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arXiv2012

Application of simplest random walk algorithms for pricing barrier options

We demonstrate effectiveness of the first-order algorithm from [Milstein, Tretyakov. Theory Prob. Appl. 47 (2002), 53-68] in application to barrier option pricing. The algorithm uses the weak Euler approximation far from barriers and a special construction motivated by linear interpolation of the price near barriers. It is easy to implement and is universal: it can be applied to various structures of the contracts including derivatives on multi-asset correlated underlyings and can deal with various type of barriers. In contrast to the Brownian bridge techniques currently commonly used for pricing barrier options, the algorithm tested here does not require knowledge of trigger probabilities nor their estimates. We illustrate this algorithm via pricing a barrier caplet, barrier trigger swap and barrier swaption.

q-fin.CP↗

arXiv2012

An FBSDE Approach to American Option Pricing with an Interacting Particle Method

In the paper, we propose a new calculation scheme for American options in the framework of a forward backward stochastic differential equation (FBSDE). The well-known decomposition of an American option price with that of a European option of the same maturity and the remaining early exercise premium can be cast into the form of a decoupled non-linear FBSDE. We numerically solve the FBSDE by applying an interacting particle method recently proposed by Fujii and Takahashi (2012d), which allows one to perform a Monte Carlo simulation in a fully forward-looking manner. We perform the fourth-order analysis for the Black-Scholes (BS) model and the third-order analysis for the Heston model. The comparison to those obtained from existing tree algorithms shows the effectiveness of the particle method.

q-fin.CP↗

arXiv2012

Multilevel Monte Carlo methods for applications in finance

Since Giles introduced the multilevel Monte Carlo path simulation method [18], there has been rapid development of the technique for a variety of applications in computational finance. This paper surveys the progress so far, highlights the key features in achieving a high rate of multilevel variance convergence, and suggests directions for future research.

q-fin.CP↗

arXiv2013

Online Portfolio Selection: A Survey

Online portfolio selection is a fundamental problem in computational finance, which has been extensively studied across several research communities, including finance, statistics, artificial intelligence, machine learning, and data mining, etc. This article aims to provide a comprehensive survey and a structural understanding of published online portfolio selection techniques. From an online machine learning perspective, we first formulate online portfolio selection as a sequential decision problem, and then survey a variety of state-of-the-art approaches, which are grouped into several major categories, including benchmarks, "Follow-the-Winner" approaches, "Follow-the-Loser" approaches, "Pattern-Matching" based approaches, and "Meta-Learning Algorithms". In addition to the problem formulation and related algorithms, we also discuss the relationship of these algorithms with the Capital Growth theory in order to better understand the similarities and differences of their underlying trading ideas. This article aims to provide a timely and comprehensive survey for both machine learning and data mining researchers in academia and quantitative portfolio managers in the financial industry to help them understand the state-of-the-art and facilitate their research and practical applications. We also discuss some open issues and evaluate some emerging new trends for future research directions.

q-fin.CP↗

arXiv2013

A Fourier Approach to the Computation of CV@R and Optimized Certainty Equivalents

We consider the class of risk measures associated with optimized certainty equivalents. This class includes several popular examples, such as CV@R and monotone mean-variance. Numerical schemes are developed for the computation of these risk measures using Fourier transform methods. This leads, in particular, to a very competitive method for the calculation of CV@R which is comparable in computational time to the calculation of V@R. We also develop methods for the efficient computation of risk contributions.

q-fin.RM↗

arXiv2013

On Reduced Form Intensity-based Model with Trigger Events

Corporate defaults may be triggered by some major market news or events such as financial crises or collapses of major banks or financial institutions. With a view to develop a more realistic model for credit risk analysis, we introduce a new type of reduced-form intensity-based model that can incorporate the impacts of both observable "trigger" events and economic environment on corporate defaults. The key idea of the model is to augment a Cox process with trigger events. Both single-default and multiple-default cases are considered in this paper. In the former case, a simple expression for the distribution of the default time is obtained. Applications of the proposed model to price defaultable bonds and multi-name Credit Default Swaps (CDSs) are provided.

q-fin.CP↗

arXiv2013

On Infectious Model for Dependent Defaults

In this paper, we propose a two-sector Markovian infectious model, which is an extension of Greenwood's model. The central idea of this model is that the causality of defaults of two sectors is in both direction, which enrich dependence dynamics. The Bayesian Information Criterion is adopted to compare the proposed model with the two-sector model in credit literature using the real data. We find that the newly proposed model is statistically better than the model in past literature. We also introduce two measures: CRES and CRVaR to give risk evaluation of our model.

q-fin.RM↗

arXiv2013

A parallel implementation of a derivative pricing model incorporating SABR calibration and probability lookup tables

We describe a high performance parallel implementation of a derivative pricing model, within which we introduce a new parallel method for the calibration of the industry standard SABR (stochastic-αβρ) stochastic volatility model using three strike inputs. SABR calibration involves a non-linear three dimensional minimisation and parallelisation is achieved by incorporating several assumptions unique to the SABR class of models. Our calibration method is based on principles of surface intersection, guarantees convergence to a unique solution and operates by iteratively refining a two dimensional grid with local mesh refinement. As part of our pricing model we additionally present a fast parallel iterative algorithm for the creation of dynamically sized cumulative probability lookup tables that are able to cap maximum estimated linear interpolation error. We optimise performance for probability distributions that exhibit clustering of linear interpolation error. We also make an empirical assessment of error propagation through our pricing model as a result of changes in accuracy parameters within the pricing model's multiple algorithmic steps. Algorithms are implemented on a GPU (graphics processing unit) using Nvidia's Fermi architecture. The pricing model targets the evaluation of spread options using copula methods, however the presented algorithms can be applied to a wider class of financial instruments.

cs.DC↗

arXiv2013

Financial Portfolio Optimization: Computationally guided agents to investigate, analyse and invest!?

Financial portfolio optimization is a widely studied problem in mathematics, statistics, financial and computational literature. It adheres to determining an optimal combination of weights associated with financial assets held in a portfolio. In practice, it faces challenges by virtue of varying math. formulations, parameters, business constraints and complex financial instruments. Empirical nature of data is no longer one-sided; thereby reflecting upside and downside trends with repeated yet unidentifiable cyclic behaviours potentially caused due to high frequency volatile movements in asset trades. Portfolio optimization under such circumstances is theoretically and computationally challenging. This work presents a novel mechanism to reach an optimal solution by encoding a variety of optimal solutions in a solution bank to guide the search process for the global investment objective formulation. It conceptualizes the role of individual solver agents that contribute optimal solutions to a bank of solutions, a super-agent solver that learns from the solution bank, and, thus reflects a knowledge-based computationally guided agents approach to investigate, analyse and reach to optimal solution for informed investment decisions. Conceptual understanding of classes of solver agents that represent varying problem formulations and, mathematically oriented deterministic solvers along with stochastic-search driven evolutionary and swarm-intelligence based techniques for optimal weights are discussed. Algorithmic implementation is presented by an enhanced neighbourhood generation mechanism in Simulated Annealing algorithm. A framework for inclusion of heuristic knowledge and human expertise from financial literature related to investment decision making process is reflected via introduction of controlled perturbation strategies using a decision matrix for neighbourhood generation.

q-fin.PM↗

arXiv2013

USLV: Unspanned Stochastic Local Volatility Model

We propose a new framework for modeling stochastic local volatility, with potential applications to modeling derivatives on interest rates, commodities, credit, equity, FX etc., as well as hybrid derivatives. Our model extends the linearity-generating unspanned volatility term structure model by Carr et al. (2011) by adding a local volatility layer to it. We outline efficient numerical schemes for pricing derivatives in this framework for a particular four-factor specification (two "curve" factors plus two "volatility" factors). We show that the dynamics of such a system can be approximated by a Markov chain on a two-dimensional space (Z_t,Y_t), where coordinates Z_t and Y_t are given by direct (Kroneker) products of values of pairs of curve and volatility factors, respectively. The resulting Markov chain dynamics on such partly "folded" state space enables fast pricing by the standard backward induction. Using a nonparametric specification of the Markov chain generator, one can accurately match arbitrary sets of vanilla option quotes with different strikes and maturities. Furthermore, we consider an alternative formulation of the model in terms of an implied time change process. The latter is specified nonparametrically, again enabling accurate calibration to arbitrary sets of vanilla option quotes.

q-fin.PR↗

arXiv2014

Ergodicity and scaling limit of a constrained multivariate Hawkes process

We introduce a multivariate Hawkes process with constraints on its conditional density. It is a multivariate point process with conditional intensity similar to that of a multivariate Hawkes process but certain events are forbidden with respect to boundary conditions on a multidimensional constraint variable, whose evolution is driven by the point process. We study this process in the special case where the fertility function is exponential so that the process is entirely described by an underlying Markov chain, which includes the constraint variable. Some conditions on the parameters are established to ensure the ergodicity of the chain. Moreover, scaling limits are derived for the integrated point process. This study is primarily motivated by the stochastic modelling of a limit order book for high frequency financial data analysis.

stat.AP↗

arXiv2013

Evaluation on the Financial Competitiveness of Chinese Listed Real Estate Companies Based on Entropy Method

The real estate is a pillar industry of China's national economy. Due to changes in policy and market conditions, the real estate companies are facing greater pressures to survive in a competitive environment. They must improve their financial competitiveness. Based on the conceptual framework of financial competitiveness, this paper presented a financial competitiveness evaluation index system, covering four aspects, including profitability, solvency, sustainable development and operational capacity. Entropy value method is applied to determine the index weight. 105 listed real estate company's financial competitiveness are evaluated, the results show that: high-scoring company has strong profitability, sustainable development and operational capacity; low-scoring company has weak profitability and poor ability of sustainable development; solvency doesn't affect the company's financial competitiveness obviously.

q-fin.GN↗

arXiv2013

Stationarity and ergodicity for an affine two factor model

We study the existence of a unique stationary distribution and ergodicity for a 2-dimensional affine process. The first coordinate is supposed to be a so-called alpha-root process with α\in(1,2]. The existence of a unique stationary distribution for the affine process is proved in case of α\in(1,2]; further, in case of α=2, the ergodicity is also shown.

math.PR↗

arXiv2013

Technical report : Risk-neutral density recovery via spectral analysis

In this paper, we propose a new method for estimating the conditional risk-neutral density (RND) directly from a cross-section of put option bid-ask quotes. More precisely, we propose to view the RND recovery problem as an inverse problem. We first show that it is possible to define restricted put and call operators that admit a singular value decomposition (SVD), which we compute explicitly. We subsequently show that this new framework allows us to devise a simple and fast quadratic programming method to recover the smoothest RND whose corresponding put prices lie inside the bid-ask quotes. This method is termed the spectral recovery method (SRM). Interestingly, the SVD of the restricted put and call operators sheds some new light on the RND recovery problem. The SRM improves on other RND recovery methods in the sense that: - it is fast and simple to implement since it requires solution of a single quadratic program, while being fully nonparametric; - it takes the bid ask quotes as sole input and does not require any sort of calibration, smoothing or preprocessing of the data; - it is robust to the paucity of price quotes; - it returns the smoothest density giving rise to prices that lie inside the bid ask quotes. The estimated RND is therefore as well-behaved as can be; - it returns a closed form estimate of the RND on the interval [0,B] of the positive real line, where B is a positive constant that can be chosen arbitrarily. We thus obtain both the middle part of the RND together with its full left tail and part of its right tail. We confront this method to both real and simulated data and observe that it fares well in practice. The SRM is thus found to be a promising alternative to other RND recovery methods.

q-fin.CP↗

arXiv2013

The Pricing of Multiple-Expiry Exotics

In this paper we extend Buchen's method to develop a new technique for pricing of some exotic options with several expiry dates(more than 3 expiry dates) using a concept of higher order binary option. At first we introduce the concept of higher order binary option and then provide the pricing formulae of $n$-th order binaries using PDE method. After that, we apply them to pricing of some multiple-expiry exotic options such as Bermudan option, multi time extendable option, multi shout option and etc. Here, when calculating the price of concrete multiple-expiry exotic options, we do not try to get the formal solution to corresponding initial-boundary problem of the Black-Scholes equation, but explain how to express the expiry payoffs of the exotic options as a combination of the payoffs of some class of higher order binary options. Once the expiry payoffs are expressed as a linear combination of the payoffs of some class of higher order binary options, in order to avoid arbitrage, the exotic option prices are obtained by static replication with respect to this family of higher order binaries.

q-fin.PR↗

arXiv2013

Pricing Corporate Defaultable Bond using Declared Firm Value

We study the pricing problem for corporate defaultable bond from the viewpoint of the investors outside the firm that could not exactly know about the information of the firm. We consider the problem for pricing of corporate defaultable bond in the case when the firm value is only declared in some fixed discrete time and unexpected default intensity is determined by the declared firm value. Here we provide a partial differential equation model for such a defaultable bond and give its pricing formula. Our pricing model is derived to solving problems of partial differential equations with random constants (de- fault intensity) and terminal values of binary types. Our main method is to use the solving method of a partial differential equation with a random constant in every subinterval and to take expectation to remove the random constants.

q-fin.PR↗

arXiv2013

An Explicit Martingale Version of Brenier's Theorem

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present martingale version. We provide the explicit martingale optimal transference plans for a remarkable class of coupling functions corresponding to the lower and upper bounds. These explicit extremal probability measures coincide with the unique left and right monotone martingale transference plans, which were introduced in \cite{BeiglbockJuillet} by suitable adaptation of the notion of cyclic monotonicity. Instead, our approach relies heavily on the (weak) duality result stated in \cite{BeiglbockHenry-LaborderePenkner}, and provides, as a by-product, an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.

q-fin.CP↗

arXiv2013

An extension of Paulsen-Gjessing's risk model with stochastic return on investments

We consider in this paper a general two-sided jump-diffusion risk model that allows for risky investments as well as for correlation between the two Brownian motions driving insurance risk and investment return. We first introduce the model and then find the integro-differential equations satisfied by the Gerber-Shiu functions as well as the expected discounted penalty functions at ruin caused by a claim or by oscillation; We also study the dividend problem for the threshold and barrier strategies, the moments and moment-generating function of the total discounted dividends until ruin are discussed. Some examples are given for special cases.

q-fin.CP↗