916 research outputs found
Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method
We study semi-linear elliptic PDEs with polynomial non-linearity and provide
a probabilistic representation of their solution using branching diffusion
processes. When the non-linearity involves the unknown function but not its
derivatives, we extend previous results in the literature by showing that our
probabilistic representation provides a solution to the PDE without assuming
its existence. In the general case, we derive a new representation of the
solution by using marked branching diffusion processes and automatic
differentiation formulas to account for the non-linear gradient term. In both
cases, we develop new theoretical tools to provide explicit sufficient
conditions under which our probabilistic representations hold. As an
application, we consider several examples including multi-dimensional
semi-linear elliptic PDEs and estimate their solution by using the Monte Carlo
method
Comparing optimal convergence rate of stochastic mesh and least squares method for Bermudan option pricing
We analyze the stochastic mesh method (SMM) as well as the least squares method (LSM) commonly
used for pricing Bermudan options using the standard two phase methodology. For both the methods, we
determine the decay rate of mean square error of the estimator as a function of the computational budget
allocated to the two phases and ascertain the order of the optimal allocation in these phases. We conclude
that with increasing computational budget, while SMM estimator converges at a slower rate compared to
LSM estimator, it converges to the true option value whereas LSM estimator, with fixed number of basis
functions, usually converges to a biased value
American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics
American options are actively traded worldwide on exchanges, thus making their accurate and efficient pricing an important problem. As most financial markets exhibit randomly varying volatility, in this paper we introduce an approximation of American option price under stochastic volatility models. We achieve this by using the maturity randomization method known as Canadization. The volatility process is characterized by fast and slow scale fluctuating factors. In particular, we study the case of an American put with a single underlying asset and use perturbative expansion techniques to approximate its price as well as the optimal exercise boundary up to the first order. We then use the approximate optimal exercise boundary formula to price American put via Monte Carlo. We also develop efficient control variates for our simulation method using martingales resulting from the approximate price formula. A numerical study is conducted to demonstrate that the proposed method performs better than the least squares regression method popular in the financial industry, in typical settings where values of the scaling parameters are small. Further, it is empirically observed that in the regimes where scaling parameter value is equal to unity, fast and slow scale approximations are equally accurate
Efficient simulation of large deviation events for sums of random vectors using saddle-point representations
We consider the problem of efficient simulation estimation of the
density function at the tails, and the probability of large
deviations for a sum of independent, identically distributed (i.i.d.),
light-tailed and nonlattice random vectors. The latter problem
besides being of independent interest, also forms a building block
for more complex rare event problems that arise, for instance, in
queuing and financial credit risk modeling. It has been extensively
studied in the literature where state-independent, exponential-twisting-based
importance sampling has been shown to be asymptotically
efficient and a more nuanced state-dependent exponential twisting
has been shown to have a stronger bounded relative error property.
We exploit the saddle-point-based representations that exist for
these rare quantities, which rely on inverting the characteristic
functions of the underlying random vectors. These representations
reduce the rare event estimation problem to evaluating certain
integrals, which may via importance sampling be represented as
expectations. Furthermore, it is easy to identify and approximate the
zero-variance importance sampling distribution to estimate these
integrals. We identify such importance sampling measures and show
that they possess the asymptotically vanishing relative error
property that is stronger than the bounded relative error
property. To illustrate the broader applicability of the proposed
methodology, we extend it to develop an asymptotically vanishing
relative error estimator for the practically important expected
overshoot of sums of i.i.d. random variables
A Deep Generative Framework for Paraphrase Generation
Paraphrase generation is an important problem in NLP, especially in question
answering, information retrieval, information extraction, conversation systems,
to name a few. In this paper, we address the problem of generating paraphrases
automatically. Our proposed method is based on a combination of deep generative
models (VAE) with sequence-to-sequence models (LSTM) to generate paraphrases,
given an input sentence. Traditional VAEs when combined with recurrent neural
networks can generate free text but they are not suitable for paraphrase
generation for a given sentence. We address this problem by conditioning the
both, encoder and decoder sides of VAE, on the original sentence, so that it
can generate the given sentence's paraphrases. Unlike most existing models, our
model is simple, modular and can generate multiple paraphrases, for a given
sentence. Quantitative evaluation of the proposed method on a benchmark
paraphrase dataset demonstrates its efficacy, and its performance improvement
over the state-of-the-art methods by a significant margin, whereas qualitative
human evaluation indicate that the generated paraphrases are well-formed,
grammatically correct, and are relevant to the input sentence. Furthermore, we
evaluate our method on a newly released question paraphrase dataset, and
establish a new baseline for future research
The implied Sharpe ratio
In an incomplete market, including liquidly-traded European options in an
investment portfolio could potentially improve the expected terminal utility
for a risk-averse investor. However, unlike the Sharpe ratio, which provides a
concise measure of the relative investment attractiveness of different
underlying risky assets, there is no such measure available to help investors
choose among the different European options. We introduce a new concept -- the
implied Sharpe ratio -- which allows investors to make such a comparison in an
incomplete financial market. Specifically, when comparing various European
options, it is the option with the highest implied Sharpe ratio that, if
included in an investor's portfolio, will improve his expected utility the
most. Through the method of Taylor series expansion of the state-dependent
coefficients in a nonlinear partial differential equation, we also establish
the behaviour of the implied Sharpe ratio with respect to an investor's
risk-aversion parameter. In a series of numerical studies, we compare the
investment attractiveness of different European options by studying their
implied Sharpe ratio.Comment: 22 pages, 6 figure
Portfolio benchmarking under drawdown constraint and stochastic sharpe ratio
We consider an investor who seeks to maximize her expected utility of wealth relative to a benchmark, or target over a finite time horizon, and under a portfolio drawdown constraint, in a market with local stochastic volatility. We propose a new investor objective paradigm which allows the investor to target the portfolio benchmark while obeying the constraint, both of which can be characterized in terms of the running maximum wealth process. In the absence of closed-form formulas for the value function and optimal portfolio strategy in the incomplete market models we consider, we obtain approximations for these quantities through the use of a coefficient expansion technique and nonlinear transformations. We utilize regularity properties of the risk tolerance function to numerically compute the estimates for our approximations. In order to achieve similar utility, compared to a constant volatility model, we illustrate that the investor must deploy a quite different portfolio strategy which depends on the current level of volatility
Study of new rare event simulation schemes and their application to extreme scenario generation
This is a companion paper based on our previous work on rare event simulation
methods. In this paper, we provide an alternative proof for the ergodicity
of shaking transformation in the Gaussian case and propose two variants of
the existing methods with comparisons of numerical performance. In numerical
tests, we also illustrate the idea of extreme scenario generation based on
the convergence of marginal distributions of the underlying Markov chains
and show the impact of the discretization of continuous time models on rare
event probability estimation
- …