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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.

This paper examines mispricing, volatility and parity on the Hang Seng Index (HSI) options and futures market. Most of the previous research has focused on futures contracts; we update this research and extend it by considering also option contracts. It is also important to examine these issues post 1997 Asian crisis. We find mispricing of HSI futures and option contracts if no transaction costs were considered. However, by incorporating transaction costs, the HSI futures are bounded within the arbitrage free region and most of the mispricing of the HSI options disappears. Additional tests on the mispricing series reveals that most of the derivative HSI contracts are positively autocorrelated and that the mispricing series for both derivative contracts are not identical among the different contract months. From our results we cannot conclude that there is causal relationship between the mispricing and the spot index volatility. Finally, our empirical results show that for HSI derivative contracts future and option parity holds, supporting our mispricing test that the HSI derivative market is efficient and has not been adversely affected by the Asian economic crisis.

This paper examines the dynamics of returns and order imbalances across the KOSPI 200 cash, futures and option markets. The information effect is more dominant than the liquidity effect in these markets. In addition, returns have more predictability power for the future movements of prices than order imbalances. Information seems to be transmitted more strongly from derivative markets to their underlying asset markets than from the underlying asset markets to their derivative markets. Finally, domestic institutional investors prefer futures, domestic individual investors prefer options, and foreign investors prefer stocks relative to other investor groups when they have new information.

We assume that the call option's value is correctly priced by Black and Scholes' option pricing model in this paper. This paper derives an exact closed-form solution for implied standard deviation under the condition that the underlying asset price equals the present value of the exercise price. The exact closed-form solution provides the true implied standard deviation and has no estimate error. This paper also develops three alternative formulas to estimate the implied standard deviation if this condition is violated. Application of the Taylor expansion on a single call option value derives the first formula. The accuracy of this formula depends on the deviation between the underlying asset price and the present value of the exercise price. Use of the Taylor formula on two call option prices with different exercise prices is used to develop the second formula, which can be used even though the underlying asset price deviates significantly from the present value of the exercise price. Extension of the second formula's approach to third options value derives the third formula. A merit of the third formula is to circumvent a required parameter used in the second formula. Simulations demonstrate that the implied standard deviations calculated by the second and third formulas provide accurate estimates of the true implied standard deviations.

There are two ad hoc approaches to Black and Scholes model. The “relative smile” approach treats the implied volatility skew as a fixed function of moneyness, whereas the “absolute smile” approach treats it as a function of the strike price. Previous studies reveal that the “absolute smile” approach is superior to the “relative smile” approach as well as to other sophisticated models for pricing options. We find that the time-to-maturity factors improve the pricing and hedging performance of the ad hoc procedures and the superiority of the “absolute smile” approach still holds even after the time-to-maturity is considered.

Theories are inconclusive about the various impacts of the introduction of basket securities on the underlying stocks. We explore those effects for the first time around the launch of options on exchange traded funds (ETFs), employing the listing of the options on the S&P 500 Depository Receipts (SPDRs) in January 2005. With known factors controlled respectively, we find that the introduction of the SPDRs options leads to lower trading volume, higher bid–ask spread, higher systematic and total risks, and lower prices for the underlying stocks, consistent with the theory that the advent of basket derivatives alters the mix of various types of portfolio traders in the related markets when they are fully integrated.

I use five separate measures of deviation from Put-Call Parity of options on a stock without splits or dividends as separate negative measures for market efficiency. I rely upon the theory of trading volume as a function of short sales costs, etc., and that of market efficiency as a function of trading volume, etc. derived by Bhattacharya (2019). I use Three-Stage Least Squares (3SLS) to estimate this structural system, separately for Nasdaq and non-Nasdaq U.S. stocks. I find, contrary to much previous theoretical and empirical work, that the impact of short sales costs & constraints on market efficiency is not significantly negative and that the impact of trading volume on market efficiency is not significantly positive, and my results are robust to various econometric specifications and financial economic assumptions.

We employ neural networks to understand volatility surface movements. We first use daily data on options on the S&P 500 index to derive a relationship between the expected change in implied volatility and three variables: the return on the index, the moneyness of the option, and the remaining life of the option. This model provides an improvement of 10.72% compared with a simpler analytic model. We then enhance the model with an additional feature: the level of the VIX index prior to the change being observed. This produces a further improvement of 62.12% and shows that the expected response of the volatility surface to movements in the index is quite different in high and low volatility environments.

In this chapter, we investigate how different measures of volatility influence bank’s capital structure beside mandatory capital requirements. We study the relationship between four volatility risk measures (volatility skew and spread, variance risk premia, and realized volatility) and bank’s market leverage and we analyze if banks adjust their capital needs in response to significant increase of risk premia discounting from traders. Among the four volatility measures, volatility skew (defined as the difference between OTM put and ATM call implied volatility and representing the perceived tail risk by traders) affects bank’s leverage the most. As volatility skew increases — hence OTM put became more expensive than ATM call — banks deleverage their assets structure. One plausible explanation relates to the higher costs of equity issuance that a bank will face during a period of distress. As the possibility to incur in expensive equity issuance increases the bank prefers to deleverage its balance sheet and create a capital buffer.

Previous studies of the limit order book report that low depths accompany wide spreads and that spreads widen and depths fall in response to higher volume, but some postulate a positive relationship between spreads and depth during normal trading periods. We calculate the option value of the limit order book at 11:00 a.m. for 10 actively traded firms listed on the Australian Stock Exchange. Simultaneously this approach enables us to consider the spread and depth of the limit order book. We find that 33.1% of the option value of the limit order book is provided at the best ask and 34.7% at the best bid. We find that the option value of the limit order book is greatest at the best bid price and the best ask price and is more stable through time than the option value of individual shares or share quantities in the book. Also, consistent with the arguments of Cohen et al. (1981), we find evidence of equilibrium in the supply and demand of liquidity.

This chapter proposes a theoretical model of initial public offering by taking into account the uncertainty in security pricing and the underwriting process. Three major factors are shown to affect the IPO pricing: (1) the uncertainty of stock price, (2) the legal requirement that restricts underwriters from selling any part of the issue above the fixed offering price, and (3) the underwriters' risk tolerance. It is shown that underpricing is negatively related to the underwriter's risk tolerance, and positively related to the length of premarket period and market price volatility.