No abstract

No abstract

This paper proposes a performance-driven gradient boosting model (pdGBM) which predicts shorthorizon price movements by combining nonlinear response functions of selected predictors. This model performs gradient descent in a constrained functional space by directly minimizing loss functions customized with different trading performance measurements. To demonstrate its practical applications, a simple trading system was designed with trading signals constructed from pdGBM predictions and fixed holding period in each trade. We tested this trading system on the highfrequency data of SPDR S&P 500 index ETF (SPY). In the out-of-sample period, it generated an average of 0.045% return per trade and an annualized Sharpe ratio close to 20 after transaction costs. Various empirical results also showed the model robustness to different parameters. These superior performances confirm the predictability of short-horizon price movements in the US equity market. We also compared the performance of this trading system with similar trading systems based on other predictive models like the gradient boosting model with L2 loss function and the penalized linear model. Results showed that pdGBM substantially outperformed all other models by higher returns in each month of the testing period. Additionally, pdGBM has many advantages including its capability of automatic predictor selection and nonlinear pattern recognition, as well as its simply structured and interpretable output function.

We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5,6,8] for short-time asymptotic expansions of heat kernels, and obtain a family of general closed-form approximate solutions for both the pricing kernel and derivative price. A bootstrap scheme allows us to extend our method to large time. We also perform analytic as well as a numerical error analysis, and compare our results to other known methods. 1 arXiv:0910.2309v2 [q-fin.PR] 21 Apr 2010We therefore allow for variable coefficients in both space and time. We assume throughout that a(x) > 0, for x > 0 and that the coefficients a, b, c are smooth functions. The perturbative method introduced in [8] for the study of parabolic equations in arbitrary dimensions was fully justified in the case when a, b, and c and all their derivatives are bounded, and are bounded away from zero: a(x) ≥ γ > 0. In this paper we complete the results of [8] with explicit formulas for the 1D case. Then we numerically test our formulas for the Black-Scholes-Merton and CEV models, obtaining an excellent agreement between our theoretical results and the numerical tests. Both the Black-Scholes-Merton model (1.2) and and the CEV model (1.3) are more general than the models considered in [8] in that their coefficients do not satisfy the assumptions of the paper, yet the numerical tests indicates that the results of that paper are still valid for the more general models considered here. This observation suggests that the theoretical framework of [8] is applicable in greater generality. We plan to study this point in a forthcoming paper.To explain our method, let us recall that, under certain conditions on the operator L and initial value h, described in details in the next section, there exists a smooth function G t (x, y) such that the solution to (1.1) has the representationThe kernel function G t (x, y) in (1.5) is the fundamental solution or the socalled Green function for the problem (1.1).Remark 1.1. Given that G t (x, y) arises in several different contexts, we will call the function G t (x, y) the transition density kernel, pricing kernel, heat kernel, or Green function interchangeably, depending on the context in which the object arises.As mentioned above, except for some very special cases no explicit formulas for G t (x, y) or U (t, x) are available. For the Black-Scholes-Merton model, a change of variables reduces the PDE to a heat equation that can then be solved explicitly. Therefore, exact formulas for the kernel G BSM and the

No abstract