We develop a conditional sampling scheme for pricing knock-out barrier options under the Linear Transformations (LT) algorithm from Imai and Tan (2006), ref. [14]. We compare our new method to an existing conditional Monte Carlo scheme from Glasserman and Staum (2001), ref. [11], and show that a substantial variance reduction is achieved. We extend the method to allow pricing knock-in barrier options and introduce a root-finding method to obtain a further variance reduction. The effectiveness of the new method is supported by numerical results.

We propose a quasi-Monte Carlo algorithm for pricing knock-out and knock-in barrier options under the Heston (1993) stochastic volatility model. This is done by modifying the LT method from Imai and Tan (2006) for the Heston model such that the first uniform variable does not influence the stochastic volatility path and then conditionally modifying its marginals to fulfill the barrier condition(s). We show that this method is unbiased and never does worse than the unconditional algorithm. In addition, the conditioning is combined with a root finding method to also force positive payouts. The effectiveness of this method is shown by extensive numerical results.

We propose an alternative to the algorithm from Cools, Kuo, Nuyens (Computing, 2010) [5], for constructing lattice rules with good trigonometric degree. The original algorithm has construction cost O(|A d (m)| + dN log N) for an N-point lattice rule in d dimensions having trigonometric degree m, where the set A d (m) has exponential size in both d and m (in the "unweighted degree" case, which is what we consider here). We reduce the cost to O(dN(log N) 2 ) with an implicit constant governing the needed precision (which is dependent on N and d).

We treat the problem of mean-variance optimal execution in markets with limited liquidity and varying volatility. When the market parameters are assumed constant, an analytical solution exists for the optimal trading rate. In general however, this problem leads to a non-linear Hamilton-Jacobi-Bellman PDE, which has to be solved numerically. Since solving such a PDE is a complex procedure, Almgren [2012] mentions a sub-optimal control that can be used as an approximation. This strategy assumes the market parameters are constant, and hence takes the analytical solution from the stationary problem, but updates the strategy each time the market parameters change. It is called the rolling horizon strategy (RHS), because it is essentially a continuously updated static control with contracting horizon. It is easy to extend to the multi-asset case as we will show. In this paper, we propose a rolling horizon Monte Carlo algorithm (RHMC). Our method chooses a trading rate based on simulations using a sub-optimal control. The potential upside of this method is that our proposed RHMC method not only uses current market information, such as the RHS, but also uses simulations to infer future market behaviour as well. Our new method is naturally formulated for the multi-asset case and allows the freedom to choose the structure of the stochastic driver processes. The results indicate that our method can significantly outperform the RHS. We also provide some insights into the RHS, showing that it converges to the optimal solution for strong risk-averse traders, at least in the setting of Almgren [2012].

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