An application to credit risk of a hybrid Monte Carlo–optimal quantization method

Abstract

In this paper we use a hybrid Monte Carlo-Optimal quantization method to approximate the conditional survival probabilities of a firm, given a structural model for its credit default, under partial information.We consider the case when the firm's value is a non-observable stochastic process (V t ) t≥0 and investors in the market have access to a process (S t ) t≥0 , whose value at each time t is related to (V s , s ≤ t). We are interested in the computation of the conditional survival probabilities of the firm given the "investor information".As an application, we analyze the shape of the credit spread curve for zero coupon bonds in two examples.

Pricing of barrier options by marginal functional quantization

“…Very recently, optimal vector quantization has become a promising tool in Numerical Probability owing to its ability to approximate either expectations or more significantly conditional expectations from some cubature formulas. This faculty to approximate conditional expectations is the crucial property used to solve some problems emerging in finance as optimal stopping problems (pricing and hedging American style options, see [2,20], stochastic control problems (see [7,19]) for portfolio management, nonlinear filtering problems (see [18,21] and [5] for an application to credit risk).…”

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

“…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.…”

Conditional survival probabilities under partial information: a recursive quantization approach with applications

“…In this paper, we both generalize and improve results derived in earlier studies. The models presented in [5] and [20] can deal with arbitrary firm-value diffusions, but are heavy. Moreover, the considered information flow is only made of a noisy version of the firm-value.…”