Pricing Asian options of discretely monitored geometric average in the regime‐switching model

This paper provides analytic pricing formulas of discretely monitored geometric Asian options under the regime‐switching model. We derive the joint Laplace transform of the discount factor, the log return of the underlying asset price at maturity, and the logarithm of the geometric mean of the asset price. Then using the change of measures and the inversion of the transform, the prices and deltas of a fixed‐strike and a floating‐strike geometric Asian option are obtained. As the numerical results, we calculate the price of a fixed‐strike and a floating‐strike discrete geometric Asian call option using our formulas and compare with the results of the Monte Carlo simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

NUMERICAL ANALYSIS ON BINOMIAL TREE METHODS FOR AMERICAN LOOKBACK OPTIONS

1　IntroductionLookback options are path-dependent options whose payoffs depend on the maximumor the minimum of the underlying asset price during the life of the options( see[6] [1 0 ][1 4] ) .Here the maximum or minimum realized asset price may be monitored either con-tinuously or discretely.An American lookback call( put) option allows to be exercised atany time prior to expiry and gives the holder the rightto buy( sell) atthe historical mini-mum( maximum) of the underlying asset price on ex…     

Heuristic option pricing with neural networks and the neuro-computer synapse 3

Today’s option and warrant pricing is based on models developed by Black, Scholes and Merton in 1973 and Cox, Ross and Rubinstein in 1979. The price movement of the underlying asset is modeled by continuous-time or discrete-time stochastic processes. Unfortunately these models are based on severely unrealistic assumptions. Permanently an unsatisfactory and quite artificial adaption to the true market conditions is necessary (future volatility of the underlying price). Here, an alternative heuristic approach with a highly accurate neural network approximation is presented. Market prices of options and warrants and the values of the influence variables form the usually very large output/ input data set. Thousands of multi-layer perceptrons with various topologies and with different weight initializations are trained with a fast sequential quadratic programming (SQP) method. The best networks are combined to an expert council network to synthesize market prices accurately. All options and warrants can be compared to single out overpriced and underpriced ones for each trading day. For each option and warrant overpriced and underpriced trading days can be used to ascertain a better buy and sell timing. Furthermore the neural model gains deep insight into the market price sen-sitivities (option Greeks), e.g., ?, Г, Θ and Ω. As an illustrative example we inves-tigate BASF stock call warrants. Time series from the beginning of 1996 to mid 1997 of 74 BASF call warrant prices at the Frankfurter Wertpapierborse (Frankfurt Stock Exchange) form the data basis. Finally a possible speed up of the training with the neuro-computer SYNAPSE 3 is briefly discussed     

Risk management of a bond portfolio using options

In this paper, we elaborate a formula for determining the optimal strike price for a bond put option, used to hedge a position in a bond. This strike price is optimal in the sense that it minimizes, for a given budget, either Value-at-Risk or Tail Value-at-Risk. Formulas are derived for both zero-coupon and coupon bonds, which can also be understood as a portfolio of bonds. These formulas are valid for any short rate model that implies an affine term structure model and in particular that implies a lognormal distribution of future zero-coupon bond prices. As an application, we focus on the Hull-White one-factor model, which is calibrated to a set of cap prices. We illustrate our procedure by hedging a Belgian government bond, and take into account the possibility of divergence between theoretical option prices and real option prices. This paper can be seen as an extension of the work of Ahn and co-workers [Ahn, D., Boudoukh, J., Richardson, M., Whitelaw, R., 1999. Optimal risk management using options. J. Financ. 54, 359-375], who consider the same problem for an investment in a share.     

期权组合最优套期保值模型及实证研究

根据实际投资中投资者可以选择不同到期日、不同敲定价格的期权组合进行套期保值的现实,本文建立了二次效用函数下期权组合最优动态套期保值模型,证明了该模型最优解存在的唯一性,并在协方差矩阵可逆和不可逆两种情形下分别给出了期权最优头寸的显式表达式。在50ETF价格先升后降、先降后升、下降和上升四种情形下,对上证50ETF期权的多种期权组合套期保值问题进行实证分析。研究结果表明:不同到期日不同敲定价格的看跌期权组合具有较好的套期保值效果。本文的研究为选择期权组合进行套期保值和解决展期期权套期保值问题提供了借鉴。     

Analytical valuation of catastrophe equity options with negative exponential jumps

A catastrophe put option is valuable in the event that the underlying asset price is below the strike price; in addition, a specified catastrophic event must have happened and influenced the insured company. This paper analyzes the valuation of catastrophe put options under deterministic and stochastic interest rates when the underlying asset price is modeled through a Lévy process with finite activity. We provide explicit analytical formulas for evaluating values of catastrophe put options. The numerical examples illustrate how financial risks and catastrophic risks affect the prices of catastrophe put options.     

The European options hedge perfectly in a Poisson-Gaussian stock market model

It is shown that n + 1 European call options written on a stock S with different strike prices (or the stock and n calls) are non-redundant assets in a model for the stock driven by a Brownian motion and n independent Poisson processes. That extends the result obtained for n = 1 by Pham and implies that the proposed model can price and perfectly hedge any integrable derivative on S.     

Estimation of risk-neutral density surfaces

Option price data is often used to infer risk-neutral densities for future prices of an underlying asset. Given the prices of a set of options on the same underlying asset with different strikes and maturities, we propose a nonparametric approach for estimating risk-neutral densities associated with several maturities. Our method uses bicubic splines in order to achieve the desired smoothness for the estimation and an optimization model to choose the spline functions that best fit the price data. Semidefinite programming is employed to guarantee the nonnegativity of the densities. We illustrate the process using synthetic option price data generated using log-normal and absolute diffusion processes as well as actual price data for options on the S&P 500 index. We also used the risk-neutral densities that we computed to price exotic options and observed that this approach generates prices that closely approximate the market prices of these options.     

使用双币种期权的风险管理

以VaR作为风险度量的工具,研究固定汇率制度下使用双币种期权对冲进行的风险管理.研究表明,可以通过确定最优敲定价格减小VaR,并对获得的结论做了比较静态分析.     

Pricing and market segmentation using opaque selling mechanisms

In opaque selling certain characteristics of the product or service are hidden from the consumer until after purchase, transforming a differentiated good into somewhat of a commodity. Opaque selling has become popular in service pricing as it allows firms to sell their differentiated products at higher prices to regular brand loyal customers while simultaneously selling to non-loyal customers at discounted prices. We develop a stylized model of consumer choice that illustrates the role of opaque selling in market segmentation. We model a firm selling a product via three selling channels: a regular full information channel, an opaque posted price channel and an opaque bidding channel where consumers specify the price they are willing to pay. We illustrate the segmentation created by opaque selling as well as compare optimal revenues and prices for sellers using regular full information channels with those using opaque selling mechanisms in conjunction with regular channels. We also study the segmentation and policy changes induced by capacity constraints.     

Risk-neutral valuation of power barrier options

Barrier options are standard exotic options traded in the financial market. These instruments are different from the vanilla options as the payoff of the option depends on whether the underlying asset price reaches a predetermined barrier level, during the life of the option. In this work, we extend the vanilla call barrier options to power call barrier options where the underlying asset price is raised to a constant power, within the standard Black–Scholes framework. It is demonstrated that the pricing of the power barrier options can be obtained from standard barrier options by a transformation which involves the power contract and a adjusted barrier. Numerical results are considered.     

Pricing perpetual American options under multiscale stochastic elasticity of variance

This paper studies pricing the perpetual American options under a constant elasticity of variance type of underlying asset price model where the constant elasticity is replaced by a fast mean-reverting Ornstein–Ulenbeck process and a slowly varying diffusion process. By using a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on the option prices and the optimal exercise prices with respect to model parameters. Our results enhance the existing option price structures in view of flexibility and applicability through the market prices of elasticity risk.     

Two Exotic Lookback Options

This paper formally analyses two exotic options with lookback features, referred to as extreme spread lookback options and look‐barrier options, first introduced by Bermin. The holder of such options receives partial protection from large price movements in the underlying, but at roughly the cost of a plain vanilla contract. This is achieved by increasing the leverage through either floating the strike price (for the case of extreme spread options) or introducing a partial barrier window (for the case of look‐barrier options). We show how to statically replicate the prices of these hybrid exotic derivatives with more elementary European binary options and their images, using new methods first introduced by Buchen and Konstandatos. These methods allow considerable simplification in the analysis, leading to closed‐form representations in the Black–Scholes framework.     

American options and stochastic interest rates

We study finite-maturity American equity options in a stochastic mean-reverting diffusive interest rate framework. We allow for a non-zero correlation between the innovations driving the equity price and the interest rate. Importantly, we also allow for the interest rate to assume negative values, which is the case for some investment grade government bonds in Europe in recent years. In this setting we focus on American equity call and put options and characterize analytically their two-dimensional free boundary, i.e. the underlying equity and the interest rate values that trigger the optimal exercise of the option before maturity. We show that non-standard double continuation regions may appear, extending the findings documented in the literature in a constant interest rate framework. Moreover, we contribute by developing a bivariate discretization of the equity price and interest rate processes that converges in distribution as the time step shrinks. This discretization, described by a recombining quadrinomial tree, allows us to compute American equity options’ prices and to analyze their free boundaries with respect to time and current interest rate. Finally, we document the existence of non-standard optimal exercise policies for American call options on a non-dividend-paying equity.

     

On martingale diffusions describing the ‘smile‐effect’ for implied volatilities

This paper discusses diffusion models describing the ‘smile‐effect’ of implied volatilities for option prices partly following the new approach of Bruno Dupire. If one restricts to the time homogeneous case, a careful study of this approach shows that the call option prices considered as a function of the price x of the underlying security, remaining time to maturity T–t and strike price K have necessarily to satisfy a certain functional equation, in order to fit into a coherent model. It is shown that for certain examples of empirically observed option prices which are reported in the literature, this functional equation does not hold. © 2000 John Wiley & Sons, Ltd.     

Option pricing under a normal mixture distribution derived from the Markov tree model

We examine a Markov tree (MT) model for option pricing in which the dynamics of the underlying asset are modeled by a non-IID process. We show that the discrete probability mass function of log returns generated by the tree is closely approximated by a continuous mixture of two normal distributions. Using this normal mixture distribution and risk-neutral pricing, we derive a closed-form expression for European call option prices. We also suggest a regression tree-based method for estimating three volatility parameters σ, σ⁺, and σ⁻ required to apply the MT model. We apply the MT model to price call options on 89 non-dividend paying stocks from the S&P 500 index. For each stock symbol on a given day, we use the same parameters to price options across all strikes and expires. Comparing against the Black–Scholes model, we find that the MT model’s prices are closer to market prices.     

Arbitrage and the tax code

We provide a detailed characterization of arbitrage-free asset prices in the presence of capital gains and income taxes. The distinguishing feature of our analysis is that we impose on the model two important features of the tax code: the limited use of capital losses and the inability to wash sell. We show that under remarkably mild conditions, the lack of pre-tax arbitrage implies the lack of post-tax arbitrage with the limited use of capital losses. The conditions are that the risk free interest rate be positive and that tax rates on interest income exceed capital gains tax rates. The result also holds when only a wash sale constraint is imposed and no investor holds a portfolio with a large capital loss. We allow investors to face different tax rates and have different bases for the calculation of capital gains taxes. The characterizations we provide have important implications for both asset pricing and portfolio choice. Our results imply that models that use arbitrage-free pre-tax models continue for derivative pricing and hedging are also arbitrage free in a world with taxes. Similarly, portfolio choice models with taxes typically specify pre-tax arbitrage free price processes and then analyze portfolio choice in the presence of taxes. In these models, it is unclear if portfolio recommendations are based on risk-return tradeoffs or on the arbitrage opportunities present in the model. Our results imply that if the above features of the tax code are modeled explicitly, then we can isolate the post-tax risk-return tradeoffs.     

Electricity swing option pricing by stochastic bilevel optimization: A survey and new approaches

We demonstrate how the problem of determining the ask price for electricity swing options can be considered as a stochastic bilevel program with asymmetric information. Unlike as for financial options, there is no way for basing the pricing method on no-arbitrage arguments. Two main situations are analyzed: if the seller has strong market power he/she might be able to maximize his/her utility, while in fully competitive situations he/she will just look for a price which makes profit and has acceptable risk. In both cases the seller has to consider the decision problem of a potential buyer – the valuation problem of determining a fair value for a specific option contract – and anticipate the buyer’s optimal reaction to any proposed strike price. We also discuss some methods for finding numerical solutions of stochastic bilevel problems with a special emphasis on using duality gap penalizations.     

Simulation methods for valuing Asian option prices in a hyperbolic asset price model

In this paper, we present a Quasi-Monte Carlo approach for pricingEuropean-style Asian options, i.e. for options whose pay-offdepends on the average price of the underlying asset where theaverage is extended over a fixed period up to the maturity date.Following a recent development in mathematical finance, we assumethat the log returns of the asset are not normally but hyperbolicallydistributed. This hypothesis is approved by several authorswith different statistic tests on real financial data. The aimof this paper is to advance the hyperbolic model to the pricingof Asian options, since there only exist pricing formulae forplain vanilla options and some types of exotic options (e.g.power call options, barrier options) so far. We show how onecan obtain prices of general Asian options in such incompletemarkets in an efficient way.     

股价遵循分数跳扩散的混合型双标的两值期权定价

应用风险中性定价原理,研究标的股价服从分数跳扩散过程的混合型双标的两值期权的定价问题,并得出定价公式,并与股价服从标准布朗运动的定价公式做出比较分析.