Exotic put options at the diffusion bond market

The direct approach is used to obtain formulas that determine the option price, portfolios (hedging strategies) and capitals for the European put options with guaranteed income for the option holder and limited payment for the investor at the diffusion (B,P)-bond market. The properties of the solution are studied. The results are specified for the bond pricing models known as the Ho-Lee and Vasicek models.     

Two-State Volatility Transition Pricing and Hedging of TXO Options

The spot markets often exhibit high and low volatilities that persist for a while. We classify the spot market volatility into two states: high and low and use the Markov chain theory to construct a Two-State volatility model for pricing and hedging Taiwan stock index options (TXO). Compared to binomial option pricing model, the Two-State model is more stable in convergence and faster in early periods of convergence but is much more time-consuming as the number of periods and computations extensively increase. The growth order of total node number for quadrinomial lattice is O(n ⁴) while it is O(n ²) for binomial lattice. Empirically, the Taiwan stock index has high-volatility = 42.85%, low-volatility = 17.39%, and probability of being in high-volatility state = 0.3487 over the in-sample period from 1/6/1990 to 04/30/2008 according to Markov chain. Using as large as 87,160 datasets of TXO covering out-of-sample period from 05/02/2008 to 03/17/2010 and strike prices from 3600 to 8700, we demonstrated that the Two-State volatility model has the most outstanding performance in high-volatility period as applying put options for pricing and hedging. However, to avoid the cost of taxes resulting from position changes, a longer-term (e.g. 10 day) hedge is more properly than a short-term (e.g. 5 day) hedge.     

Numerical analysis for Spread option pricing model of markets with finite liquidity: first-order feedback model

In this paper, we discuss the numerical analysis and the pricing and hedging of European Spread options on correlated assets when, in contrast to the standard framework and consistent with a market with imperfect liquidity, the option trader's trading in the stock market has a direct impact on one of the stocks price. We consider a first-order feedback model which leads to a linear partial differential equation. The Peaceman–Rachford scheme is applied as an alternating direction implicit method to solve the equation numerically. We also discuss the stability and convergence of this numerical scheme. Finally, we provide a numerical analysis of the effect of the illiquidity in the underlying asset market on the replication of an European Spread option; compared to the Black–Scholes case, a trader generally buys less stock to replicate a call option.     

Approximate Option Pricing

As increasingly large volumes of sophisticated options are traded in world financial markets, determining a ``fair' price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n -period option on a stock is the expected time-discounted value of the future cash flow on an n -period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte Carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this article we show that pricing an arbitrary path-dependent option is \#-P hard. We show that certain types of path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these we design deterministic polynomial-time approximate algorithms. We show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation of the value of an otherwise identical n -period American put option. In contrast to Monte Carlo methods, our algorithms have guaranteed error bounds that are polynomially small (and in some cases exponentially small) in the maturity n . For the error analysis we derive large-deviation results for random walks that may be of independent interest. Received August 13, 1996; revised April 2, 1997.     

Generalized route planning model for hazardous material transportation with VaR and equity considerations

Recently, the Value-at-Risk (VaR) framework was introduced for the routing problem of a single hazmat trip. In this paper, we extend the VaR framework in two important ways. First, we show how to apply the VaR concept to a more realistic multi-trip multi-hazmat type framework, which determines routes that minimize the global VaR value while satisfying equity constraints. Second, we show how to embed the algorithm for the single hazmat trip problem into a Lagrangian relaxation framework to obtain an efficient solution method for this general case. We test our computational experience based on a real-life hazmat routing scenario in the Albany district of New York State. Our results indicate that one can achieve a high degree of risk dispersion while controlling the VaR value within the desired confidence level.     

Derivative Asset Pricing with Transaction Costs: An Extension

This paper examines the optimal perfect hedging (super-replication) of an option by a cash-plus-riskless asset portfolio within the context of the binomial model. The cases discussed here were not covered by the earlier studies of Boyle and Vorst (1992) and Bensaid, Lesne, Pagès and Scheinkman (1992). It is argued that these cases are empirically important, and that there is some indication that they are encountered very often in practice in the Swiss options market. A new algorithm is developed to compute the option price lower bound (bid price) for such cases. It is then shown that, for most such cases, the portfolio hedging the short call when replication is not optimal coincides with the Merton (1973) lower bound.     

Contracts choice for supply chain under inflation

Inflation causes an increase in the retail price and a decrease in the market demand, both arise from the problem of seasonal product management and occur during the long production lead time. As an effective tool for hedging against the risks, option contracts, including call, put, and bidirectional option contracts, have been proved to benefit two members in a one‐supplier and one‐retailer supply chain under inflation. The aim of this paper is to examine the effect of different option contacts on the decisions and performances for both the supplier and the retailer under inflation. Our results suggest that the retailer prefers adopting portfolio contracts with bidirectional options under inflation, whereas the supplier is inclined to provide call option contracts under inflation. Our study also reveals that call option contracts are implemented ultimately by the supply chain under inflation because of the supplier's market dominant position.     

Minimax hedging strategy

We present several variants of a robust risk management strategy based on minimax for the writer of a European call option on a stock and show that it performs at least as well as the standard hedging strategy, delta hedging. When using the minimax strategy, the hedger specifies a worst case scenario in terms of the price of the underlying stock. The minimax strategy recommends the number of shares in the underlying stock the hedger should hold in order to minimize the hedging error against the worst case occurring. The minimax hedging error may correspond to an extreme point of the price range being considered or to a mid-range solution. Simulation and empirical results suggest that the minimax strategy is particularly powerful for hedging the risk of writing an option when the price of the underlying stock is both highly volatile and crosses over the exercise frequently.     

VaR and ES for linear portfolios with mixture of elliptic distributions risk factors

In this paper, we generalize the Linear VaR method from portfolios with normally distributed risk factors to portfolios with mixture of elliptically distributed ones. We treat both the Expected Shortfall and the Value-at-Risk of such portfolios. Special attention is given to the particular case of a mixture of multivariate t-distributions. This is a part of J. SADEFO-KAMDEM PhD Thesis[12] of the Université de Reims, France . It has been presented at the workshop on modelling and computation in Financial Engineering at Bad Herrenalb, Germany May 6-8, 2003. The author is an associate professor at the Department of mathematics, université d’Evry Val d’Essonne.     

Intradaily dynamic portfolio selection

A portfolio selection model which allocates a portfolio of currencies by maximizing the expected return subject to Value-at-Risk (VaR) constraint is designed and implemented. Based on an econometric implementation using intradaily data, the optimal portfolio allocation is forecasted at regular time intervals. For the estimation of the conditional variance from which the VaR is computed, univariate and multivariate GARCH models are used. Model evaluation is done using two economic criteria and two statistical tests. The result for each model is given by the best forecasted intradaily investment recommendations in terms of the optimal weights of the currencies in the risky portfolio. The results show that estimating the VaR from multivariate GARCH models improves the results of the forecasted optimal portfolio allocation, compared to using a univariate model.     

Optimal Algorithms for <Emphasis Type="Italic">k</Emphasis>-Search with Application in Option Pricing

In the k-search problem, a player is searching for the k highest (respectively, lowest) prices in a sequence, which is revealed to her sequentially. At each quotation, the player has to decide immediately whether to accept the price or not. Using the competitive ratio as a performance measure, we give optimal deterministic and randomized algorithms for both the maximization and minimization problems, and discover that the problems behave substantially different in the worst-case. As an application of our results, we use these algorithms to price “lookback options”, a particular class of financial derivatives. We derive bounds for the price of these securities under a no-arbitrage assumption, and compare this to classical option pricing. J. Lorenz is partially supported by UBS AG. K. Panagiotou is partially supported by the SNF, grant number: 200021-107880/1.     

Optimal divestment time in supply chain redesign under oligopoly: evidence from shale oil production plants

This paper investigates the optimal timing for the closure of production plants in the context of supply chain redesign. We consider producers operating under oligopoly and analyze their optimal decisions with the aid of real options. We contextualize this study in the energy supply chain, where revenue is heavily influenced by the Organization of the Petroleum Exporting Countries (OPEC). It is optimal to divest from a production if certain values pass below the liquidation cost. The optimal time to divest is, then, the optimal time to exercise a perpetual American put option on the total value of the project, where the strike price is the liquidation value and expenses. The OPEC oligpolistic power is modeled with the aid of a hidden market in a Hidden Markov Model, since the state of the Markov chain is not directly visible to the production manager. The reported results can be used by supply chain managers in decision making on production network redesign in the oligopolistic environment considering present value of plants and cash flows in the supply chain. The advantage of the modeling approach proposed is its analytical tractability, especially for project valuation and real‐option problems.     

Hedging an option portfolio with minimum transaction lots: A fuzzy goal programming problem

Options are designed to hedge against risks to their underlying assets such as stocks. One method of forming option-hedging portfolios is using stochastic programming models. Stochastic programming models depend heavily on scenario generation, a challenging task. Another method is neutralizing the Greek risks derived from the Black–Scholes formula for pricing options. The formula expresses the option price as a function of the stock price, strike price, volatility, risk-free interest rate, and time to maturity. Greek risks are the derivatives of the option price with respect to these variables. Hedging Greek risks requires no human intervention for generating scenarios. Linear programming models have been proposed for constructing option portfolios with neutralized risks and maximized investment profit. However, problems with these models exist. First, feasible solutions that can perfectly neutralize the Greek risks might not exist. Second, models that involve multiple assets and their derivatives were incorrectly formulated. Finally, these models lack practicability because they consider no minimum transaction lots. Considering minimum transaction lots can exacerbate the infeasibility problem. These problems must be resolved before option hedging models can be applied further. This study presents a revised linear programming model for option portfolios with multiple underlying assets, and extends the model by incorporating it with a fuzzy goal programming method for considering minimum transaction lots. Numerical examples show that current models failed to obtain feasible solutions when minimum transaction lots were considered. By contrast, while the proposed model solved the problems efficiently.     

A Dimension Reduction Shannon-Wavelet Based Method for Option Pricing

We present a robust and highly efficient dimension reduction Shannon-wavelet method for computing European option prices and hedging parameters under a general jump-diffusion model with square-root stochastic variance and multi-factor Gaussian interest rates. Within a dimension reduction framework, the option price can be expressed as a two-dimensional integral that involves only (i) the value of the variance at the terminal time, and (ii) the time-integrated variance process conditional on this value. A Shannon wavelet inverse Fourier technique is developed to approximate the conditional density of the time-integrated variance process. Furthermore, thanks to the excellent approximation properties of Shannon wavelets, the overall pricing procedure is reduced to the evaluation of just a single integral that involves only the density of the terminal variance value. This single integral can be accurately evaluated, since the density of the variance at the terminal time is known in closed-form. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and impressive efficiency of the method.     

Modeling and Simulation of an Artificial Stock Option Market

Since their introduction in 1973, options have become an important and very popular financial instrument. However, despite much research performed on the subject, the effects of option trading on the underlying asset market are still debated. Both empirical and theoretical studies have failed to point out how price volatility and volumes of the underlying asset are affected. In this paper we present the first study on the effects of an option market related to an underlying stock market, using an artificial financial market based on heterogeneous agents. We modeled a realistic European option using two market models. The microstructure of the first model is kept as simple as possible, being composed only of random traders. The second model is more complex and realistic, involving the presence of various kinds of trading strategies (random, fundamentalist and chartist). We show that the introduction of options, in the proposed models, tends to decrease the volatility of the underlying stock price. Moreover, the traders’ wealth can be strongly affected by the use of option hedging.     

An Integer Programming Model for Pricing American Contingent Claims under Transaction Costs

We study the problem of computing the lower hedging price of an American contingent claim in a finite-state discrete-time market setting under proportional transaction costs. We derive a new mixed-integer linear programming formulation for calculating the lower hedging price. The linear programming relaxation of the formulation is exact in frictionless markets. Our results imply that it might be optimal for the holder of several identical American claims to exercise portions of the portfolio at different time points in the presence of proportional transaction costs while this incentive disappears in their absence.     

Real‐time waiting‐price trading interval in a heterogeneous options market: a Bernoulli distribution

Options pricing remains an open research question that is challenging for both theoreticians and practitioners. Unlike many classical binomial models that assume a “representative agent,” the model suggested herein considers two players who are heterogeneous with respect to their estimations of the distribution of the underlying asset price on expiration day, and with respect to their levels of willingness to make a transaction (eagerness level). A two‐player binomial model is developed to find the real‐time optimal option price in two stages. First, we determine a primary feasible pricing domain. We then find a narrower feasible domain, termed the “waiting‐price trading interval,” meaning the region within which the players may either wait for better offers (due to a change in market conditions or player beliefs), or make an immediate transaction. The suggested model is formulated by a nonlinear optimization problem and the optimal price is shown to be unique. We demonstrate that the counter player's eagerness level has a significant effect on the proposed optimal option price. Using empirical analysis, several known lattice‐based models for option pricing, such as CRR and Tian, are compared with the current model (herein, S‐H) in which the price offered by the model player takes into account the subjective beliefs of the opposing market player. The comparison shows significant advantages to the S‐H model in terms of the expected profit on expiration day.     

Valuation of American Continuous-Installment Options

We present three approaches to value American continuous-installment options written on assets without dividends or with continuous dividend yield. In an American continuous-installment option, the premium is paid continuously instead of up-front. At or before maturity, the holder may terminate payments by either exercising the option or stopping the option contract. Under the usual assumptions, we are able to construct an instantaneous riskless dynamic hedging portfolio and derive an inhomogeneous Black–Scholes partial differential equation for the initial value of this option. This key result allows us to derive valuation formulas for American continuous-installment options using the integral representation method and consequently to obtain closed-form formulas by approximating the optimal stopping and exercise boundaries as multipiece exponential functions. This process is compared to the finite difference method to solve the inhomogeneous Black–Scholes PDE and a Monte Carlo approach.     

Effective Risk Assessment in Resilient Communication Networks

The paper discusses business impact analysis in the context of resilient communication networks. It is based on the total (aggregated) penalty that may be paid by an operator when the services (identified with transport demands) provided are interrupted due to network failures. The level of penalty is expressed as a commonly accepted business risk measure, Value-at-Risk (\(VaR\)). First, the main concern over \(VaR\), namely the theoretical lack of subadditivity, is discussed. The study shows that, in practice, disadvantages do not appear in resilient network design, and \(VaR\) can be used without the need to apply more complex and less informative measures. Second, a method for calculating the upper bound of the total penalty is presented. The assessment is performed for unprotected and protected services with a broad variety of compensation policies used to translate technical loss to monetarily expressed penalty. The proposed bounds are experimentally shown to be effective in comparison with alternative calculation methods, and also in the case when some of the assumptions taken during the modelling stage are not met.     

Option Pricing With Modular Neural Networks

This paper investigates a nonparametric modular neural network (MNN) model to price the S&P-500 European call options. The modules are based on time to maturity and moneyness of the options. The option price function of interest is homogeneous of degree one with respect to the underlying index price and the strike price. When compared to an array of parametric and nonparametric models, the MNN method consistently exerts superior out-of-sample pricing performance. We conclude that modularity improves the generalization properties of standard feedforward neural network option pricing models (with and without the homogeneity hint).