From Time Series of Returns to Option Prices: The Stochastic Discount Factor Approach

This paper establishes the existence of a solution to the optimization problem. Supposing that risk assets pay continuous dividend regarded as the function of time. It is established that the behaviour model of the stock pricing process is jump-diffusion driven by a count process. We give a characterization of the optimal portfolio by means of the value function and the equivalent martingale measure defined by the utility function. The unique equivalent martingale measure, the unique optimal consumption and portfolio pair and the corresponding wealth process are deduced. We provide a simple characterization of an equilibrium market and discuss existence and uniqueness of equilibrium in the economy.

ABSTRACTThis article considers modelling nonnormality in return with stable Paretian (SP) innovations in generalized autoregressive conditional heteroskedasticity (GARCH), exponential generalized autoregressive conditional heteroskedasticity (EGARCH) and Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroskedasticity (GJR-GARCH) volatility dynamics. The forecasted volatilities from these dynamics have been used as a proxy to the volatility parameter of the Black–Scholes (BS) model. The performance of these proxy-BS models has been compared with the performance of the BS model of constant volatility. Using a cross section of S&P500 options data, we find that EGARCH volatility forecast with SP innovations is an excellent proxy to BS constant volatility in terms of pricing. We find improved performance of hedging for an illustrative option portfolio. We also find better performance of spectral risk measure (SRM) than value-at-risk (VaR) and expected shortfall (ES) in estimating option portfolio risk in case of the proxy-BS models under SP innovations.Abbreviation: generalized autoregressive conditional heteroskedasticity (GARCH), exponential generalized autoregressive conditional heteroskedasticity (EGARCH) and Glosten-Jagannathan-Runkle generalized autoregressive conditional heteroskedasticity (GJR-GARCH)

This paper study the problem of wealth optimization.It is established that the behavior model of the stock pricing process is jump-diffusion driven by a count process and stochastic volatility. Supposing that risk assets pay continuous dividend regarded as the function of time. It is proved that the existence of an optimal portfolio and unique equivalent martingale measure by stochastic analysis methods. The unique equivalent martingale measure ,the optimal wealth process, the value function and the optimal portfolio are deduced. Key words: Jump-Diffusion process; Stochastic volatility; Dividends; Incomplete financial market; Wealth optimization

This paper studies the problem of consumption optimization and equilibrium in discontinuous time financial markets. It is established that the behavior model of the stock pricing process is jump-diffusion driven by a count process. It is proved that the existence of unique optimal consumption and portfolio pair and unique equivalent martingale measure by stochastic analysis methods. The unique equivalent martingale measure, the unique optimal consumption and portfolio pair and the corresponding wealth process are deduced. Finally we provide a simple characterization of an equilibrium market.

This paper study the problem of the portfolio selection. It is established that the behavior model of the stock pricing process is jump-diffusion driven by a count process and stochastic volatility. Supposing that risk assets pay continuous dividend regarded as the function of time. It is proved that the existence of an optimal portfolio and unique equivalent martingale measure by stochastic analysis methods. The unique equivalent martingale measure, the optimal wealth process, the value function and the optimal portfolio are deduced.

Pricing American options when the underlying asset follows GARCH processes

An Empirical Study on the Stock Price Reaction to Earnings Announcements using the Stochastic Discount Factor Approach

The analysis and interpretation of risk play a crucial role in different areas of modern finance. This includes pricing of financial products, capital allocation and derivation of economic capital. Key to this analysis is the quantification of the risk via risk measures. A promising approach is to define risk measures by a set of desirable properties. This leads to the main topic of this research; the characterization of so called convex risk measures. First, we review the concept of convex risk measures on Lebesgue spaces and provide a structural basis for the following parts of the thesis by stating and proving the different characterization results and adjusting them to our definitions and notation. A key result of this thesis is the characterization of linear combinations and convolutions of convex risk measures. We study different dual correspondences, which are induced by the Fenchel-Legendre transform. In our case we investigated three different duality correspondences, which are sum and inf-convolution, difference and deconvolution and multiplication of scalars and epi-multiplication. These results are used to characterize linear combinations and convolutions of convex risk measures. We investigate when certain combinations of risk measures belong to the class of convex risk measures and investigate the basic properties. Furthermore, two applications based on theoretical results of the first part of the thesisare derived. In the first application we study the pricing and hedging problem for contingent claims in an incomplete market as a trade-off between a trader and a regulator. In our model the regulator allows the trader to take some risk, but insists that the residual risk, which is not hedged away, has to be covered. To achieve this, the regulator introduces an extra bank account, which serves as a capital reserve to cover for eventual losses of the trader, and is dependent on the risk of the trader’s portfolio. The risk attitudes of the trader and the regulator are reflected by different risk measures. We derive risk measure price and the risk indifference price. In both cases, the resulting risk measure is given by a weighted sum of the regulator’s and trader’s risk measures. This new operator is also a convex risk measure as we have proven in the first part of the thesis. In the second application we consider the problem of partial hedging of a contingent claim. Under the assumption of a complete market, it is always possible to replicate the claim. In this case, the claim can be priced using the unique equivalent martingale measure. The question is of a different nature when the initial capital is less than the expectation under the equivalent martingale measure. Under this condition we derive a suitable hedging strategy such that the risk of the difference of the hedging portfolio and the claim is minimized. As risk measure we consider Average Value at Risk and a simple spectral risk measure. We discovered that overhedging may arise. Nevertheless this happens only in special situations, for example, when the level of Average Value at Risk is high or the initial capital is close to the value which is required for a perfect hedging strategy. The results are illustrated by solving the problem for a call and a put option in the Black-Scholes model

This paper constitutes the first empirical investigation of the Canadian government bond options traded at the Montreal Exchange. Thanks to the availability of transactions data, this market offers an opportunity unique in North America for empirical testing of bond option pricing models. We investigated the validity of a variant of the Schaefer and Schwartz (1987) time‐dependent variance model between May 1986 and December 1988, proposing and testing a closed‐form approximation of the model, which performed very well. We revealed the existence of pricing biases which are function of the options' maturity, exercise price, and underlying bond duration. The maturity bias only affected options with fifteen days or less to maturity and was consistent with the presence of jumps in the yield process. Such jumps are not allowed in the context of our model. We also reported a volatility smile along the exercise price dimension, which has been widely documented in the equity options literature. Options on bonds of short duration were shown to be overpriced, relative to options on long duration bonds. This may be seen as evidence that bond yields are mean reverting. We demonstrated that the sources of bias documented in this study tended to manifest themselves when the options were highly illiquid, and that they affected only a small fraction of our sample. Abstracting from transactions costs, simulated hedging strategies designed to take advantage of deviations from model prices were on average profitable.RésuméCet article constitue la première étude empirique portant sur les options sur obligations du gouvernement du Canada transigées à la Bourse de Montréal. En raison de l'existence d'une base de données transactionnelles, ce marchéoffre une opportunitéunique en Amérique du nord pour tester des modèles d'évaluation d'options sur obligations. Cette étude examine la validitéd'une variante du modèle à dépendance temporelle de Schaefer et Schwartz (1987) pour la période allant de mai 1986 à décembre 1988. Nous proposons et testons une formule approximative de ce modèle qui affiche une très bonne performance. Nous révélons l'existence de biais dans ce marchéqui sont fonction de la maturitéet du prix de levée des options, ainsi que de la durée des obligations sous‐jacentes. Le biais relatif à la maturitén'affecte que les options de quinze jours avant échéance ou moins et pourrait ětre causépar la présence de soubresauts dans le processus gouvernant les rendements. De tels soubresauts ne sont pas admissibles dans le contexte de notre modèle. Nous rapportons également l'existence d'un sourire de volatilitéqui a étélargement documentédans la littérature portant sur le marchédes options sur actions ordinaires. Les options sur obligations de courte durée s'avèrent sur‐évaluées relativement aux options sur obligations de longue durée. Cela pourrait signifier la présence de “mean reversion” dans les rendements. Nous démontrons que les sources de biais identifiées dans la présente étude se manifestent principalement lorsque les options sont très peu traitées et que celles‐ci n'affectent qu'une petite partie de notre échantillon. Hormis les coǔts de transactions, nos simulations d'opérations de contrepartie visant à tirer profit des opportunités d'arbitrage identifiées par le modèle se sont avérées, en moyenne, profitables.

We consider the problem of pricing contingent claims on a stock whose price process is modelled by a geometric Levy process, in exact analogy with the ubiquitous geometric Brownian motion model. Because the noise process has jumps of random sizes, such a market is incomplete and there is not a unique equivalent martingale measure. We study several approaches to pricing options which all make use of an equivalent martingale measure that is in different respects closest to the underlying canonical measure, the main ones being the Follmer-Schweizer minimal measure and the martingale measure which has minimum relative entropy with respect to the canonical measure. It is shown that the minimum relative entropy measure is that constructed via the Esscher transform, while the Follmer-Schweizer measure corresponds to another natural analogue of the classical Black-Scholes measure.

Portfolio Efficiency and Discount Factor Bounds with Conditioning Information: An Empirical Study

Investment Opportunities in Central and Eastern European Equity Markets

A market is considered where trading can take place only at discrete time points, the trading frequency cannot grow without bound, and the number of states of nature is finite. The main objectives of the paper are to show that the market can be completed also with highly correlated risky assets, and to describe an efficient algorithm to compute a self-financing hedging strategy. The algorithm consists off-line of a backwards recursion and on-line of the solution, in each period, of a system of linear equations; it is a consequence of a proof where, using a well-known mathematical property, it is shown that uniqueness of the martingale measure implies completeness also in our setting. The significance of ‘multistate’ models versus the familiar binomial model is discussed and it is shown how the evolution of prices of the (correlated) risky assets can be chosen so that a given probability measure is already the unique equivalent martingale measure.

In this paper, volatility is estimated and then forecast using unobserved components‐realized volatility (UC‐RV) models as well as constant volatility and GARCH models. With the objective of forecasting medium‐term horizon volatility, various prediction methods are employed: multi‐period prediction, variable sampling intervals and scaling. The optimality of these methods is compared in terms of their forecasting performance. To this end, several UC‐RV models are presented and then calibrated using the Kalman filter. Validation is based on the standard errors on the parameter estimates and a comparison with other models employed in the literature such as constant volatility and GARCH models. Although we have volatility forecasting for the computation of Value‐at‐Risk in mind the methodology presented has wider applications. This investigation into practical volatility forecasting complements the substantial body of work on realized volatility‐based modelling in business. Copyright © 2007 John Wiley & Sons, Ltd.

Volatility plays an important role in the field of financial econometrics as one of the risk indicators. Many various models address the problem of modeling the volatilities of financial asset returns. This study provides a new empirical performance comparison of the four different GARCH-type models, namely GARCH, GARCH-M, GJR-GARCH, and log-GARCH models based on simulated data and real data such as the DJIA, S&P 500, and S&P CNX Nifty indices on a daily period from January 2000 to December 2017. We also investigate the estimation results obtained using Solver’Excel and verify those results against the results obtained using a Markov chain Monte Carlo method. The simulation study showed that the GARCH model is outperformed by other models. Meanwhile, the empirical study provides evidence that the GJR-GARCH model provides the best fitting, followed by the GARCH-M, GARCH, and log-GARCH models. Furthermore, this study recommends the use of Excel’s Solver in practice when the parameter estimates for GARCH-type model do not close to zero.