﻿LEAVING aside the courageous efforts of people around Monthly Review and Science and Society there has, by and large, been hardly any Marxist tradition among American intellectuals. However, during the last few years the situation has considerably changed and there is now a growing interest in Marxism, particularly among the younger social scientists of America. The book under review falls within this trend.

Value and Capital

In this paper, we present a new strategy for pricing average value options, i.e. options whose payoff depends on the average price of the underlying asset over a fixed period leading up to the maturity date. Such options are of particular interest and importance for thinly-traded assets (e.g. crude oil), since price manipulation is inhibited, and both the investor and issuer enjoy a welcome degree of protection from the vagaries of the market. These options are often implicit in a bond contract, although they also appear in a straightforward form. Our results suggest that the price of an average-value option will always be lower than that of a standard European option. Our pricing strategy involves Monte Carlo simulation with variance reduction elements and offers an enhanced pricing method to both arbitragers and hedgers, as well as to the issuers of such bonds.

A Pricing Method for Options Based on Average Asset Values

Introduction To Stochastic Calculus With Applications

In relation with Monte Carlo methods to solve some integro-differential equations, we study the approximation problem of $\mathbb{E}g(X_T)$ by $\mathbb{E}g(\overline{X}_T^n)$, where $(X_t, 0 \leq t \leq T)$ is the solution of a stochastic differential equation governed by a Levy process $(Z_t), (\overline{X}_t^n)$ is defined by the Euler discretization scheme with step $T/n$. With appropriate assumptions on $g(\cdot)$, we show that the error of $\mathbb{E}g(X_T) - \mathbb{E}g(\overline{X}_T^n)$ can be expanded in powers of $1/n$ if the Levy measure of $Z$ has finite moments of order high enough. Otherwise the rate of convergence is slower and its speed depends on the behavior of the tails of the Levy measure.

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The Euler scheme for Lévy driven stochastic differential equations

In a recent paper, Longstaff and Schwartz (2001) suggest a method to American option valuation based on simulation. The method is termed the Least Squares Monte Carlo (LSM) method, and although it has become widely used, not much is known about the properties of the estimator. This paper corrects this shortcoming using theory from the literature on seminonparametric series estimators. A central part of the LSM method is the approximation of a set of conditional expectation functions. We show that the approximations converge to the true expectation functions under general assumptions in a multiperiod, multidimensional setting. We obtain convergence rates in the two-period, multidimensional case, and we discuss the relation between the optimal rate of convergence and the properties of the conditional expectation. Furthermore, we show that the actual price estimates converge to the true price. This provides the mathematical foundation for the use of the LSM method in derivatives research.

Convergence of the Least Squares Monte Carlo Approach to American Option Valuation

This paper analyses the robustness of Least-Squares Monte Carlo, a technique proposed by Longstaff and Schwartz (2001) for pricing American options. This method is based on least-squares regressions in which the explanatory variables are certain polynomial functions. We analyze the impact of different basis functions on option prices. Numerical results for American put options show that this approach is quite robust to the choice of basis functions. For more complex derivatives, this choice can slightly affect option prices.

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On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives

This paper reformulates the stochastic string model of Santa-Clara and Sornette using stochastic calculus with continuous semimartingales. We present some new results, such as: (a) the dynamics of the short-term interest rate, (b) the PDE that must be satisfied by the bond price, and (c) an analytic expression for the price of a European bond call option. Additionally, we clarify some important features of the stochastic string model and show its relevance to price derivatives and the equivalence with an infinite dimensional HJM model to price European options.

Stochastic string models with continuous semimartingales

This article prices cap and swaptions in the Spanish market using the Vasicek, Cox, Ingersoll, and Ros and Hull and White (HW) models. Derivatives prices obtained with the Vasicek and CIR models estimated from time series data are very similar, but they differ substantially from the values given by the HW model fited to the term structureof interest rate swap yields (especially for at-the-money and out-of-the-money options). When the former models are estimated cross-sectionally, they produce option proces similar to those of the HW model. In samples of caps and swaptions, the Vasicek model estimated cross-sectionally, they produce option prices similar to those of the HW model. In samples of caps and swaptions, the Vasicek model estimated cross-sectionally outperforms the HW model. The Vasicek and CIR models estimated from time series produce very large pricing errors.

Consistent versus Non-Consistent Term Structure Models: Some Evidence from the Spanish Market

This paper analyses the robustness of Least - Squares Monte Carlo, a technique proposed by Longstaff and Schwartz (2001) for pricing American options. This method is based on least - squares regressions in which the explanatory variables are certain polynomial functions. We analyze the impact of different basis functions on option prices. Numerical results for American put options show that this approach is quite robust to the choice of basis functions. For more complex derivatives, this choice can slightly affect option prices.

/pdf/on-the-robustness-of-least-squares-monte-carlo-lsm-for-31vaz3tiqj.pdf

On the Robustness of Least - Squares Monte Carlo (LSM) for Pricing American Derivatives

This article prices caps and swaptions in the Spanish market using the Vasicek, Cox, Ingersoll, and Ross and Hull and White (HW) models. Derivative prices obtained with the Vasicek and CIR models estimated from time series data are very similar, but they differ substantially from the values given by the HW model fitted to the term structure of interest rate swap yields (especially for at-the-money and out-of-the-money options). When the former models are estimated cross-sectionally, they produce option prices similar to those of the HW model. In samples of caps and swaptions, the Vasicek model estimated cross-sectionally outperforms the HW model. The Vasicek and CIR models estimated from time series produce very large pricing errors.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1031350_code240848.pdf?abstractid=242010&mirid=2

Javier F. Navas

Papers

On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives

Stochastic string models with continuous semimartingales

Consistent versus Non-Consistent Term Structure Models: Some Evidence from the Spanish Market

On the Robustness of Least - Squares Monte Carlo (LSM) for Pricing American Derivatives

Consistent Versus Non-Consistent Term Structure Models: Some Evidence from the Spanish Market