Pricing of barrier options by marginal functional quantization

Abstract

This paper is devoted to the pricing of Barrier options by optimal quadratic quantization method. From a known useful representation of the premium of barrier options one deduces an algorithm similar to one used to estimate nonlinear filter using quadratic optimal functional quantization. Some numerical tests are fulfilled in the Black-Scholes model and in a local volatility model and a comparison to the so called Brownian Bridge method is also done.

Monte Carlo Variance Reduction by Conditioning for Pricing with Underlying a Continuous-Time Finite State Markov Process

“…. , N, according to (13) and (14), from the definition ofṼ (15), and from the equalities (21), and by recalling that X T = X ν T in (11) and X T = X ν T +1 in (12), one obtains immediately the following corollary, where, in line with the other analogous vectors defined previously,Ṽ…”

“…We may thus be confident that also in other situations, where a comparison with a benchmark is no longer possible, our approach performs well. In subsection 6.1 we use a "Kushner-type" approximation according to [10]; other spatial discretization/quantization methods may also be used, in particular optimal quantization methods according to [2] (for specific financial application of optimal quantization, see also [4], [15]). Referring to [13], we report prices of zero-coupon bonds that are computed according to our MC with conditioning, and we compare them with the exact values of the continuous-time counterpart, with those obtained from the analytical formula (6) (to allow for this comparison we consider a time homogeneous case), and also with the values obtained from other computational methods, namely plain MC, a recombining binomial tree model applied to the continuous-time counterpart, and the algorithm described in [8] with the discrete-time Markov chain obtained via a deterministic time discretization.…”

“…While in the present paper we consider the midpoint between the upper and lower bounds as the value to be computed analytically in our (hybrid) MC method with conditioning (see (15) and Corollary 4.2), in [13] the authors consider just the upper bound that in general does not differ much from the midpoint. In the tables below, prices for various maturities T and different initial values for the short rater 0 = r 0 =r, as well as for different values of the parameters in (45), are reported from [13].…”

Recursive marginal quantization of higher-order schemes

“…We refer to Glasserman [2003] and Sagna [2011] for the continuous monitoring case using the Rayleigh distribution.…”

Conditional hitting time estimation in a nonlinear filtering model by the Brownian bridge method

“…Remark that the use of the marginal functional quantization method can not be justified from the theoretical point of view since we do not know yet the rate of convergence of the marginals of the quantized process to the marginals of the initial process. However, this method has proved its efficiency from the numerical viewpoint when used to estimate barrier option by optimal quantization, see [14] (not that the considered marginal quantization in [14] is a little bit different from the one considered in this paper, nevertheless the numerical results are the same, up to at least a 10 −3 absolute error order). Let us come back to the problem of interest and let the functionals π y,m and ̟ y,m be defined for every bounded measurable function f by π y,m f = E f (X tm )L m y K m and ̟ y,m f = E f (X tm )L m y .…”