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European Options (european + option)

Distribution by Scientific Domains

Business, Economics, Finance and Accounting	85%

Selected Abstracts

MSM Estimators of European Options on Assets with Jumps

MATHEMATICAL FINANCE, Issue 2 2001
João Amaro de Matos
This paper shows that, under some regularity conditions, the method of simulated moments estimator of European option pricing models developed by Bossaerts and Hillion (1993) can be extended to the case where the prices of the underlying asset follow Lévy processes, which allow for jumps, with no losses on their asymptotic properties, still allowing for the joint test of the model. [source]

A Dynamic Investment Model with Control on the Portfolio's Worst Case Outcome

MATHEMATICAL FINANCE, Issue 4 2003
Yonggan Zhao
This paper considers a portfolio problem with control on downside losses. Incorporating the worst-case portfolio outcome in the objective function, the optimal policy is equivalent to the hedging portfolio of a European option on a dynamic mutual fund that can be replicated by market primary assets. Applying the Black-Scholes formula, a closed-form solution is obtained when the utility function is HARA and asset prices follow a multivariate geometric Brownian motion. The analysis provides a useful method of converting an investment problem to an option pricing model. [source]

Numerical valuation of options under Kou's model

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2007
Jari ToivanenArticle first published online: 6 AUG 200
Numerical methods are developed for pricing European and American options under Kou's jump-diffusion model which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a European option is given by a partial integro-differential equation (PIDE) while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy to implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. Also for pricing American options similar iterations can be employed. A numerical experiment demonstrates that the described method is very efficient as accurate option prices can be computed in a few milliseconds on a PC. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

The valuation of European options when asset returns are autocorrelated

THE JOURNAL OF FUTURES MARKETS, Issue 1 2006
Szu-Lang Liao
This article derives the closed-form formula for a European option on an asset with returns following a continuous-time type of first-order moving average process, which is called an MA(1)-type option. The pricing formula of these options is similar to that of Black and Scholes, except for the total volatility input. Specifically, the total volatility input of MA(1)-type options is the conditional standard deviation of continuous-compounded returns over the option's remaining life, whereas the total volatility input of Black and Scholes is indeed the diffusion coefficient of a geometric Brownian motion times the square root of an option's time to maturity. Based on the result of numerical analyses, the impact of autocorrelation induced by the MA(1)-type process is significant to option values even when the autocorrelation between asset returns is weak. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:85,102, 2006 [source]

HEDGING STRATEGIES AND MINIMAL VARIANCE PORTFOLIOS FOR EUROPEAN AND EXOTIC OPTIONS IN A LÉVY MARKET

MATHEMATICAL FINANCE, Issue 4 2010
Wing Yan Yip
This paper presents hedging strategies for European and exotic options in a Lévy market. By applying Taylor's theorem, dynamic hedging portfolios are constructed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importance as such other derivatives might be readily available. Moment swaps or power jump assets are not typically liquidly traded. It is shown how minimal variance portfolios can be used to hedge the higher order terms in a Taylor expansion of the pricing function, investing only in a risk-free bank account, the underlying asset, and potentially variance swaps. The numerical algorithms and performance of the hedging strategies are presented, showing the practical utility of the derived results. [source]

THE DEPENDENCE STRUCTURE OF RUNNING MAXIMA AND MINIMA: RESULTS AND OPTION PRICING APPLICATIONS

MATHEMATICAL FINANCE, Issue 1 2010
Umberto Cherubini
We provide general results for the dependence structure of running maxima (minima) of sets of variables in a model based on (i) Markov dynamics; (ii) no Granger causality; (iii) cross-section dependence. At the time series level, we derive recursive formulas for running minima and maxima. These formulas enable to use a "bootstrapping" technique to recursively recover the pricing kernels of barrier options from those of the corresponding European options. We also show that the dependence formulas for running maxima (minima) are completely defined from the copula function representing dependence among levels at the terminal date. The result is applied to multivariate discrete barrier digital products. Barrier Altiplanos are simply priced by (i) bootstrapping the price of univariate barrier products; (ii) evaluating a European Altiplano with these values. [source]

OPTIMAL RISK SHARING FOR LAW INVARIANT MONETARY UTILITY FUNCTIONS

MATHEMATICAL FINANCE, Issue 2 2008
E. Jouini
We consider the problem of optimal risk sharing of some given total risk between two economic agents characterized by law-invariant monetary utility functions or equivalently, law-invariant risk measures. We first prove existence of an optimal risk sharing allocation which is in addition increasing in terms of the total risk. We next provide an explicit characterization in the case where both agents' utility functions are comonotone. The general form of the optimal contracts turns out to be given by a sum of options (stop-loss contracts, in the language of insurance) on the total risk. In order to show the robustness of this type of contracts to more general utility functions, we introduce a new notion of strict risk aversion conditionally on lower tail events, which is typically satisfied by the semi-deviation and the entropic risk measures. Then, in the context of an AV@R-agent facing an agent with strict monotone preferences and exhibiting strict risk aversion conditional on lower tail events, we prove that optimal contracts again are European options on the total risk. [source]

TERM STRUCTURES OF IMPLIED VOLATILITIES: ABSENCE OF ARBITRAGE AND EXISTENCE RESULTS

MATHEMATICAL FINANCE, Issue 1 2008
Martin Schweizer
This paper studies modeling and existence issues for market models of stochastic implied volatility in a continuous-time framework with one stock, one bank account, and a family of European options for all maturities with a fixed payoff function h. We first characterize absence of arbitrage in terms of drift conditions for the forward implied volatilities corresponding to a general convex h. For the resulting infinite system of SDEs for the stock and all the forward implied volatilities, we then study the question of solvability and provide sufficient conditions for existence and uniqueness of a solution. We do this for two examples of h, namely, calls with a fixed strike and a fixed power of the terminal stock price, and we give explicit examples of volatility coefficients satisfying the required assumptions. [source]

OPTIONS AND EFFICIENCY IN MULTIDATE SECURITY MARKETS

MATHEMATICAL FINANCE, Issue 4 2005
Alexandre M. Baptista
This paper extends the work of Ross (1976; Q. J. Econ. (90)1, 75,89) to multidate security markets. First, we show that if a primitive security separates states at the terminal date, then there exist multiperiod European options on that security generating dynamically complete markets. Second, we show that if a primitive security conditionally separates states at the terminal date, then there exist multiperiod European options on that security generating generically dynamically complete markets provided that certain conditions hold. Third, we show that there are economies for which the minimum number of multiperiod European options on a primitive security generating generically dynamically complete markets is relatively large. Finally, we show that in these economies, a relatively small number of multiperiod European options on possibly different portfolio strategies of primitive securities generates generically dynamically complete markets. [source]

Numerical valuation of options under Kou's model

Pricing American options by canonical least-squares Monte Carlo

THE JOURNAL OF FUTURES MARKETS, Issue 2 2010
Qiang Liu
Options pricing and hedging under canonical valuation have recently been demonstrated to be quite effective, but unfortunately are only applicable to European options. This study proposes an approach called canonical least-squares Monte Carlo (CLM) to price American options. CLM proceeds in three stages. First, given a set of historical gross returns (or price ratios) of the underlying asset for a chosen time interval, a discrete risk-neutral distribution is obtained via the canonical approach. Second, from this canonical distribution independent random samples of gross returns are taken to simulate future price paths for the underlying. Third, to those paths the least-squares Monte Carlo algorithm is then applied to obtain early exercise strategies for American options. Numerical results from simulation-generated gross returns under geometric Brownian motions show that the proposed method yields reasonably accurate prices for American puts. The CLM method turns out to be quite similar to the nonparametric approach of Alcock and Carmichael and simulations done with CLM provide additional support for their recent findings. CLM can therefore be viewed as an alternative for pricing American options, and perhaps could even be utilized in cases when the nature of the underlying process is not known. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:175,187, 2010 [source]

Black-Scholes-Merton revisited under stochastic dividend yields

THE JOURNAL OF FUTURES MARKETS, Issue 7 2006
Abraham LiouiArticle first published online: 9 MAY 200
European options are priced in a framework à la Black-Scholes-Merton, which is extended to incorporate stochastic dividend yield under a stochastic mean,reverting market price of risk. Explicit formulas are obtained for call and put prices and their Greek parameters. Some well-known properties of the Black-Scholes-Merton formula fail to hold in this setting. For example, the delta of the call can be negative and even greater than one in absolute terms. Moreover, call prices can be a decreasing function of the underlying volatility although the latter is constant. Finally, and most importantly, option prices highly depend on the features of the market price of risk, which does not need to be specified at all in the standard Black-Scholes-Merton setting. The results are simulated in order to assess the economic impact of assuming that the dividend yield is deterministic when it is actually stochastic, as well as to assess the economic importance of the features of the market price of risk. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:703,732, 2006 [source]

The valuation of European options when asset returns are autocorrelated

A two-mean reverting-factor model of the term structure of interest rates

THE JOURNAL OF FUTURES MARKETS, Issue 11 2003
Manuel Moreno
This article presents a two-factor model of the term structure of interest rates. It is assumed that default-free discount bond prices are determined by the time to maturity and two factors, the long-term interest rate, and the spread (i.e., the difference) between the short-term (instantaneous) risk-free rate of interest and the long-term rate. Assuming that both factors follow a joint Ornstein-Uhlenbeck process, a general bond pricing equation is derived. Closed-form expressions for prices of bonds and interest rate derivatives are obtained. The analytical formula for derivatives is applied to price European options on discount bonds and more complex types of options. Finally, empirical evidence of the model's performance in comparison with an alternative two-factor (Vasicek-CIR) model is presented. The findings show that both models exhibit a similar behavior for the shortest maturities. However, importantly, the results demonstrate that modeling the volatility in the long-term rate process can help to fit the observed data, and can improve the prediction of the future movements in medium- and long-term interest rates. So it is not so clear which is the best model to be used. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23: 1075,1105, 2003 [source]