Related papers: Explicit Heston Solutions and Stochastic Approxima…
In the first part of this thesis, we focus on American options in the Heston model. We first give an analytical characterization of the value function of an American option as the unique solution of the associated (degenerate) parabolic…
A number of optimal decision problems with uncertainty can be formulated into a stochastic optimal control framework. The Least-Squares Monte Carlo (LSMC) algorithm is a popular numerical method to approach solutions of such stochastic…
The multi-path Traveling Salesman Problem with stochastic travel costs arises in hybrid vehicle routing applications designed for Smart City and City Logistics, where multiple paths exist between each pair of locations. Travel times along…
Stochastic volatility models have existed in Option pricing theory ever since the crash of 1987 which violated the Black-Scholes model assumption of constant volatility. Heston model is one such stochastic volatility model that is widely…
It is shown how to obtain accurate values for American options using Monte Carlo simulation. The main feature of the novel algorithm consists of tracking the boundary between exercise and hold regions via optimization of a certain payoff…
We present an iterative sampling method which delivers upper and lower bounding processes for the Brownian path. We develop such processes with particular emphasis on being able to unbiasedly simulate them on a personal computer. The…
This report investigates the computation of option Greeks for European and Asian options under the Heston stochastic volatility model on GPU. We first implemented the exact simulation method proposed by Broadie and Kaya and used it as a…
Parametric estimation of stochastic differential equations (SDEs) has been a subject of intense studies already for several decades. The Heston model for instance is driven by two coupled SDEs and is often used in financial mathematics for…
We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on \Ito stochastic differential equations in infinite dimensional spaces, and prove strong and weak…
Under the assumption of no-arbitrage, the pricing of American and Bermudan options can be casted into optimal stopping problems. We propose a new adaptive simulation based algorithm for the numerical solution of optimal stopping problems in…
In many contemporary optimization problems such as those arising in machine learning, it can be computationally challenging or even infeasible to evaluate an entire function or its derivatives. This motivates the use of stochastic…
Experiments in predator-prey systems show the emergence of long-term cycles. Deterministic model typically fails in capturing these behaviors, which emerge from the microscopic interplay of individual based dynamics and stochastic effects.…
Optimization of expensive computer models with the help of Gaussian process emulators in now commonplace. However, when several (competing) objectives are considered, choosing an appropriate sampling strategy remains an open question. We…
In this paper, we present a reduced basis method for pricing European and American options based on the Black-Scholes and Heston model. To tackle each model numerically, we formulate the problem in terms of a time dependent variational…
This contribution examines optimization problems that involve stochastic dominance constraints. These problems have uncountably many constraints. We develop methods to solve the optimization problem by reducing the constraints to a finite…
We consider assets for which price $X_t$ and squared volatility $Y_t$ are jointly driven by Heston joint stochastic differential equations (SDEs). When the parameters of these SDEs are estimated from $N$ sub-sampled data $(X_{nT}, Y_{nT})$,…
The logistic specification has been used extensively in non-Bayesian statistics to model the dependence of discrete outcomes on the values of specified covariates. Because the likelihood function is globally weakly concave estimation by…
When the nonconvex problem is complicated by stochasticity, the sample complexity of stochastic first-order methods may depend linearly on the problem dimension, which is undesirable for large-scale problems. To alleviate this linear…
A number of optimization approaches have been proposed for optimizing nonconvex objectives (e.g. deep learning models), such as batch gradient descent, stochastic gradient descent and stochastic variance reduced gradient descent. Theory…
We consider minimization of stochastic functionals that are compositions of a (potentially) non-smooth convex function $h$ and smooth function $c$ and, more generally, stochastic weakly-convex functionals. We develop a family of stochastic…