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Riding on a Smile

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In this paper, transaction costs as small nonlinear price impact are introduced into American option pricing under the Heston stochastic volatility model, forming a gap between option prices of the holder and writer. Through the use of a dynamic hedging strategy together with a known option independent of transaction costs, we derive two different nonlinear pricing partial differential equation systems for the holder and writer, respectively. A numerical algorithm is designed to solve the systems so that American option prices as well as the optimal exercise boundary can be simultaneously obtained. Examples are presented to illustrate the effect of transaction costs on both option prices and optimal exercise prices.
Article
In the field of option pricing, scholars have been exploring the constant settings of volatility and interest rate ever since the Black-Scholes model was put forward. This research introduces a floating interest rate into the local volatility model, analyzes the model’s pricing effects, and compares them from the aspects of an in-sample pricing error and out-of-sample pricing error respectively using data of CSI (China Securities Index) SH-SZ (Shanghai-Shenzhen) 300 stock index options. The empirical results show that the Surface Stochastic Volatility Inspire model is better than the Stochastic Volatility Inspire model.
Chapter
This chapter simulates the economic impact of three regional innovation models, i.e., Hubei model (science discovery-hub), Sichuan model (technology development-specialization), and Guangdong model (production application-aggregation), based on the formulation in Chap. 8. The three simulation scenarios of three regional innovation models and one baseline scenario are reported; and the simulate industries, simulate variables, and simulate time are introduced. This chapter describes the settings of R&D investment type and aggregation degree in four scenarios. The macroeconomy indicators including gross domestic product, domestic demand, domestic price, wage rate, total investment, and exports in four scenarios are simulated, compared, and explained. The incomes, demands, and savings of residents, enterprises, and government in four scenarios are simulated, compared, and explained. The industrial indicators including industrial output, export, and capital return rate of 32 industrial sectors in four scenarios are simulated, compared, and explained. The industrial structures representing the ratios of agriculture, manufacture, and service industries are also analyzed.
Article
The implied volatility in the Black-Scholes framework is not a constant but a function of both the strike price (“smile/skew”) and the time to expiry. A popular approach to recovering the volatility surface is through the use of deterministic volatility function models via Dupire’s equation. A new method for volatility surface calibration based on the Mellin transform is proposed. An explicit formula for the volatility surface is obtained in terms of the Mellin transform of the call option price with respect to the strike price, and a numerical algorithm is provided. Results of numerical simulations are presented and the stability of the method is numerically verified. The proposed Mellin transform approach provides a simpler and more direct fitting of generalised forms of the volatility surface given previously in the literature.
Article
European futures options are not traded on the Chinese exchanges and that generates difficulties to calibrate fundamental market parameters, such as the implied volatilities. We propose an efficient willow tree method to resolve the problem of calibrating the implied volatility from American-style options. The proposed willow tree construction is independent of the volatility itself so as to minimize the cost of the calibration. We also apply the proposed method to calibrate the implied volatilities of most frequently traded options in the Chinese market, sugar and soybean meal, based on the daily closing prices, and construct the corresponding implied volatility surfaces. The results indicate the seasonality in the volatility of commodity spot prices and futures prices in China. Moreover, based on the implied volatility distortion close to the option maturity observed in our empirical results, we suggest a minimum tick price scheme to avoid the distortion and decrease of hedging costs.
Article
We study the sensitivity of the leverage effect to changes of the volatility and the price, showing the existence of an analytical link between the latter and the price‐leverage covariation in settings with, respectively, stochastic and level‐dependent volatility. From the financial standpoint, the results we obtain allow for the interpretation of the price‐leverage covariation as a gauge of the responsiveness of the leverage effect to price and volatility changes. The empirical study of S&P500 high‐frequency prices over the period March 2018–April 2018, carried out by means of nonparametric Fourier estimators, supports this interpretation of the role of the price‐leverage covariation.
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Deposit insurance is a financial tool that guarantees the bank’s depositors from banks’ failure to maintain their assets. The dynamic of banks’ assets leads to the uncertainty of banks’ ability to pay their debts to the third party when due. Employing a multi-state regime-switching volatility in the insurance pricing, is aimed to fit the stochastic occurrence on the asset’s volatility. Some numerical results by using Monte Carlo and quasi-Monte Carlo, are performed to obtain the price of deposit insurance efficiently. Comparison between the obtained results by the Monte Carlo and quasi-Monte Carlo methods and the standard Black–Scholes model (no-switching) are also presented. In our finding, the quasi-Monte Carlo method has a faster convergence, better smoothness, and greater accuracy compared to the Monte Carlo method.
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In this article, we present a new class of pricing models that extend the application of Wishart processes to the so-called stochastic local volatility (or hybrid) pricing paradigm. This approach combines the advantages of local and stochastic volatility models. Despite the growing interest on the topic, however, it seems that no particular attention has been paid to the use of multidimensional specifications for the stochastic volatility component. Our work tries to fill the gap: we introduce two hybrid models in which the stochastic volatility dynamics is described by means of a Wishart process. The proposed parametrizations not only preserve the desirable features of existing Wishart-based models but significantly enhance the ability of reproducing market prices of vanilla options.
Article
In the pricing of exotic options, model risk arises when different models yield different prices, even though they are calibrated to the same observable prices of plain vanilla options. We analyze model risk in the case of barrier options and bonus certificates. This study uses an empirical data set of over 40,000 certificates to analyze the real market extent of model risk for traded barrier options. In particular, applying the local volatility model, the Heston model, and the Bates model, theoretical model risk amounts to about 8.5% (median) of the barrier option value. In contrast, the median empirical model risk, based on the range of market prices, is only 2.2%. We find evidence that the majority of issuers prefer stochastic volatility over local volatility models. Model risk is a factor priced into issuers’ margin policy—that is, they let retail customers pay for their model risk.
In this work, we formulate a pricing model for European options with transaction costs under Heston-type stochastic volatility. The resulting pricing partial differential equations (PDEs) are a pair of nonlinear convection-diffusion-reaction equations with mixed derivative terms, for the writing and holding prices, respectively. The equations are solved numerically by the explicit Euler method. Numerical experiments are presented to illustrate the order of convergence and the effect of the transaction costs on option prices.
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