31,278 research outputs found
Sequential Monte Carlo pricing of American-style options under stochastic volatility models
We introduce a new method to price American-style options on underlying
investments governed by stochastic volatility (SV) models. The method does not
require the volatility process to be observed. Instead, it exploits the fact
that the optimal decision functions in the corresponding dynamic programming
problem can be expressed as functions of conditional distributions of
volatility, given observed data. By constructing statistics summarizing
information about these conditional distributions, one can obtain high quality
approximate solutions. Although the required conditional distributions are in
general intractable, they can be arbitrarily precisely approximated using
sequential Monte Carlo schemes. The drawback, as with many Monte Carlo schemes,
is potentially heavy computational demand. We present two variants of the
algorithm, one closely related to the well-known least-squares Monte Carlo
algorithm of Longstaff and Schwartz [The Review of Financial Studies 14 (2001)
113-147], and the other solving the same problem using a "brute force" gridding
approach. We estimate an illustrative SV model using Markov chain Monte Carlo
(MCMC) methods for three equities. We also demonstrate the use of our algorithm
by estimating the posterior distribution of the market price of volatility risk
for each of the three equities.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS286 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Estimation of Risk-Neutral Density Surfaces
Option price data is often used to infer risk-neutral densities for future prices of an underlying asset. Given the prices of a set of options on the same underlying asset with different strikes and maturities, we propose a nonparametric approach for estimating risk-neutral densities associated with several maturities. Our method uses bicubic splines in order to achieve the desired smoothness for the estimation and an optimization model to choose the spline functions that best fit the price data. Semidefinite programming is employed to guarantee the nonnegativity of the densities. We illustrate the process using synthetic option price data generated using log-normal and absolute diffusion processes as well as actual price data for options on the S&P500 index. We also used the risk-neutral densities that we computed to price exotic options and observed that this approach generates prices that closely approximate the market prices of these options.
Preliminary remarks on option pricing and dynamic hedging
An elementary arbitrage principle and the existence of trends in financial
time series, which is based on a theorem published in 1995 by P. Cartier and Y.
Perrin, lead to a new understanding of option pricing and dynamic hedging.
Intricate problems related to violent behaviors of the underlying, like the
existence of jumps, become then quite straightforward by incorporating them
into the trends. Several convincing computer experiments are reported.Comment: 1st International Conference on Systems and Computer Science,
Villeneuve d'Ascq : France (2012
Multidimensional Quasi-Monte Carlo Malliavin Greeks
We investigate the use of Malliavin calculus in order to calculate the Greeks
of multidimensional complex path-dependent options by simulation. For this
purpose, we extend the formulas employed by Montero and Kohatsu-Higa to the
multidimensional case. The multidimensional setting shows the convenience of
the Malliavin Calculus approach over different techniques that have been
previously proposed. Indeed, these techniques may be computationally expensive
and do not provide flexibility for variance reduction. In contrast, the
Malliavin approach exhibits a higher flexibility by providing a class of
functions that return the same expected value (the Greek) with different
accuracies. This versatility for variance reduction is not possible without the
use of the generalized integral by part formula of Malliavin Calculus. In the
multidimensional context, we find convenient formulas that permit to improve
the localization technique, introduced in Fourni\'e et al and reduce both the
computational cost and the variance. Moreover, we show that the parameters
employed for variance reduction can be obtained \textit{on the flight} in the
simulation. We illustrate the efficiency of the proposed procedures, coupled
with the enhanced version of Quasi-Monte Carlo simulations as discussed in
Sabino, for the numerical estimation of the Deltas of call, digital Asian-style
and Exotic basket options with a fixed and a floating strike price in a
multidimensional Black-Scholes market.Comment: 22 pages, 6 figure
One numerical procedure for two risk factors modeling
We propose a numerical procedure for the pricing of financial contracts whose contingent claims are exposed to two sources of risk: the stock price and the short interest rate. More precisely, in our pricing framework we assume that the stock price dynamics is described by the Cox, Ross Rubinstein (CRR, 1979) binomial model under a stochastic risk free rate, whose dynamics evolves over time accordingly to the Black, Derman and Toy (BDT, 1990) one-factor model. To this aim, we set the hypothesis that the instantaneous correlation between the trajectories of the future stock price (conditional on the current value of the short rate) and of the future short rate is zero. We then apply the resulting stock price dynamics to evaluate the price of a simple contract, i.e. of a stock option. Finally, we compare the derived price to the price of the same option under different pricing models, as the traditional Black and Scholes (1973) model. We expect that, the difference in the two prices is not sensibly large. We conclude showing in which cases it should be helpful to adopt the described model for pricing purposes.option pricing; stochastic short rate model; binomial tree
A Hedged Monte Carlo Approach to Real Option Pricing
In this work we are concerned with valuing optionalities associated to invest
or to delay investment in a project when the available information provided to
the manager comes from simulated data of cash flows under historical (or
subjective) measure in a possibly incomplete market. Our approach is suitable
also to incorporating subjective views from management or market experts and to
stochastic investment costs. It is based on the Hedged Monte Carlo strategy
proposed by Potters et al (2001) where options are priced simultaneously with
the determination of the corresponding hedging. The approach is particularly
well-suited to the evaluation of commodity related projects whereby the
availability of pricing formulae is very rare, the scenario simulations are
usually available only in the historical measure, and the cash flows can be
highly nonlinear functions of the prices.Comment: 25 pages, 14 figure
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