Characteristic functions and option valuation in a Markov chain market

We introduce an approach for valuing some path-dependent options in a discrete-time Markov chain market based on the characteristic function of a vector of occupation times of the chain. A pricing kernel is introduced and analytical formulas for the prices of Asian options and occupation time call options are derived.     

The Path Integral Approach to Financial Modeling and Options Pricing

In this paper we review some applications of the path integral methodology of quantum mechanics to financial modeling and options pricing. A path integral is defined as a limit of the sequence of finite-dimensional integrals, in a much the same way as the Riemannian integral is defined as a limit of the sequence of finite sums. The risk-neutral valuation formula for path-dependent options contingent upon multiple underlying assets admits an elegant representation in terms of path integrals (Feynman–Kac formula). The path integral representation of transition probability density (Green's function) explicitly satisfies the diffusion PDE. Gaussian path integrals admit a closed-form solution given by the Van Vleck formula. Analytical approximations are obtained by means of the semiclassical (moments) expansion. Difficult path integrals are computed by numerical procedures, such as Monte Carlo simulation or deterministic discretization schemes. Several examples of path-dependent options are treated to illustrate the theory (weighted Asian options, floating barrier options, and barrier options with ladder-like barriers).     

Valuation of electricity swing options by multistage stochastic programming

Electricity swing options are Bermudan-style path-dependent derivatives on electrical energy. We consider an electricity market driven by several exogenous risk factors and formulate the pricing problem for a class of swing option contracts with energy and power limits as well as ramping constraints. Efficient numerical solution of the arising multistage stochastic program requires aggregation of decision stages, discretization of the probability space, and reparameterization of the decision space. We report on numerical results and discuss analytically tractable limiting cases.     

The numerical approximation of nonlinear Black–Scholes model for exotic path-dependent American options with transaction cost

In this paper, a new second-order exponential time differencing (ETD) method based on the Cox and Matthews approach is developed and applied for pricing American options with transaction cost. The method is seen to be strongly stable and highly efficient for solving the nonlinear Black–Scholes model. Furthermore, it does not incur unwanted oscillations unlike the ETD–Crank–Nicolson method for exotic path-dependent American options. The computational efficiency and reliability of the new method are demonstrated by numerical examples and by comparing it with the existing methods.     

Monte Carlo option pricing with asymmetric realized volatility dynamics

What are the advances introduced by realized volatility models in pricing options? In this short paper we analyze a simple option pricing framework based on the dually asymmetric realized volatility model, which emphasizes extended leverage effects and empirical regularity of high volatility risk during high volatility periods. We conduct a brief empirical analysis of the pricing performance of this approach against some benchmark models using data from the S&P 500 options in the 2001-2004 period. The results indicate that as expected the superior forecasting accuracy of realized volatility translates into significantly smaller pricing errors when compared to models of the GARCH family. Most importantly, our results indicate that the presence of leverage effects and a high volatility risk are essential for understanding common option pricing anomalies.     

Approximate Option Pricing

As increasingly large volumes of sophisticated options are traded in world financial markets, determining a ``fair' price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n -period option on a stock is the expected time-discounted value of the future cash flow on an n -period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte Carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this article we show that pricing an arbitrary path-dependent option is \#-P hard. We show that certain types of path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these we design deterministic polynomial-time approximate algorithms. We show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation of the value of an otherwise identical n -period American put option. In contrast to Monte Carlo methods, our algorithms have guaranteed error bounds that are polynomially small (and in some cases exponentially small) in the maturity n . For the error analysis we derive large-deviation results for random walks that may be of independent interest. Received August 13, 1996; revised April 2, 1997.     

Fuzzy pricing of geometric Asian options and its algorithm

Owing to the fluctuations of the financial market, input data in the options pricing formula cannot be expected to be precise. This paper discusses the problem of pricing geometric Asian options under the fuzzy environment. We present the fuzzy price of the geometric Asian option under the assumption that the underlying stock price, the risk-free interest rate and the volatility are all fuzzy numbers. This assumption makes the financial investors to pick any geometric Asian option price with an acceptable belief degree. In order to obtain the belief degree, the interpolation search algorithm has been proposed. Some numerical examples are presented to illustrate the rationality and practicability of the model and the algorithm. Finally, an empirical study is performed based on the real data. The empirical study results indicate that the proposed fuzzy pricing model of geometric Asian option is a useful tool for modeling the imprecise problem in the real world.     

Option pricing under the Merton model of the short rate

Previous option pricing research typically assumes that the risk-free rate or the short rate is constant during the life of the option. In this study, we incorporate the stochastic nature of the short rate in our option valuation model and derive explicit formulas for European call and put options on a stock when the short rate follows the Merton model. Using our option model as a benchmark, our numerical analysis indicates that, in general, the Black–Scholes model overvalues out-of-the-money calls, moderately overvalues at-the-money calls, and slightly overvalues in-the-money calls. Our analysis is directly extensible to American calls on non-dividend-paying stocks and to European puts by virtue of put-call parity.     

Model Risk for European-Style Stock Index Options

In empirical modeling, there have been two strands for pricing in the options literature, namely the parametric and nonparametric models. Often, the support for the nonparametric methods is based on a benchmark such as the Black-Scholes (BS) model with constant volatility. In this paper, we study the stochastic volatility (SV) and stochastic volatility random jump (SVJ) models as parametric benchmarks against feedforward neural network (FNN) models, a class of neural network models. Our choice for FNN models is due to their well-studied universal approximation properties of an unknown function and its partial derivatives. Since the partial derivatives of an option pricing formula are risk pricing tools, an accurate estimation of the unknown option pricing function is essential for pricing and hedging. Our findings indicate that FNN models offer themselves as robust option pricing tools, over their sophisticated parametric counterparts in predictive settings. There are two routes to explain the superiority of FNN models over the parametric models in forecast settings. These are nonnormality of return distributions and adaptive learning     

Isogeometric analysis in option pricing

ABSTRACT

Isogeometric analysis is a recently developed computational approach that integrates finite element analysis directly into design described by non-uniform rational B-splines (NURBS). In this paper, we show that price surfaces that occur in option pricing can be easily described by NURBS surfaces. For a class of stochastic volatility models, we develop a methodology for solving corresponding pricing partial integro-differential equations numerically by isogeometric analysis tools and show that a very small number of space discretization steps can be used to obtain sufficiently accurate results. Presented solution by finite element method is especially useful for practitioners dealing with derivatives where closed-form solution is not available.     

Time Changes, Laplace Transforms and Path-Dependent Options

Path-dependent options have become increasingly popular over the last few years, in particular in FX markets, because of the greater precision with which they allow investors to choose or avoid exposure to well-defined sources of risk. The goal of the paper is to exhibit the power of stochastic time changes and Laplace transform techniques in the evaluation and hedging of path-dependent options in the Black–Scholes–Merton setting. We illustrate these properties in the specific case of Asian options and continuously (de-) activating double-barrier options and show that in both cases, the pricing and, just as important, the hedging results are more accurate than the ones obtained through Monte Carlo simulations.     

Robust Artificial Neural Networks for Pricing of European Options

The option pricing ability of Robust Artificial Neural Networks optimized with the Huber function is compared against those optimized with Least Squares. Comparison is in respect to pricing European call options on the S&P 500 using daily data for the period April 1998 to August 2001. The analysis is augmented with the use of several historical and implied volatility measures. Implied volatilities are the overall average, and the average per maturity. Beyond the standard neural networks, hybrid networks that directly incorporate information from the parametric model are included in the analysis. It is shown that the artificial neural network models with the use of the Huber function outperform the ones optimized with least squares. JEL Classification: G13, G14     

A neural network model for estimating option prices

A neural network model that processes financial input data is developed to estimate the market price of options at closing. The network's ability to estimate closing prices is compared to the Black-Scholes model, the most widely used model for the pricing of options. Comparisons reveal that the mean squared error for the neural network is less than that of the Black-Scholes model in about half of the cases examined. The differences and similarities in the two modeling approaches are discussed. The neural network, which uses the same financial data as the Black-Scholes model, requires no distribution assumptions and learns the relationships between the financial input data and the option price from the historical data. The option-valuation equilibrium model of Black-Scholes determines option prices under the assumptions that prices follow a continuous time path and that the instantaneous volatility is nonstochastic.     

期权定价多元树的计算

树形结构法是期权定价的基本方法之一,其中二元树形法目前得到了广泛的应用．本文的主题是多元树形结构法,文中对其算法及构造进行了讨论,同时,还找出了二个新的五元树形结构模型和一个七元树形结构模型．与二元树形结构法相比,多元树形结构法能有更快的收敛速度和更高的计算精度,然而期权的内在收益函数(payoff function)的不光滑性会降低树形结构法的计算效率．本文讨论了克服这一缺陷的方法,最后文中对在定价计算中的效率问题进行了实证研究．     

Valuation of European continuous-installment options

This paper is concerned with the valuation of European continuous-installment options where the aim is to determine the initial premium given a constant installment payment plan. The distinctive feature of this pricing problem is the determination, along with the initial premium, of an optimal stopping boundary since the option holder has the right to stop making installment payments at any time before maturity. Given that the initial premium function of this option is governed by an inhomogeneous Black-Scholes partial differential equation, we can obtain two alternative characterizations of the European continuous-installment option pricing problem, for which no closed-form solution is available. First, we formulate the pricing problem as a free boundary problem and using the integral representation method, we derive integral expressions for both the initial premium and the optimal stopping boundary. Next, we use the linear complementarity formulation of the pricing problem for determining the initial premium and the early stopping curve implicitly with a finite difference scheme. Finally, the pricing problem is posed as an optimal stopping problem and then implemented by a Monte Carlo approach.     

Efficient solutions for discrete Asian options

While in the literature most studies on pricing focus on continuous Asian options, in this paper we provide efficient solutions for both European and American discrete average price Asian options. The method used for deriving the approximation formula for European Asian options is based on the idea of Bouaziz et al. (J Bank Finance 18:823–839, 1994) and Taso et al. (J Futures Mark 23:487–516, 2003) in which the Taylor expansion is used to obtain the approximation formula for continuous average strike Asian options. By using the Taylor expansion to the second order, a simple and accurate solution can be obtained. The approximation formula for the European Asian option can further be used to enhance the efficiency of the pricing of the American Asian options when using the numerical method.     

Two-State Volatility Transition Pricing and Hedging of TXO Options

The spot markets often exhibit high and low volatilities that persist for a while. We classify the spot market volatility into two states: high and low and use the Markov chain theory to construct a Two-State volatility model for pricing and hedging Taiwan stock index options (TXO). Compared to binomial option pricing model, the Two-State model is more stable in convergence and faster in early periods of convergence but is much more time-consuming as the number of periods and computations extensively increase. The growth order of total node number for quadrinomial lattice is O(n ⁴) while it is O(n ²) for binomial lattice. Empirically, the Taiwan stock index has high-volatility = 42.85%, low-volatility = 17.39%, and probability of being in high-volatility state = 0.3487 over the in-sample period from 1/6/1990 to 04/30/2008 according to Markov chain. Using as large as 87,160 datasets of TXO covering out-of-sample period from 05/02/2008 to 03/17/2010 and strike prices from 3600 to 8700, we demonstrated that the Two-State volatility model has the most outstanding performance in high-volatility period as applying put options for pricing and hedging. However, to avoid the cost of taxes resulting from position changes, a longer-term (e.g. 10 day) hedge is more properly than a short-term (e.g. 5 day) hedge.     

Using computational methodology to price European options with actual payoff distributions

Most option pricing methods use mathematical distributions to approximate underlying asset behavior. However, pure mathematical distribution approaches have difficulty approximating the real distribution. This study first introduces an innovative computational method for pricing European options based on the real payoff distribution of the underlying asset. This computational approach can also be applied to applications related to expected value that require real distributions rather than mathematical distributions. This study makes the following contributions: (a) solving the risk neutral issue related to price options with real payoff distributions; (b) proposing a simple method for adjusting standard deviation based on the need to apply short term volatility to real world applications; (c) demonstrating an option pricing algorithm that is easy to apply to cross field applications.     

Timer formulas and decidable metric temporal logic

We define a quantitative temporal logic that is based on a simple modality within the framework of monadic predicate logic. Its canonical model is the real line (and not an ω-sequence of some type). It can be interpreted either by behaviors with finite variability or by unrestricted behaviors. For finite variability models it is as expressive as any logic suggested in the literature. For unrestricted behaviors our treatment is new. In both cases we prove decidability and complexity bounds using general theorems from logic (and not from automata theory). The technical proof uses a sublanguage of the metric monadic logic of order, the language of timer normal form formulas. Metric formulas are reduced to timer normal form and timer normal form formulas allow elimination of the metric.     

Polynomial Algorithms for Pricing Path-Dependent Interest Rate Instruments

In this paper we study algorithms for pricing of interest rate instruments using recombining tree (scenario lattice) interest models. The price is defined as expected discounted cash flow. If the cash-flow generated by the instrument depends on the full or partial history of interest rates (path-dependent contracts), then pricing algorithms are typically of exponential complexity. We show that for some models, including product form cash-flows, additive cash-flows, delayed cash-flows and limited path-dependent cash-flows, polynomial pricing algorithms exist.