# Non-parametric liquidity-adjusted VaR model: a stochastic programming approach

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## Abstract

This paper proposes a Stochastic Programming (SP) approach for the calculation of the liquidity-adjusted Value-at-Risk (LVaR). The model presented in this paper offers an alternative to Almgren and Chriss’s mean-variance approach (1999 and 2000). In this research, a two-stage stochastic programming model is developed with the intention of deriving the optimal trading strategies that respond dynamically to a given market situation. The sample paths approach is adopted for scenario generation. The scenarios are thus represented by a collection of simulated sample paths rather than the ‘tree structure’ usually employed in stochastic programming. Consequently, the SP LVaR presented in this paper can be considered as a non-parametric approach, which is in contrast to Almgren and Chriss’s parametric solution. Initially, a set of numerical experiments indicates that the LVaR figures are quite similar for both approaches when all the underlying financial assumptions are identical. Following this sanity check, a second set of numerical experiments shows how the randomness of the different types (i.e., Bid and Ask spread) can be easily incorporated into the problem due to the stochastic programming formulation and how optimal and adaptive trading strategies can be derived through a two-stage structure (i.e., a ‘recourse’ problem). Hence, the results presented in this paper allow the introduction of new dimensionalities into the computation of LVaR by incorporating different market conditions.

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