Problem bank

Problem 333 of 333HardStochastic CalculusP333

Expected inventory penalty for a market maker

  1. A market maker skews her quotes to push her inventory back toward zero, so her position in shares follows

    dXt=−κXt dt+σ dWt.dX_t = -\kappa X_t\,dt + \sigma\,dW_t.

    The risk desk charges her a running penalty of ∫tTXs2 ds\int_t^T X_s^2\,ds. Define

    g(t,x)=E[∫tTXs2 ds ∣ Xt=x].g(t, x) = E\Big[\int_t^T X_s^2\,ds \,\Big|\, X_t = x\Big].

    (a) Show that gg solves a PDE of Feynman-Kac type, and give its terminal condition.

    (b) Solve the PDE with the guess g=A(τ)x2+B(τ)g = A(\tau)x^2 + B(\tau), where τ=T−t\tau = T - t. Check the answer by computing the expectation directly.

    (c) Take κ=2\kappa = 2 per hour, σ=10\sigma = 10 shares per hour\sqrt{\text{hour}}, x=30x = 30 shares and one hour until the close. Evaluate gg and compare it with the case κ=0\kappa = 0.