating the same argument as in the single-asset case applied to the last equation in (2), we find that g must be linear, meaning

\((4)\ \ \ \ \ \ \xi f^{12}(v) + f^{21}\left( \xi v \right) = c\left(\xi\right) v,\ \ \ \ \forall v,\,\xi \ge 0\)

Interestingly, (4) forces both f^{12} and f^{21} to be linear. To see this, differentiate in ξ and set ξ=0 to find

\((5)\ \ \ \ \ \ \begin{equation*} \begin{split} &\xi f^{12}(v) + f^{21}(\xi v) = c(\xi) v\\ \\ &f^{12}(v) + v\left(f^{21}\right)’(\xi v) = c’(\xi) v\\ \\ &\xi = 0\implies f^{12}(v) = \left(c’(0) - \left(f^{21}\right)’(0)\right)v, \end{split} \end{equation*}\)

so that f^{12} is linear, and, setting ξ=1 in (4), so is f^{21}. (See reference 2 in the footnotes for a different proof.)

Quantitatively Yours,

1. J.Gatheral, “No-Dynamic-Arbitrage and Market Impact” (Quant. Finance (10) 2010)

2. Michael Schneidera, Fabrizio Lillo, “Cross-impact and no-dynamic-arbitrage” https://arxiv.org/abs/1612.07742