Exponential Utility Ansatz in an Inventory-Control HJB Equation
Summary
The document examines a market-making style Hamilton–Jacobi–Bellman equation with a diffusive asset price, inventory changes from bid and ask fills, and exponential terminal utility. It tests a separable value-function ansatz that expresses wealth through a reduced function of time, price, and inventory. Applying the chain rule transforms the equation into one for that reduced function and produces exponential terms for the fill events.
The question is why the transformed arrival-rate terms appear with the opposite signs from a claimed expression. The answer identifies a sign error in differentiating the negative exponential value function: its derivative with respect to the ansatz variable is negative, not positive. That correction reverses the signs in the normalized terms. This is a concise algebraic correction, not a derivation of an optimal quoting policy or a numerical solution. It assumes the stated HJB form and ansatz, and does not discuss boundary conditions for inventory, the arrival-rate specification, or model calibration.
Key ideas
- The ansatz factors wealth dependence from the value function using exponential utility.
- Differentiating a negative exponential introduces a negative derivative that must be carried through the HJB substitution.
- The corrected derivative changes the signs of the normalized arrival-rate terms.
- The document resolves an algebraic sign issue but does not solve or calibrate the full market-making model.
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# Ansatz and HJB equation
# Ansatz and HJB equation
Suppose we have an HJB equation of the form $$ \frac{\partial v}{\partial t}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}v}{\partial s^{2}}+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[v(t,s,x+s+\delta^{a},q-1)-v(t,s,x,q)\right]\right\}+max_{\delta^{b}}\left\{ \lambda^{b}(\delta^{b})\left[v(t,s,x-s+\delta^{b},q+1)-v(t,s,x,q)\right]\right\} $$ with terminal condition $$ v(T,s,x,q)=-e^{-\gamma(x+qs)} $$ We will search a solution of the form $$ v(t,s,x,q)=-e^{-\gamma\left(x+\theta(t,s,q)\right)}=f(x,\theta(t,s,q)) $$ by direct substitution into HJB equation and application of the chain rule we get $$ \frac{\partial f(x,\theta(t,s,q))}{\partial\theta(t,s,q)}\frac{\partial\theta(t,s,q)}{\partial t}+\frac{1}{2}\sigma^{2}\left[\frac{\partial f(x,\theta(t,s,q))}{\partial\theta(t,s,q)}\frac{\partial^{2}\theta(t,s,q)}{\partial s^{2}}+\frac{\partial^{2}f(x,\theta(t,s,q))}{\partial\theta(t,s,q)^{2}}\left(\frac{\partial\theta(t,s,q)}{\partial s}\right)^{2}\right]+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[f(x+s+\delta^{a},\theta(t,s,q-1))-f(x,\theta(t,s,q))\right]\right\} +max_{\delta^{b}}\left\{ \lambda^{b}(\delta^{b})\left[f(x-s+\delta^{b},\theta(t,s,q+1))-f(x,\theta(t,s,q))\right]\right\} $$ taking derivatives of $f$ $$ \gamma f(x,\theta(t,s,q))\frac{\partial\theta(t,s,q)}{\partial t}+\frac{1}{2}\sigma^{2}\left[\gamma f(x,\theta(t,s,q))\frac{\partial^{2}\theta(t,s,q)}{\partial s^{2}}-\gamma^{2}f(x,\theta(t,s,q))\left(\frac{\partial\theta(t,s,q)}{\partial s}\right)^{2}\right]+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[f(x+s+\delta^{a},\theta(t,s,q-1))-f(x,\theta(t,s,q))\right]\right\}+max_{\delta^{b}}\left\{ \lambda^{b}(\delta^{b})\left[f(x-s+\delta^{b},\theta(t,s,q+1))-f(x,\theta(t,s,q))\right]\right\} $$ dividing by $\gamma f(x,\theta(t,s,q))$ $$ \frac{\partial\theta(t,s,q)}{\partial t}+\frac{1}{2}\sigma^{2}\left[\frac{\partial^{2}\theta(t,s,q)}{\partial s^{2}}-\gamma\left(\frac{\partial\theta(t,s,q)}{\partial s}\right)^{2}\right]+max_{\delta^{a}}\left\{ \frac{\lambda^{a}(\delta^{a})}{\gamma}\left[e^{-\gamma\left(s+\delta^{a}+\theta(t,s,q-1)-\theta(t,s,q)\right)}-1\right]\right\}+max_{\delta^{b}}\left\{ \frac{\lambda^{b}(\delta^{b})}{\gamma}\left[e^{\gamma\left(s-\delta^{b}-\theta(t,s,q+1)+\theta(t,s,q)\right)}-1\right]\right\} $$ Is this correct? It is claimed that with this ansatz we should instead have $$ \frac{\partial\theta(t,s,q)}{\partial t}+\frac{1}{2}\sigma^{2}\left[\frac{\partial^{2}\theta(t,s,q)}{\partial s^{2}}-\gamma\left(\frac{\partial\theta(t,s,q)}{\partial s}\right)^{2}\right]+max_{\delta^{a}}\left\{ \frac{\lambda^{a}(\delta^{a})}{\gamma}\left[1-e^{-\gamma\left(s+\delta^{a}+\theta(t,s,q-1)-\theta(t,s,q)\right)}\right]\right\} +max_{\delta^{b}}\left\{ \frac{\lambda^{b}(\delta^{b})}{\gamma}\left[1-e^{\gamma\left(s-\delta^{b}-\theta(t,s,q+1)+\theta(t,s,q)\right)}\right]\right\} $$ Not really sure why signs are different, but think I am missing something really trivial.
## Answer by sle (score 1)
https://quant.stackexchange.com/a/69323
derivatives had a wrong sign (had to be γ(-f) instead of γf for example)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.