Pricing Of Derivative Securities Research Articles

Psychological Barriers and Option Pricing

Psychological barriers are prevalent among various asset classes, and it is important to consider their impact on the prices of derivative securities. This paper demonstrates the potential existence of such barriers on the SP this paper also evaluates this model's empirical performance relative to the Black-Scholes and constant elasticity of variance (CEV) models. The in-sample calibration result of the threshold model is found to be superior. Furthermore, it is found that the model provides an efficient hedging method in terms of dollar-value hedging errors.

A Dual Interpretation of the Case-Shiller Index and Its Implications to Home Appraisals

The S&P/Case-Shiller® Home Price Indices are designed to measure change in the total value of all existing single-family housing stocks. This paper utilizes duality in linear programming to explore a close connection between the methodology of the indices and the classic no-arbitrage (NA) condition in financial markets. In essence, the interpretation of the NA is the “dual” problem to the primal minimization problem by which a regression is used to estimate the indices. The variables of the duality of the NA maximization problem present a term structure of discount factors, the reciprocal of which are backward-looking indices. The insight induces a new methodology for appraising single-family housing and exposes its connection to the commonly-used appraisal method. It is well known that the pricing of derivative securities is based on arbitrage arguments. In view of the index family being the underlying asset of the recent home price derivatives, the intimate connection between its estimation methodology and the classical definition of the NA condition is a point of interest.

Money Mathematics: Examining Ethics Education in Quantitative Finance

The field of quantitative analysis is often mistaken to be a discipline free from ethical burdens. The quantitative financial analyst or “quant” profession holds a position of significant responsibility as the keeper of mathematical models used in complex derivative security pricing and risk management. Despite this responsibility very few postgraduate programs address the teaching of ethics and professional standards in their curriculum, and the credibility of the profession has suffered as a result of several high-profile financial losses. Some of these failures could have been avoided and their impacts diminished if ethical considerations were integrated with quantitative method. Appropriate development in ethics education for quants is needed to identify points in the decision-making process where ethical questions can arise, and to explain how quants can protect stakeholders from the costs of unethical behaviour. An approach to ethics education needs to be flexible and allow for different methods to infuse ethical coverage into the course. Such an approach will go some way towards aligning the profession with other specialisations in banking and avoid the need for complex and unnecessary regulation.

Use of Bayesian Estimates to determine the Volatility Parameter Input in the Black-Scholes and Binomial Option Pricing Models

The valuation of options and many other derivative instruments requires an estimation of exante or forward looking volatility. This paper adopts a Bayesian approach to estimate stock price volatility. We find evidence that overall Bayesian volatility estimates more closely approximate the implied volatility of stocks derived from traded call and put options prices compared to historical volatility estimates sourced from IVolatility.com (“IVolatility”). Our evidence suggests use of the Bayesian approach to estimate volatility can provide a more accurate measure of ex-ante stock price volatility and will be useful in the pricing of derivative securities where the implied stock price volatility cannot be observed.

A Closer Look at Barrier Exchange Options

A barrier exchange option is an exchange option that is knocked out the first time the prices of two underlying assets become equal. Lindset, S., & Persson, S.‐A. (2006) present a simple dynamic replication argument to show that, in the absence of arbitrage, the current value of the barrier exchange option is equal to the difference in the current prices of the underlying assets and that this pricing formula applies irrespective of whether the option is European or American. In this study, we take a closer look at barrier exchange options and show, despite the simplicity of the pricing formula presented by Lindset, S., & Persson, S.‐A. (2006), that the barrier exchange option in fact involves a surprising array of key concepts associated with the pricing of derivative securities including: put–call parity, barrier in–out parity, static vs. dynamic replication, martingale pricing, continuous vs. discontinuous price processes, and numeraires. We provide valuable intuition behind the pricing formula which explains its apparent simplicity. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 33:29–43, 2013

On pricing and hedging options in regime-switching models with feedback effect

We study the pricing and hedging of European-style derivative securities in a Markov, regime-switching, model with a feedback effect depending on the economic condition. We adopt a pricing kernel which prices both financial and economic risks explicitly in a dynamically incomplete market and we provide an equilibrium analysis. A martingale representation for a European-style index option's price is established based on the price kernel. The martingale representation is then used to construct the local risk-minimizing strategy explicitly and to characterize the corresponding pricing measure.

Dividends in the Theory of Derivative Securities Pricing

This paper develops the fundamental aspects of the theory of martingale pricing of derivative securities in a setting where the cumulative gains processes are Ito processes while the cumulative dividend processes of both the underliers and the derivative securities are general enough to cover all cases encountered in practical applications. A key ingredient is a general formula for how to change the unit of account of a cumulative dividend process. The formula is inconsistent with parts of the earlier literature. It obeys a unit-invariance rule for trading strategies, satisfies a consistency property when the unit is changed twice in a row, gives the correct results in well-known and uncontroversial special cases, and fits perfectly into a generalization of the martingale valuation theory. Using that generalized theory, we show that the value of a dividend process equals the value of a claim to the nominal amount of dividends yet to be accumulated plus the value of a flow of interest on the cumulative dividends at each point in time.

MOMENT EXPLOSIONS AND STATIONARY DISTRIBUTIONS IN AFFINE DIFFUSION MODELS

Many of the most widely used models in finance fall within the affine family of diffusion processes. The affine family combines modeling flexibility with substantial tractability, particularly through transform analysis; these models are used both for econometric modeling and for pricing and hedging of derivative securities. We analyze the tail behavior, the range of finite exponential moments, and the convergence to stationarity in affine models, focusing on the class of canonical models defined by Dai and Singleton (2000). We show that these models have limiting stationary distributions and characterize these limits. We show that the tails of both the transient and stationary distributions of these models are necessarily exponential or Gaussian; in the non-Gaussian case, we characterize the tail decay rate for any linear combination of factors. We also give necessary and sufficient conditions for a linear combination of factors to be Gaussian. Our results follow from an investigation into the stability properties of the systems of ordinary differential equations associated with affine diffusions.

Dimension Reduction Techniques in Quasi-Monte Carlo Methods for Option Pricing

Many problems in mathematical finance (e.g., the pricing of financial derivative securities) can be formulated as high-dimensional integrals, which are often tackled by quasi-Monte Carlo methods. Quasi-random points have the special property that the one-dimensional projections and the earlier dimensional projections are extremely well distributed. This property strongly motivates us to reduce the effective dimensions of the problems. This paper deals with multiasset path-dependent options. The goal of this paper is twofold: (1) to design new methods for dimension reduction and to offer further insight into dimension reduction techniques, and (2) to investigate the practical performance of various dimension reduction strategies in conjunction with different quasi-random points (digital nets and good lattice points). We propose two groups of two-stage procedures for the constructions of correlated multidimensional Brownian motions, which provide a general framework to combine and use different dimension reduction techniques and optimize the use of quasi-random points. We find the inherent weights that characterize the weighted property and the dimension structure of the underlying functions (e.g., the sensitivity indices, the effective dimensions, etc.), and show how each dimension reduction strategy affects them. Numerical experiments on multiasset path-dependent options demonstrate that significant improvements can be achieved by proper use of dimension reduction strategies and quasi-random points. It is also shown that different quasi-random point sets have different behavior with respect to dimension reduction strategies.

Refined and Enhanced Fast Fourier Transform Techniques, with an Application to the Pricing of Barrier Options

The fast Fourier transform (FFT) technique is now a standard tool for the numerical calculation of prices of derivative securities. Unfortunately, in many important situations, such as the pricing of contingent claims of European type near expiry, and the pricing of barrier options close to the barrier, the standard implementation of this technique leads to serious systematic errors. We propose a new, fast and efficient, variant of the FFT technique, which is free of these problems, and is as easy to implement as the most common version of FFT. As an example, we show how our method leads to a pricing algorithm for down-and-out barrier put options that is the most efficient one to date, both in terms of the speed and in terms of the accuracy of the computations.

Using neural network to forecast stock index option price: a new hybrid GARCH approach

This study aims to apply a new hybrid approach to estimate volatility in neural network option-pricing model. The analytical results also indicate that the new hybrid method can be used to forecast the prices of derivative securities. Owing to combines the grey forecasting model with the GARCH to improve the estimated ability, the empirical evidence shows that the new hybrid GARCH model outperforms the other approaches in the neural network option-pricing model.

A computational study on general equilibrium pricing of derivative securities

This paper analyses the accuracy of replicating portfolio methods in predicting asset prices. In a two-period, general equilibrium model with incomplete financial markets and heterogeneous agents, a computational study is conducted under various distributional assumptions. The focus is on the price of a call option on an underlying risky asset. There is evidence that the value of the (approximate) replicating portfolio is a good approximation for the general equilibrium price for CRRA preferences, but not for CARA preferences. Furthermore, there is strong evidence that the introduction of the call option reduces market incompleteness, but that the price of the underlying asset is unchanged. There is, however, inconclusive evidence on the welfare effects of the option.

A Computational Study on General Equilibrium Pricing of Derivative Securities

This paper analyses the accuracy of replicating portfolio methods in predicting asset prices. In a two-period, general equilibrium model with incomplete financial markets and heterogeneous agents, a computational study is conducted under various distributional assumptions. We focus on the price of a call option on an underlying risky asset. There is evidence that the value of the (approximate) replicating portfolio is a good approximation for the general equilibrium price for CRRA preferences, but not for CARA preferences. Furthermore, there is strong evidence that the introduction of the call option reduces market incompleteness and that the price of the underlying asset is unchanged. There is, however, inconclusive evidence on the welfare effects of the option.

Asset Price Bubbles in Incomplete Markets

This paper studies asset price bubbles in a continuous time model using the local martingale framework. Providing careful definitions of the asset's market and fundamental price, we characterize all possible price bubbles in an incomplete market satisfying the free lunch with vanishing risk (NFLVR) and dominance assumptions. We show that the two leading models for bubbles as either charges or as strict local martingales, respectively, are equivalent. We propose a new theory for bubble birth which involves a nontrivial modification of the classical martingale pricing framework. This modification involves the market exhibiting different local martingale measures across time - a possibility not previously explored within the classical theory. Finally, we investigate the pricing of derivative securities in the presence of asset price bubbles, and we show that: (i) European put options can have no bubbles, (ii) European call options and discounted forward prices have bubbles whose magnitudes are related to the asset's price bubble, (iii) with no dividends, American call options may be exercised early, (iv) European put-call parity in market prices must always hold, regardless of bubbles, and (v) futures price bubbles can exist and they are independent of the underlying asset's price bubble. Many of these results stand in contrast to those of the classical theory. We propose, but do not implement, some new tests for the existence of asset price bubbles using derivative securities.

Evaluation of Derivative Security Prices in the Heath-Jarrow-Morton Framework as Path Integrals using Fast Fourier Transform Techniques

This paper considers the evaluation of derivative security prices within the Heath-Jarrow-Morton framework of stochastic interest rates, such as bond options. Within this framework, the stochastic dynamics driving prices are in general non-Markovian. Hence, in principle the partial differential equations governing prices require an infinite dimensinal state space. We discuss a class of forward rate volatility functions which allow the stochastic dynamics to be expressed in Markovian form and hence obtain a finite dimensional state space for the partial differential equations governing prices. By applying to the Markovian form, the transformed suggested by Eydeland (1994), the pricing problem can be set up as a path integral in function space. These integrals are evaluated using fast fourier transform techniques. We apply the technique to the pricing of American bond options and compare the computational time with other methods currently employed such as the method of lines and more traditional partial differential equation solution techniques.

Implied derivative security prices based two-factor interest model: a UK application

In this paper the extended Box Method recently introduced to finance is used to value bond and option prices based on the two-factor CKLS interest rate model. The two-factor CKLS model is estimated using the one-year Eurodollar rate for the UK as the long rate and either the one-week, or one-month Euro dollar rate for the UK as the short rate. Overall, it is found that both and option prices are sensitive to the model used.

Satisfying convex risk limits by trading

A random variable, representing the final position of a trading strategy, is deemed acceptable if under each of a variety of probability measures its expectation dominates a floor associated with the measure. The set of random variables representing pre-final positions from which it is possible to trade to final acceptability is characterized. In particular, the set of initial capitals from which one can trade to final acceptability is shown to be a closed half-line [ (0),1). Methods for computing (0) are provided, and the application of these ideas to derivative security pricing is developed.

Jackknifing Bond Option Prices

In continuous time specifications, the prices of interest rate derivative securities depend crucially on the mean reversion parameter of the associated interest rate diffusion equation. This parameter is well known to be subject to estimation bias when standard methods like maximum likelihood (ML) are used. The estimation bias can be substantial even in very large samples and it translates into a bias in pricing bond options and other derivative securities that is important in practical work. The present paper proposes a very general method of bias reduction for pricing bond options that is based on Quenouille's (1956) jackknife. We show how the method can be applied directly to the options price itself as well as the coefficients in continuous time models. The method is implemented and evaluated here in the Cox, Ingersoll and Ross (1985) model, although it has much wider applicability. A Monte Carlo study shows that the proposed procedure achieves substantial bias reductions in pricing bond options with only mild increases in variance that do not compromise the overall gains in mean squared error. Our findings indicate that bias correction in estimation of the drift can be more important in pricing bond options than correct specification of the diffusion. Thus, even if ML or approximate ML can be used to estimate more complicated models, it still appears to be of equal or greater importance to correct for the effects on pricing bond options of bias in the estimation of the drift. An empirical application to U.S. interest rates highlights the differences between bond and option prices implied by the jackknife procedure and those implied by the standard approach. These differences are large and suggest that bias reduction in pricing options is important in practical applications.

Mixture of models: A simple recipe for a … Hangover?

The idea of using a of derivative security prices computed using di?erent (the so-called mixture of models, or ensemble of models, approach) has been put forth recently by a number of authors. Some view it as a simple way to add stochastic volatility to virtually any model, and others advocate it on the grounds that it provides a simple and tractable method for capturing certain market characteristics, most importantly volatility smile. Ease of calibration to market prices of vanilla and exotic instruments is also cited as the approach's redeeming quality. While not disputing the fact that such are easy to calibrate, we explain that these are under-specified (leading to multiple possible prices of derivatives). We also demonstrate that the weighted average valuation formula, the main selling point of the mixture of models approach, is self-inconsistent and cannot be used for valuation.

Applications of δ-function perturbation to the pricing of derivative securities

In the recent econophysics literature, the use of functional integrals is widespread for the calculation of option prices. In this paper, we extend this approach in several directions by means of δ-function perturbations. First, we show that results about infinitely repulsive δ-function are applicable to the pricing of barrier options. We also introduce functional integrals over skew paths that give rise to a new European option formula when combined with δ-function potential. We propose accurate closed-form approximations based on the theory of comonotonic risks in case the functional integrals are not analytically computable.