We develop a new algorithm that allows us to compute pairwise-correlation sensitivities in a Monte Carlo framework by modifying only one trajectory at a time, resulting in a significant decrease in Brownian noise, computing time and memory requirements. We apply this algorithm to the case of the risk management of a large portfolio of options on baskets of equities, but the same algorithm can be used for computing correlation sensitivities in any Monte Carlo framework. We show how this idea can also simplify the computation of correlation Greeks in the framework of adjoint algorithmic differentiation.
The Cox-Ingersoll-Ross model is widely used in financial engineering for the pricing of interest rate derivatives. Various Euler-Maruyama-and Milstein-type discretization schemes have been developed to approximate its solution, among which the full truncation Euler-Maruyama method is popular in practice and is known to be p-strongly pound convergent with order 1/2 under certain parameter conditions. In this paper, we investigate the projected Euler-Maruyama method for solving the Cox-Ingersoll-Ross model. By combining the projection technique with the normalized error analysis introduced by Cozma and Reisinger, we establish the p-strong pound convergence of the projected Euler-Maruyama method with order 1/2 over a significantly wide range of parameter settings.
Trades subject to a collateral agreement that allows cash in multiple currencies need to be discounted with the so-called cheapest-to-deliver curve, which embeds the optionality inherent in the choice of collateral currencies. These curves can be computed essentially exactly via a Monte Carlo simulation, but analytical approximations allow for faster calculation. We revisit this long-standing problem and propose an approximation that improves on the existing methods in terms of speed, accuracy and extendibility to any number of currencies.
We discuss applications of policy gradient methods for the optimal execution of an asset position via limit orders. We study two examples in-depth: a parametric limit order book (LOB) model and a realistic generative adversarial neural network (GAN) LOB model. In the first case, we apply a zeroth-order gradient estimator to a suitable parameterization of candidate policies and propose modifications to lower the variance in the estimate, including conditional sampling and a backward-in-time recursion. In the second case, we adapt a recently published LOB-GAN model to obtain a differentiable map from the parameters to the objective. We then alter a standard policy gradient method with a pathwise gradient estimator to overcome issues with the nonconvexity and roughness of the loss landscape, studying different initializations using inexact dynamic programming and second-order optimization steps, as well as regularization of the learnt policies. In both cases, we are able to learn effective trading strategies.
The paper presents a Bayesian framework for the calibration of financial models using neural stochastic differential equations (neural SDEs), for which we also formulate a global universal approximation theorem based on Barron-type estimates. The method is based on the specification of a prior distribution on the neural network weights and a well-chosen likelihood function. The resulting posterior distribution can be seen as a mixture of different classical neural SDE models yielding robust bounds on the implied volatility surface. A methodology for learning the change of measure between the risk-neutral and the historical measure is necessary to take into consideration both historical financial time series data and option price data. The key ingredient for the robust numerical optimization of our neural networks is a Langevin-type algorithm, commonly used in the Bayesian approaches to draw posterior samples.
In this paper we propose a machine learning algorithm for time-inconsistent portfolio optimization. The proposed algorithm builds on neural-network-based trading schemes, in which the asset allocation at each time point is determined by a neural network. The loss function is given by an empirical version of the objective function of the portfolio optimization problem. Moreover, various trading constraints are naturally fulfilled by choosing appropriate activation functions in the output layers of the neural networks. Besides this, our main contribution is to add options to a portfolio of risky assets and a risk-free bond and to use additional neural networks to determine the amount allocated to the options as well as their strike prices. We consider objective functions that are more in line with the rational preferences of an investor than the classical mean-variance, apply realistic trading constraints and model the assets with a correlated jump-diffusion stochastic differential equation. With an incomplete market and a more involved objective function, we show that it is beneficial to add options to the portfolio. Moreover, we show that adding options leads to a more consistent stock allocation with less demand for drastic reallocations.
Motivated by the equations of cross valuation adjustments (XVAs) accounting for the fungibility of capital at risk with variation margin, we introduce a simulation/ regression scheme for a class of anticipated backward stochastic differential equations, where the coefficient entails a conditional expected shortfall of the martingale part of the solution. The scheme is explicit in time and uses neural network least-squares and quantile regressions for the embedded conditional expectations and expected shortfall computations. An a posteriori Monte Carlo validation procedure allows assessment of the scheme's regression error at each time step. The superiority of this scheme with respect to Picard iterations is illustrated in the context of a high-dimensional market and a default risk XVA use case.
In this paper we consider the numerical solution of the two-dimensional time-dependent partial integro-differential complementarity problem (PIDCP) that holds for the value of American-style options under the two-asset Kou jump-diffusion model. Following the method of lines (MOL), we derive an efficient numerical method for the pertinent PIDCP. Here, for the discretization of the nonlocal double integral term, an extension is employed of the fast algorithm by Toivanen (2008) in the case of the one-asset Kou jump-diffusion model. For the temporal discretization, we study a useful family of second-order diagonally implicit Runge-Kutta (DIRK) methods. Their adaptation to the semidiscrete two-dimensional Kou PIDCP is obtained by means of an effective iteration introduced by d'Halluin, Forsyth Labahn (2004) and d'Halluin, Forsyth Vetzal (2005). Ample numerical experiments are presented showing that the proposed numerical method achieves a favourable, second-order convergence behaviour to the American two-asset option value as well as to its Greeks Delta and Gamma.
This paper is devoted to the price-storage dynamics in natural gas markets. A novel stochastic path-dependent volatility model is introduced with path-dependence in both price volatility and storage increments. Model calibrations are conducted for both the price and storage dynamics. Further, we discuss the pricing problem of discrete-time swing options using the dynamic programming principle, and a deep learning-based method is proposed for numerical approximations. A numerical algorithm is provided, followed by a convergence analysis result for the deep-learning approach.
We propose a novel data-driven neural network (NN) optimization framework for solving an optimal stochastic control problem under stochastic constraints. Customized activation functions for the output layers of the NN are applied, which permits training via standard unconstrained optimization. The optimal solution yields a multi-period asset allocation and decumulation strategy for a holder of a defined contribution (DC) pension plan. The objective function of the optimal control problem is based on expected wealth withdrawn (EW) and expected shortfall (ES) that directly targets left-tail risk. The stochastic bound constraints enforce a guaranteed minimum withdrawal each year. We demonstrate that the data-driven approach is capable of learning a near-optimal solution by benchmarking it against the numerical results from a Hamilton-Jacobi-Bellman (HJB) Partial Differential Equation (PDE) computational framework.
This article presents a simple but effective and efficient approach to improve the accuracy and stability of Least-Squares Monte Carlo for American-style option pricing as well as expected exposure calculation in valuation adjustments. The key idea is to construct the ansatz of conditional expected continuation payoff using the finite difference solution from one dimension, to be used in linear regression. This approach bridges between solving backward partial differential equations and Monte Carlo simulation, aiming at achieving the best of both worlds. Independent of model settings, the ansatz is proved to serve as a control variate to reduce the least-squares errors. We illustrate the technique with realistic examples including Bermudan options, worst of issuer callable notes and expected positive exposure on European options. The method can be considered as a generic numerical scheme across various asset classes, in particular, as an accurate method for pricing and risk-managing American-style derivatives under arbitrary dimensions.
In this paper, we develop a novel method based on Malliavin calculus to find an approximation for the convexity adjustment for various classical interest rate products. Malliavin calculus provides a simple way to get a template for the convexity adjustment. We find the approximation for Futures, OIS Futures, FRAs, and CMSs under a general family of the one-factor Cheyette model. We have also seen the excellent quality of the numerical accuracy of the formulas obtained.
We propose a non-parametric extension with leverage functions to the Andersen commodity curve model. We calibrate this model to market data for WTI and NG including option skew at the standard maturities. While the model can be calibrated by an analytical formula for the deterministic rate case, the stochastic rate case demands estimation of an expectation for which we employ Monte Carlo simulation. We find that the market smile is captured for the deterministic rate case; and with relatively low number of paths, for the stochastic rate case. Since there is typically at most one standard maturity with liquid volatility data for each futures contract, there is flexibility on the shape of nonstandard maturity implied volatility and how the total implied variance accumulates. We equip the model with different total implied variance accumulators to demonstrate that flexibility.
We present an algorithm for the calibration of local volatility from market option prices through deep self-consistent learning, by approximating both market option prices and local volatility using deep neural networks. Our method uses the initial-boundary value problem of the underlying Dupire's partial differential equation solved by the parameterized option prices to bring corrections to the parameterization in a self-consistent way. By exploiting the differentiability of neural networks, we can evaluate Dupire's equation locally at each strike-maturity pair; while by exploiting their continuity, we sample strike-maturity pairs uniformly from a given domain, going beyond the discrete points where the options are quoted. Moreover, the absence of arbitrage opportunities are imposed by penalizing an associated loss function as a soft constraint. For comparison with existing approaches, the proposed method is tested on both synthetic and market option prices, which shows an improved performance in terms of reduced interpolation and reprice errors, as well as the smoothness of the calibrated local volatility. An ablation study has been performed, asserting the robustness and significance of the proposed method.
The stochastic alpha-beta-rho (SABR) model has been widely adopted in options trading. In particular, the normal (beta = 0) SABR model is a popular model choice for interest rates because it allows negative asset values. The option price and delta under the SABR model are typically obtained via asymptotic implied volatility approximation, but the results are often inaccurate and arbitrageable. Using a recently discovered price transition law, we propose a Gaussian quadrature integration scheme to price options under the normal SABR model. The compound Gaussian quadrature sum over only 49 points can calculate a very accurate price and delta that are arbitrage-free.
We adopt the least squares Monte Carlo (LSMC) method to price time-capped American options. The cap can be an independent random variable or dependent on the asset price at a random time. We investigate various time caps. In particular, we give an algorithm for pricing the American options capped by the first drawdown epoch, focusing on the geometric Levy market. We prove that our estimator converges to the true price as the discretization step tends to zero and the number of trajectories tends to infinity.
This paper leverages the equal risk pricing (ERP) framework for the valuation of illiquid financial derivatives. Such a method sets the derivative price as the premium, which leads to equal residual optimal hedging risk for agents hedging the long and short positions on the derivative. The inclusion of multiple hedging instruments such as liquid vanilla options in the hedging procedure ensures that information contained in the price of these liquid assets is transferred to the price of the illiquid asset. This property leads to genuine market-consistent pricing, which allows the use of observables (liquid instrument prices) to determine nonobservable prices (ie, that of illiquid derivatives) to be maximized. The ERP problem is solved numerically through deep reinforcement learning. Several numerical experiments are provided to study the properties of equal risk prices of derivatives when multiple options are used as hedging instruments. Notably, the equal risk prices produced with option hedges are typically lower than those obtained through hedges relying exclusively on the underlying asset, which is caused by a reduction in the level of market incompleteness when options become available for trading.
