A finite dimensional approximation for pricing moving average options

We propose a method for pricing American options whose pay-off depends on the moving average of the underlying asset price. The method uses a finite dimensional approximation of the infinite-dimensional dynamics of the moving average process based on a truncated Laguerre series expansion. The resulting problem is a finite-dimensional optimal stopping problem, which we propose to solve with a least squares Monte Carlo approach. We analyze the theoretical convergence rate of our method and present numerical results in the Black-Scholes framework

Pricing path-dependent Bermudan options using Wiener chaos expansion: an embarrassingly parallel approach

In this work, we propose a new policy iteration algorithm for pricing Bermudan options when the payoff process cannot be written as a function of a lifted Markov process. Our approach is based on a modification of the well-known Longstaff Schwartz algorithm, in which we basically replace the standard least square regression by a Wiener chaos expansion. Not only does it allow us to deal with a non Markovian setting, but it also breaks the bottleneck induced by the least square regression as the coefficients of the chaos expansion are given by scalar products on the L^2 space and can therefore be approximated by independent Monte Carlo computations. This key feature enables us to provide an embarrassingly parallel algorithm.Comment: The Journal of Computational Finance, Incisive Media, In pres

Gradient-Bounded Dynamic Programming with Submodular and Concave Extensible Value Functions

We consider dynamic programming problems with finite, discrete-time horizons and prohibitively high-dimensional, discrete state-spaces for direct computation of the value function from the Bellman equation. For the case that the value function of the dynamic program is concave extensible and submodular in its state-space, we present a new algorithm that computes deterministic upper and stochastic lower bounds of the value function similar to dual dynamic programming. We then show that the proposed algorithm terminates after a finite number of iterations. Finally, we demonstrate the efficacy of our approach on a high-dimensional numerical example from delivery slot pricing in attended home delivery.Comment: 6 pages, 2 figures, accepted for IFAC World Congress 202

Some numerical methods for solving stochastic impulse control in natural gas storage facilities

The valuation of gas storage facilities is characterized as a stochastic impulse control problem with finite horizon resulting in Hamilton-Jacobi-Bellman (HJB) equations for the value function. In this context the two catagories of solving schemes for optimal switching are discussed in a stochastic control framework. We reviewed some numerical methods which include approaches related to partial differential equations (PDEs), Markov chain approximation, nonparametric regression, quantization method and some practitioners’ methods. This paper considers optimal switching problem arising in valuation of gas storage contracts for leasing the storage facilities, and investigates the recent developments as well as their advantages and disadvantages of each scheme based on dynamic programming principle (DPP

A Closed-Form Approximation of Likelihood Functions for Discretely Sampled Diffusions: the Exponent Expansion

In this paper we discuss a closed-form approximation of the likelihood functions of an arbitrary diffusion process. The approximation is based on an exponential ansatz of the transition probability for a finite time step $\Delta t$, and a series expansion of the deviation of its logarithm from that of a Gaussian distribution. Through this procedure, dubbed {\em exponent expansion}, the transition probability is obtained as a power series in $\Delta t$. This becomes asymptotically exact if an increasing number of terms is included, and provides remarkably accurate results even when truncated to the first few (say 3) terms. The coefficients of such expansion can be determined straightforwardly through a recursion, and involve simple one-dimensional integrals. We present several examples of financial interest, and we compare our results with the state-of-the-art approximation of discretely sampled diffusions [A\"it-Sahalia, {\it Journal of Finance} {\bf 54}, 1361 (1999)]. We find that the exponent expansion provides a similar accuracy in most of the cases, but a better behavior in the low-volatility regime. Furthermore the implementation of the present approach turns out to be simpler. Within the functional integration framework the exponent expansion allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. This is illustrated with the application to simple path-dependent interest rate derivatives. Finally we discuss how these results can also be used to increase the efficiency of numerical (both deterministic and stochastic) approaches to derivative pricing.Comment: 28 pages, 7 figure

A Forward Equation for Barrier Options under the Brunick&Shreve Markovian Projection

We derive a forward equation for arbitrage-free barrier option prices, in terms of Markovian projections of the stochastic volatility process, in continuous semi-martingale models. This provides a Dupire-type formula for the coefficient derived by Brunick and Shreve for their mimicking diffusion and can be interpreted as the canonical extension of local volatility for barrier options. Alternatively, a forward partial-integro differential equation (PIDE) is introduced which provides up-and-out call prices, under a Brunick-Shreve model, for the complete set of strikes, barriers and maturities in one solution step. Similar to the vanilla forward PDE, the above-named forward PIDE can serve as a building block for an efficient calibration routine including barrier option quotes. We provide a discretisation scheme for the PIDE as well as a numerical validation.Comment: 20 pages, Quantitative Finance Volume 16, 2016 - Issue