Barrier options are a widely used class of path-dependent derivative securities. These options either cease to exist or come into existence when some pre-specified asset price barrier is hit during the option’s life. Merton (1973) has derived a down-and-out call price by solving the corresponding partial differential equation with some boundary conditions. Reiner & Rubinstein (1991) published c...

Exotic options are popular financial derivatives that play essential roles in financial markets. How to price them efficiently and accurately is very important both in theory and practice. The lattice model is usually used to price them. The prices computed by the lattice converge to the theoretical value under the continuous-time model. But the lattice model may produce quite slow convergence;...

We propose a new approach to the model calibration problem, which takes into account the multiplicity of solutions. Starting from a prior distribution on model parameters and a set of observed option prices, we propose a probabilistic construction which yields an arbitrage free pricing rule consistent with these observed option prices. Our approach yields a simple Monte Carlo algorithm for simu...

We address the problem of valuation and hedging of Asian basket and spread options derivatives common in energy markets. We extend the Generalized LogNormal approach, introduced in Borovkova et al. (2007), to Asian basket options and apply it to energy option markets. We provide closed form formulae for the option price and the greeks, which is extremely useful for option traders. Inverting the...

The QLBS model is a discrete-time option hedging and pricing model that is based on Dynamic Programming (DP) and Reinforcement Learning (RL). It combines the famous Q-Learning method for RL with the Black-Scholes (-Merton) model’s idea of reducing the problem of option pricing and hedging to the problem of optimal rebalancing of a dynamic replicating portfolio for the option, which is made of a...

We present an algorithm for American option pricing based on stochastic approximation techniques. Option pricing algorithms generally involve some sort of discretization, either on the state space or on the underlying functional space. Our work, which is an application of a more general perturbed gradient algorithm introduced recently by the authors, consists in approximating the value function...

There is the need for applying numerical methods to problems that cannot be solved analytically and as the spatial dimension of the problem is increased the need for computational recourses increase exponentially, a phenomenon known as the “curse of dimensionality”. In the BlackScholes-Merton framework the American option pricing problem has no closed form solution and a numerical procedure has...

Modern option pricing techniques are often considered among the most mathematically complex of all applied areas of financial engineering. In particular these techniques derive their impetus from four milestones of option pricing models: Bachelier model, Samuelson model, Black-Scholes-Merton model and Levy model. In this paper we evaluate all related option pricing models based on these milesto...

This paper proves the convergence of applying the radial basis functions as a global spatial approximation method for solving the option pricing models. The computational advantage of this method is illustrated by giving numerical examples on solving both the European and American options pricing models whereas the latter is a free boundary value problem.

The paper is focused on numerical approximation of early exercise boundary within American put option pricing problem. Assuming non-dividend paying, American put option leads to two disjunctive regions, a continuation one and a stopping one, which are separated by an early exercise boundary. We present variational formulation of American option problem with special attention to early exercise a...