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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…

Sequential Monte Carlo (SMC) methods have successfully been used in many applications in engineering, statistics and physics. However, these are seldom used in financial option pricing literature and practice. This paper presents SMC method…

In this paper we consider the pricing of options on interest rates such as caplets and swaptions in the L\'evy Libor model developed by Eberlein and \"Ozkan (2005). This model is an extension to L\'evy driving processes of the classical…

A reinforcement algorithm introduced by H.A. Simon \cite{Simon} produces a sequence of uniform random variables with memory as follows. At each step, with a fixed probability $p\in(0,1)$, $\hat U_{n+1}$ is sampled uniformly from $\hat U_1,…

We propose a very efficient method for pricing various types of lookback options under Markov models. We utilize the model-free representations of lookback option prices as integrals of first passage probabilities. We combine efficient…

We present the method of moments approach to pricing barrier-type options when the underlying is modelled by a general class of jump diffusions. By general principles the option prices are linked to certain infinite dimensional linear…

This paper studies Vickrey first-price auctions under binary feedback. Leveraging the enhanced performance of machine learning algorithms, the new algorithm uses past information to improve the regret bounds of the BROAD-OMD algorithm.…

This paper explores a surprising equivalence between two seemingly-distinct convex optimization methods. We show that simulated annealing, a well-studied random walk algorithms, is directly equivalent, in a certain sense, to the central…

This paper examines a class of barrier options-multi-step barrier options, which can have any finite number of barriers of any level. We obtain a general, explicit expression of option prices of this type under the Black-Scholes model.…

We introduce a novel approach to perform first-order optimization with orthogonal and unitary constraints. This approach is based on a parametrization stemming from Lie group theory through the exponential map. The parametrization…

We calculate crossing probabilities and one-sided last exit time densities for a class of moving barriers on an interval $[0,T]$ via Schwartz distributions. We derive crossing probabilities and first hitting time densities for another class…

Random walks with a general, nonlinear barrier have found recent applications ranging from reionization topology to refinements in the excursion set theory of halos. Here, we derive the first-crossing distribution of random walks with a…

A class of algorithms in discrete space and continuous time for Brownian first passage time estimation is considered. A simple algorithm is derived that yields exact mean first passage times (MFPT) for linear potentials in one dimension,…

As is known, an option price is a solution to a certain partial differential equation (PDE) with terminal conditions (payoff functions). There is a close association between the solution of PDE and the solution of a backward stochastic…

For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For…

Many approaches for conducting Bayesian inference on discretely observed diffusions involve imputing diffusion bridges between observations. This can be computationally challenging in settings in which the temporal horizon between…

This paper is a supplement to our recent paper ``Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in L\'evy models". We introduce the class of regime-switching L\'evy models with memory,…

There is only one technique for prior-free optimal mechanism design that generalizes beyond the structurally benevolent setting of digital goods. This technique uses random sampling to estimate the distribution of agent values and then…

Recent work has shown that machine-learned predictions can provably improve the performance of classic algorithms. In this work, we propose the first minimum-cost network flow algorithm augmented with a dual prediction. Our method is based…

Options have provided a field of much study because of the complexity involved in pricing them. The Black-Scholes equations were developed to price options but they are only valid for European styled options. There is added complexity when…