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We extend the one-factor stochastic volatility model to incorporate super-linearly growing coefficient terms with a Markov-switching framework. Since the proposed model is intractable analytically, we develop various mathematical techniques to investigate the convergence in probability of the numerical solutions to the true solution under the local Lipschitz condition. Finally, we perform simulation examples to demonstrate the convergence result and justify the result for the Monte Carlo evaluation of some option contracts written on an underlying interest rate whose prices are governed by this model.

In value-at-risk (VaR) methodology of option risk measurement, the determination of market values of the current option positions under various market scenarios is critical. Under the full revaluation and factor sensitivity approach which are accepted by regulators, accurate revaluation and precise factor sensitivity calculation of options in response to significant moves in market variables are important for measuring option risks in terms of VaR figures. This paper provides a method for pricing equity options in the constant elasticity variance (CEV) model environment using the Lie-algebraic technique when the model parameters are time-dependent. Analytical solutions for option values incorporating time-dependent model parameters are obtained in various CEV processes. The numerical results, which are obtained by employing a very efficient computing algorithm similar to the one proposed by Schroder [11], indicate that the option values are sensitive to the time-dependent volatility term structures. It is also possible to generate further results using various functional forms for interest rate and dividend term structures. From the analytical option pricing formulae, one can achieve more accuracy to compute factor sensitivities using more realistic term-structures in volatility, interest rate and dividend yield. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black–Scholes model, more precise risk management in equity options can be achieved by incorporating term-structures of interest rates, volatility and dividend into the CEV option valuation model.

This paper provides a method for pricing options in the constant elasticity of variance (CEV) model environment using the Lie-algebraic technique when the model parameters are time-dependent. Analytical solutions for the option values incorporating time-dependent model parameters are obtained in various CEV processes with different elasticity factors. The numerical results indicate that option values are sensitive to volatility term structures. It is also possible to generate further results using various functional forms for interest rate and dividend term structures. Furthermore, the Lie-algebraic approach is very simple and can be easily extended to other option pricing models with well-defined algebraic structures.

In this paper we consider the evaluation of sensitivities of options on spots and forward contracts in commodity and energy markets. We derive different expressions for these sensitivities, based on techniques from the recently introduced Malliavin approach [8, 9]. The Malliavin approach provides representations of the sensitivities in terms of expectations of the payoff and a random variable only depending on the underlying dynamics. We apply Monte–Carlo methods to evaluate such expectations, and to compare with numerical differentiation. We propose to use a refined quasi Monte–Carlo method based on adaptive techniques to reduce variance. Our approach gives a significant improvement of convergence.

This paper proposes a numerical approach for computing bounds for the arbitrage-free prices of an option when some options are available for trading. Convex duality reveals a close relationship with recently proposed calibration techniques and implied trees. Our approach is intimately related to the uncertain volatility model of Avellaneda, Levy and Parás, but it is more general in that it is not based on any particular form of the asset price process and does not require the seller's price of an option to be a differentiable function of the cash-flows of the option. Numerical tests on S&P500 options demonstrate the accuracy and robustness of the proposed method.

We present a quasi-analytic perturbation expansion for multivariate N-dimensional Gaussian integrals. The perturbation expansion is an infinite series of lower-dimensional integrals (one-dimensional in the simplest approximation). This perturbative idea can also be applied to multivariate Student-t integrals. We evaluate the perturbation expansion explicitly through 2nd order, and discuss the convergence, including enhancement using Padé approximants. Brief comments on potential applications in finance are given, including options, models for credit risk and derivatives, and correlation sensitivities.

This paper demonstrates the simple incorporation of any shape of risk aversion into an asset allocation framework. Indeed, the relevant literature about risk aversion shows mixed evidence regarding the shape of this important but subjective variable. Our setting builds on, and can be compared with, the well-known constant relative risk aversion (CRRA) framework and mostly preserves the tractability of the affine-CRRA framework. Our numerical analysis exhibits some link between measures of risk aversions and empirical studies of asset allocation.

When the discretely adjusted option hedges are constructed by the continuous-time Black–Scholes delta, then the hedging errors appear. The first objective of the paper is to consider a discrete-time adjusted delta, such that the hedging error can be reduced. Consequently, a partial differential equation for option valuation associated with the problem is derived and solved.

The second objective is to compare the obtained results with the results given by the Black–Scholes formula. The obtained option values may be higher than those given by the Black–Scholes formula, however, unless the option is near expiry, the difference is relatively small.

When one invests in portfolios of derivatives (such as options), the delta-gamma approximation (DGA) is often used as a risk management strategy to reduce the risk associated with the underlying asset price. However, this approximation is locally accepted only for small changes of the underlying asset price. When these changes become large, the option prices estimated by the DGA may significantly differ from those of the market (or those that are estimated using, for instance, the Black–Scholes model), depending mainly on the time-to-maturity, implied volatility, and moneyness. Hence, in practice, before the change of the underlying asset price becomes large, rebalancing operations are demanded to minimize the losses occurred due to the error introduced by the DGA. The frequency of rebalancing may be high when the rate at which the underlying asset price significantly changes. Nonetheless, frequent rebalancing may be unattainable, as there are associated transaction costs. Hence, there is a trade-off between the losses resulting from the inaccurate performance of the DGA and the transaction costs incurring from frequent hedging operations. In the present work, we show two approaches that can outperform the DGA, in this way to increase the accuracy of estimating the option prices with the ultimate goal of reducing the losses due to the estimation error. The first method is similar to the DGA but we change the reference value that the DGA uses (that is, the initial price of the underlying asset) to the underlying asset price forecasted for the time horizon. We coin this method as the extended delta-gamma approximation (EDGA). The second method that we consider in this work is the locally weighted regression (LWR) that locally regresses the option prices from the changes of the underlying asset prices, with the same reference value that is employed in the EDGA method. Finally, we compare the performance of the two methods presented in this work to that of some existing methods.

In contrast to their role in theory, options are in practice not only traded for hedging purposes. Many investors also use them for speculation purposes. For these investors the Black–Scholes price serves only as an orientation, their decisions to buy, hold or hedge an option are also based on subjective beliefs and on their personal utility functions (in the widest possible sense). The aim of this paper is to present a general framework to include different types of investors, especially hedgers, pure speculators and speculators following strategies with bounded risk. We derive their subjective values of an option endogenously from the solution of their decision problems.