This project presents an optimal high-frequency market making strategy using the Hamilton-Jacobi-Bellman (HJB) equation.
The project includes implementations for the following three scenarios:
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Constant market parameter mid-price process. One is a geometric Brownian motion the other Merton's jump diffusion process
$\frac{dS_t}{S_t} = \mu dt + \sigma dW_t$ $\frac{dS_t}{S_t} = \mu dt + \sigma dW_t + \xi dN_{t}^a - \xi dN_{t}^b$
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Stochastic volatility model(Heston model) and with jump diffusion process
- $\begin{aligned} d S_t & =\mu d t+\sqrt{v_t} d B_t^1+\varepsilon^{+} d M_t^{+}-\varepsilon^{-} d M_t^{-}, \ d v_t & =k\left(\theta-v_t\right) d t+\sigma \sqrt{v_t} d B_t^2, \ d B_t^1 d B_t^2 & =\rho d t \end{aligned}$
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Implied alpha and stochastic limit order book (LOB) model
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$d S_t=\left(v+\alpha_t\right) d t+\sigma d W_t$ -
It is a measure of the adverse selection of a the market maker. The implied alpha follows the process:
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$d \alpha_t=-\zeta \alpha_t d t+\sigma_\alpha d B_t+\epsilon^{+} d \bar{M}_t^{+}-\epsilon^{-} d \bar{M}_t^{-}+\tilde{\epsilon}^{+} d Z_t^{+}-\tilde{\epsilon}^{-} d Z_t^{-}$ -
The stochastic limit order book model is a model that is used to model the dynamics of the limit order book.
The model is based on the following assumptions: -
$\left{\begin{array}{l} d \kappa_t^{-}=\beta_\kappa\left(\theta_\kappa-\kappa_t^{-}\right) d t+\eta_\kappa d \bar{M}t^{-}+\nu\kappa d \bar{M}t^{+}+\tilde{\eta}\kappa d Z_t^{-}+\tilde{\nu}\kappa d Z_t^{+}, \ d \kappa_t^{+}=\beta\kappa\left(\theta_\kappa-\kappa_t^{+}\right) d t+\nu_\kappa d \bar{M}t^{-}+\eta\kappa d \bar{M}t^{+}+\tilde{\nu}\kappa d Z_t^{-}+\tilde{\eta}_\kappa d Z_t^{+}, \end{array}\right.$
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The full source code is written in C++ and is not open source due to its application for real market making purposes.