Question

Trading two consecutive orders This exercise reconciles two points of view on consecutive trades: (a) One can treat the second order as the first order's continuation. (b) One can treat the second order as a separate trade from an external source. One shows that the two points of view lead to the same optimal trading strategy. Consider a generalized OW model with parameters β>0,λ=eγ satisfying the no price-manipulation condition 2β+γ′>0. For a given trading process Q, the price impact is the solution to the SDE dIt​=−βt​It​dt+λt​dQt​ Consider a deterministic alpha signal αt​. 1. In the continuation approach, Q0​=Q~​ and I0​=I~ are non-zero and capture the first order's effect. Derive the optimal impact state It0​. 2. In the separate approach, Q0​=0 and I0​=0. The external impact Iˉ captures the first order's effect. The first order's external impact satisfies ∀t>0;dIˉt​=−βt​Iˉt​dt Derive the optimal impact state It1​. 3. Reconcile the two points of view.

Answer 1

The student's question focuses on finding the optimal trading strategy for two consecutive orders within a financial mathematics model and then reconciling the two approaches taken to analyze the trades. New equilibrium states are compared with the original equilibrium to show that both strategies result in the same optimal outcome.

Step-by-step explanation:

The question revolves around the optimal trading strategy within a generalized Optimal Weight (OW) model in the context of financial mathematics, where trading consecutive orders are contemplated from two perspectives: (a) the continuation approach (treating the second order as the continuation of the first), and (b) the separate approach (treating the second order as a new trade from an external source). The optimal impact states It0 and It1 must be derived under the mathematical framework given by the parameters of the model, considering the dynamic nature of demand and supply in market equilibrium dynamics. Eventually, the two perspectives need to be reconciled, showing that both approaches yield the same optimal strategy.

Following the reconciliation, we compare the new equilibrium price and quantity to the original equilibrium price. The new equilibrium (E2) is concluded to occur at a lower quantity and a lower price than the original equilibrium (Eo). This insight aligns with the economic theory of supply and demand, where shifts in supply and demand curves affect the equilibrium price and quantity.