Pricing Of Derivative Securities Research Articles

On asset pricing in a binomial model with fixed and proportional transaction costs, portfolio constraints and dividends

AbstractWe extend the classical binomial model proposed by Cox, Ross, and Rubinstein for derivative security pricing to encompass both fixed and proportional transaction costs, portfolio constraints including margin requirements, and dividend-paying assets. Our focus is on studying option hedging within this enriched framework. Initially, we establish the existence of a hedging strategy in this context. Subsequently, we determine the optimal hedging strategy and its associated initial cost by decomposing the problem into a sequence of hedging problems. To illustrate our approach, we present a numerical example within a 3-period binomial model.

Uncertainty, Spillovers, and Forecasts of the Realized Variance of Gold Returns

Using data for the group of G7 countries and China for the sample period 1996Q1 to 2020Q4, we study the role of uncertainty and spillovers for the out-of-sample forecasting of the realized variance of gold returns and its upside (good) and downside (bad) counterparts. We go beyond earlier research in that we do not focus exclusively on U.S.-based measures of uncertainty, and in that we account for international spillovers of uncertainty. Our results, based on the Lasso estimator, show that, across the various model configurations that we study, uncertainty has a more systematic effect on out-of-sample forecast accuracy than spillovers. Our results have important implications for investors in terms of, for example, pricing of related derivative securities and the development of portfolio-allocation strategies.

Efficient Markets and Contingent Claims Valuation: An Information Theoretic Approach.

This research article shows how the pricing of derivative securities can be seen from the context of stochastic optimal control theory and information theory. The financial market is seen as an information processing system, which optimizes an information functional. An optimization problem is constructed, for which the linearized Hamilton–Jacobi–Bellman equation is the Black–Scholes pricing equation for financial derivatives. The model suggests that one can define a reasonable Hamiltonian for the financial market, which results in an optimal transport equation for the market drift. It is shown that in such a framework, which supports Black–Scholes pricing, the market drift obeys a backwards Burgers equation and that the market reaches a thermodynamical equilibrium, which minimizes the free energy and maximizes entropy.

Trading with small nonlinear price impact

We study portfolio choice with small nonlinear price impact on general market dynamics. Using probabilistic techniques and convex duality, we show that the asymptotic optimum can be described explicitly up to the solution of a nonlinear ODE, which identifies the optimal trading speed and the performance loss due to the trading friction. Previous asymptotic results for proportional and quadratic trading costs are obtained as limiting cases. As an illustration, we discuss how nonlinear trading costs affect the pricing and hedging of derivative securities and active portfolio management.

The term structure of volatility predictability

Volatility forecasting is crucial for portfolio management, risk management, and pricing of derivative securities. Still, little is known about the accuracy of volatility forecasts and the horizon of volatility predictability. This paper aims to fill these gaps in the literature. We begin this paper by introducing the notions of spot and forward predicted volatilities and propose describing the term structure of volatility predictability by spot and forward forecast accuracy curves. Then, we perform a comprehensive study of the term structure of volatility predictability in stock and foreign exchange markets. Our results quantify the volatility forecast accuracy across horizons in two major markets and suggest that the horizon of volatility predictability is significantly longer than that reported in earlier studies. Nevertheless, the aforesaid horizon is observed to be much shorter than the longest maturity of traded derivative contracts.

Pointwise Arbitrage Pricing Theory in Discrete Time

We develop a robust framework for pricing and hedging of derivative securities in discrete-time financial markets. We consider markets with both dynamically and statically traded assets and make minimal measurability assumptions. We obtain abstract (pointwise) fundamental theorem of asset pricing and pricing–hedging duality. Our results are general and, in particular, cover both the so-called model independent case as well as the classical probabilistic case of Dalang–Morton–Willinger. Our analysis is scenario-based: a model specification is equivalent to a choice of scenarios to be considered. The choice can vary between all scenarios and the set of scenarios charged by a given probability measure. In this way, our framework interpolates between a model with universally acceptable broad assumptions and a model based on a specific probabilistic view of future asset dynamics.

The Term Structure of Volatility Predictability

Volatility forecasting is crucial for portfolio management, risk management, and pricing of derivative securities. Still, little is known about the accuracy of volatility forecasts and the horizon of volatility predictability. This paper aims to fill these gaps in the literature. We begin this paper by introducing the notions of the spot and forward predicted volatilities and propose to describe the term structure of volatility predictability by the spot and forward forecast accuracy curves. Then we perform a comprehensive study on the term structure of volatility predictability in the stock and foreign exchange markets. Our results quantify the volatility forecast accuracy across horizons in the two major markets and suggest that the horizon of volatility predictability is significantly longer than that reported in the earlier studies. Nevertheless, the horizon of volatility predictability is found to be much shorter than the longest maturity of traded derivative contracts.

The Limits to Volatility Predictability: Quantifying Forecast Accuracy Across Horizons

Volatility forecasting is crucial for portfolio management, risk management, and pricing of derivative securities. Still, little is known about how far ahead one can forecast volatility. First, in this paper we introduce the notions of the spot and forward predicted volatilities and propose to describe the term structure of volatility predictability by the spot and forward forecast accuracy curves. Then, by employing a few popular time-series volatility models, we perform a comprehensive empirical study on the horizon of volatility predictability. Our results suggest that, whereas the spot volatility can be predicted over horizons that extend to 35 weeks, the horizon of the forward volatility predictability is rather short and limited to approximately 7.5 weeks. Finally, we suggest a plausible explanation for why standard models fail to provide sensible longer-horizon volatility forecasts.

Taylor expansion for derivative securities pricing as a precondition for strategic market decisions

The strategy of managing the pricing processes, in particular managing the dynamics of the price of the underlying asset and its volatility, the prices of indices, shares, options, the magnitude of financial flows, in the method of calculating the company’s rating based on the available quotations of securities, is developed. The article deals with the study of pricing and calculating the volatility of European options with general local-stochastic volatility, applying Taylor series methods for degenerate diffusion processes. The application of this idea requires new approaches caused by degradation difficulties. Price approximation is obtained by solving the Cauchy problem of partial differential equations diffusion with inertia, and the volatility approximation is completely explicit, that is, it does not require special functions. If the payoff of options is a function of only x, then the Taylor series expansion does not depend on t and an analytical expression of the fundamental solution is considerably simplified. Applied an approach to the pricing of derivative securities on the basis of classical Taylor series expansion, when the stochastic process is described by the diffusion equation with inertia (degenerate parabolic equation). Thus, the approximate value of options can be calculated as effectively as the Black-Scholes pricing of derivative securities. On the basis of the solved problem, it is possible to calculate their turns step-by-step. This enables to predict the dynamics of the pricing of derivatives and to create a strategy of behavior at options according to the passage of the process. For each approximation, price volatility is calculated, which makes it possible to take into account all changes in the market and to calculate possible situations. The step-by-step finding of the change in yield and volatility in the relevant analysis enables us to make informed strategic decisions by traders in the financial markets.

A Comparative Study on Barrier Option Pricing using Antithetic and Quasi Monte-Carlo Simulations

Monte-Carlo simulations have been utilized greatly in the pricing of derivative securities. Over the years, several variance reduction techniques have been developed to curb the instability, as well as, increase the simulation e?ciencies of the Monte-Carlo methods. Our approach in this research work will consider the use of antithetic variate techniques to estimate the fair prices of barrier options. Next, we use the quasi-Monte Carlo method, together with Sobol sequence to estimate the values of the same option. An extended version of the Black-Scholes model will serve as basis for the exact prices of these exotic options. The resulting simulated prices will be compared to the exact prices. The research concludes by showing some results which proves that when random numbers are generated via low discrepancy sequences in contrast to the normal pseudo-random numbers, a more efficient simulation method is ensued. This is further applicable in pricing complex derivatives without closed formsolutions.

Modelling Crypto-Currencies Financial Time-Series

This paper studies the behaviour of crypto currencies financial time-series of which Bitcoin is the most prominent example. The dynamic of those series is quite complex displaying extreme observations, asymmetries, and several nonlinear characteristics which are difficult to model. We develop a new dynamic model able to account for long-memory and asymmetries in the volatility process as well as for the presence of time-varying skewness and kurtosis. The empirical application, carried out on 606 crypto currencies, indicates that a robust filter for the volatility of crypto currencies is strongly required. Forecasting results show that the inclusion of time{varying skewness systematically improves volatility, density, and quantile predictions at different horizons. Going forward, as this new and unexplored market will develop, our results will be important for asset allocation, risk management, and pricing of derivative securities.

Efficient Computation of Option Prices and Greeks by Quasi--Monte Carlo Method with Smoothing and Dimension Reduction

Discontinuities and high dimensionality are common in the problems of pricing and hedging of derivative securities. Both factors have a tremendous impact on the accuracy of the quasi--Monte Carlo (QMC) method. An ideal approach to improve the QMC method is to transform the functions to make them smoother and having smaller effective dimension. This paper develops a two-step procedure to tackle the challenging problems of both the discontinuity and the high dimensionality concurrently. In the first step, we adopt the smoothing method to remove the discontinuities of the payoff function, improving the smoothness. In the second step, we propose a general dimension reduction method (called the CQR method) to reduce the effective dimension such that the better quality of QMC points in their initial dimensions can be fully exploited. The CQR method is based on the combination of the $k$-means clustering algorithm and the QR decomposition. The $k$-means clustering algorithm, a classical algorithm of machine lear...

Hedging with small uncertainty aversion

We study the pricing and hedging of derivative securities with uncertainty about the volatility of the underlying asset. Rather than taking all models from a prespecified class equally seriously, we penalise less plausible ones based on their “distance” to a reference local volatility model. In the limit for small uncertainty aversion, this leads to explicit formulas for prices and hedging strategies in terms of the security’s cash gamma.

Credit risk and contagion via self-exciting default intensity

Recent empirical evidences indicate that default rates are influenced not only by the observable or latent risk factors, but also depend on the history of past defaults. Motivated by this empirical finding, we consider in this paper a reduced-form, intensity-based credit risk model, which allows for both frailty and default contagion, using a so-called “self-exciting” intensity, in the sense that the default intensity varies not only with the risk factors, but also depends on the previous default history of all the firms. With “self-exciting” default intensity, we are able to obtain closed-form expressions for the pricing of credit derivative securities in our model. The estimation of parameters using the EM algorithm is considered as well.

Hedging with Small Uncertainty Aversion

We study the pricing and hedging of derivative securities with uncertainty about the volatility of the underlying asset. Rather than taking all models from a prespecified class equally seriously, we penalise less plausible ones based on their distance to a reference local volatility model. In the limit for small uncertainty aversion, this leads to explicit formulas for prices and hedging strategies in terms of the security’s cash gamma.

Modelling the volatility of the Dow Jones Islamic Market World Index using a fractionally integrated time-varying GARCH (FITVGARCH) model

Appropriate modelling of the process of volatility has implications for portfolio selection, the pricing of derivative securities and risk management. Further, a large body of research has suggested that both long memory and structural changes simultaneously characterize the structure of financial returns volatility. Given this, in this article, we aim to model conditional volatility of the returns of the Dow Jones Islamic Market World Index (DJIM), interest on which has come to the fore following the need for renovation of the conventional financial system, in the wake of the recent global financial crisis. To model the conditional volatility of the DJIM returns, accounting for both long memory and structural changes, we allow the parameters in the conditional variance equation of the fractionally integrated generalized autoregressive conditional heteroscedasticity (FIGARCH) model to be time dependent, such that the parameters evolve smoothly over time based on a logistic smooth transition function, yiel...

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 S&P 500 Index and examines their impact on this index's rate of return and volatility. It focuses on deriving analytic European option prices under the assumption that the dynamics of stock prices follow a threshold model; 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. © 2013 Wiley Periodicals, Inc. Jrl Fut Mark 35:52–74, 2015

Game theoretic analysis of incomplete markets: emergence of probabilities, nonlinear and fractional Black–Scholes equations

Expanding the ideas of the author's paper 'Nonexpansive maps and option pricing theory' (Kibernetica 34:6 (1998), 713-724) we develop a pure game-theoretic approach to option pricing, by-passing stochastic modeling. Risk neutral probabilities emerge automatically from the robust control evaluation. This approach seems to be especially appealing for incomplete markets encompassing extensive, so to say untamed, randomness, when the coexistence of infinite number of risk neutral measures precludes one from unified pricing of derivative securities. Our method is robust enough to be able to accommodate various markets rules and settings including path dependent payoffs, American options and transaction costs. On the other hand, it leads to rather simple numerical algorithms. Continuous time limit is described by nonlinear and/or fractional Black-Scholes type equations.

Incomplete‐Market Equilibria Solved Recursively on an Event Tree

ABSTRACTBecause of non‐traded human capital, real‐world financial markets are massively incomplete, while the modeling of imperfect, dynamic financial markets remains a wide‐open and difficult field. Some 30 years after Cox, Ross, and Rubinstein (1979) taught us how to calculate the prices of derivative securities on an event tree by simple backward induction, we show how a similar formulation can be used in computing heterogeneous‐agents incomplete‐market equilibrium prices of primitive securities. Extant methods work forward and backward, requiring a guess of the way investors forecast the future. In our method, the future is part of the current solution of each backward time step.

Functional dynamic factor models with application to yield curve forecasting

Accurate forecasting of zero coupon bond yields for a continuum of maturities is paramount to bond portfolio management and derivative security pricing. Yet a universal model for yield curve forecasting has been elusive, and prior attempts often resulted in a trade-off between goodness of fit and consistency with economic theory. To address this, herein we propose a novel formulation which connects the dynamic factor model (DFM) framework with concepts from functional data analysis: a DFM with functional factor loading curves. This results in a model capable of forecasting functional time series. Further, in the yield curve context we show that the model retains economic interpretation. Model estimation is achieved through an expectation-maximization algorithm, where the time series parameters and factor loading curves are simultaneously estimated in a single step. Efficient computing is implemented and a data-driven smoothing parameter is nicely incorporated. We show that our model performs very well on forecasting actual yield data compared with existing approaches, especially in regard to profit-based assessment for an innovative trading exercise. We further illustrate the viability of our model to applications outside of yield forecasting.