Exotic Options Research Articles

Martingale optimal transport in the discrete case via simple linear programming techniques

We consider the problem of finding consistent upper price bounds and super replication strategies for exotic options, given the observation of call prices in the market. This field of research is called model-independent finance and has been introduced by Hobson 1998. Here we use the link to mass transport problems. In contrast to existing literature we assume that the marginal distributions at the two time points we consider are discrete probability distributions. This has the advantage that the optimization problems reduce to linear programs and can be solved rather easily when assuming a general martingale Spence Mirrlees condition. We will prove the optimality of left-monotone transport plans under this assumption and provide an algorithm for its construction. Our proofs are simple and do not require much knowledge of probability theory. At the end we present an example to illustrate our approach.

Applications of the Fractional Diffusion Equation to Option Pricing and Risk Calculations

In this article, we first provide a survey of the exponential option pricing models and show that in the framework of the risk-neutral approach, they are governed by the space-fractional diffusion equation. Then, we introduce a more general class of models based on the space-time-fractional diffusion equation and recall some recent results in this field concerning the European option pricing and the risk-neutral parameter. We proceed with an extension of these results to the class of exotic options. In particular, we show that the call and put prices can be expressed in the form of simple power series in terms of the log-forward moneyness and the risk-neutral parameter. Finally, we provide the closed-form formulas for the first and second order risk sensitivities and study the dependencies of the portfolio hedging and profit-and-loss calculations upon the model parameters.

Efficient multiple control variate method with applications to exotic option pricing

The Monte Carlo simulation method is still the only feasible approach to handle high dimensional problems encountered in many areas so far. The main drawback of this method is its slow convergence. A variance reduction technique is one of the main methods to speed up Monte Carlo simulations. In this paper, we reconsider the multiple control variate method and provide sufficient conditions to ensure that the variance of an m-variate control variate estimator is smaller than that of a k-variate control variate estimator for any k where . The results can be applied to a wide range of high dimensional complex problems where exact solutions do not exist. As nontrivial examples, the results are applied to problems of options pricing under the Black-Scholes-Merton’s model. For arithmetic Asian and basket options, more efficient new control variate estimators are constructed. Numerical results show that the constructed multiple control variate estimators are more efficient than estimators with fewer control variates in reducing variances.

Exotic options pricing under special Lévy process models: A biased control variate method approach

Option pricing plays an important role in financial engineering. No explicit formulas can be derived for many exotic options when the underlying asset prices follow more realistic models. The Monte Carlo simulation method is the only feasible approach to obtain numerical values of these options usually. To overcome the slow convergence – the main drawback for the Monte Carlo method, variance reduction and quasi-Monte Carlo methods are proposed. This paper proposes the application of biased control variate method to speed up the evaluation of exotic options prices by simulations under a special type of Lévy processes. We construct very efficient biased control variates for both fixed and floating strike lookback options, as well as barrier options.

Option pricing with orthogonal polynomial expansions

AbstractWe derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein–Stein, and Hull–White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier‐transform‐based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.

Asian-barrier option pricing formulas of uncertain financial market

Abstract Barrier option is an exotic option on an underlying asset whose payoff depends on whether or not the underlying asset’s price has reached predetermined barrier level. This paper mainly investigates Asian-barrier option pricing problem under uncertain financial market. Assume that stock price follows an uncertain differential equation, some Asian-barrier option pricing formulas are derived. Moreover, several numerical examples are given to illustrate the effectiveness of the proposed model.

American and exotic options in a market with frictions

In a market with frictions, bid and ask prices are described by sublinear pricing functionals, which can be defined recursively using coherent risk measures. We prove the convergence of bid and ask prices for various European and American possible path-dependent options, in particular plain vanilla, Asian, lookback and barrier options in a binomial model with transaction costs. We perform several numerical experiments to confirm the theoretical findings. We apply the results to real market data of American options and compute an implied liquidity to describe the bid–ask spread. This method describes liquidity over time very well, compared to the classical approach of describing bid and ask prices by quoting bid and ask implied volatilities.

Pricing of Proactive Hedging European Option with Dynamic Discrete Position Strategy

Proactive hedging European option is an exotic option for hedgers in the options market proposed recently by Wang et al. It extends the classical European option by requiring option holders to continuously trade in underlying assets according to a predesigned trading strategy, to proactively hedge part of the potential risk from underlying asset price changes. To generalize this option design for practical application, in this study, a proactive hedging option with discrete trading strategy is developed and its pricing formula is deducted assuming the underlying asset price follows Geometric Fractional Brownian Motion. Simulation studies show that proactive hedging option with discrete trading strategy still enjoys strong price advantage compared to the classical European option for majority of parameter space. The observed price advantage is stronger when the underlying asset has more volatility or when the asset price follows closer to Geometric Brownian Motion. Additionally, we found that a higher frequency trading strategy has stronger price advantage if there is no trading cost. The findings in this research strongly facilitate the practical application of the proactive hedging option, making this lower-cost trading tool more feasible.

On the pricing of exotic options: A new closed-form valuation approach

Abstract We provide a novel method to estimate in a closed-form solution the option prices of various exotic options, using techniques based on Laplace–Beltrami operator for estimating diffusion boundary times. We estimate exit times and their expectations, the hitting probabilities, boundary local times until the first hitting and other probabilistic quantities and moment generating functions related to local hitting times. Our findings maybe of paramount importance for traders, investors, speculators and more broadly speaking for financial institutions.

The Pleasure and Pain of Investing: Parallels with Exotic Options

I bring together important ideas from behavioral economics, namely prospect theory, anchoring, and memory, with some elementary concepts from quantitative finance, namely the valuation of exotic options, to show how one may model the “pleasure and pain“ caused by investment in financial products. I shall qualify the downside of monitoring your profit and loss too frequently, and we will see how fortune can favor the brave.

Institucionalni i regulatorni aspekti primene finansijskih instrumenata za upravljanje rizikom u Republici Srbiji

This paper presents an analysis of institutional and regulatory framework for the implementation of a new group of financial instruments for risk management purposes that should enable faster economic development of the Republic of Serbia. The development of a country's economy is measured, among other things, in terms of the degree of development of its financial market, which consists of several segments. One of them is the trade of derivative securities. American authors point out that the economies of developing countries with more developed financial markets show better results and more stable economic growth compared to those which do not have or have neglected the development of financial markets and instruments. Investors are interested in hedging (securing) their investments, i.e. with the purpose of reducing or eliminating financial and business risks. The development and application of exotic options as modern financial instruments for risk management purposes is an important requirement for attracting foreign investments, as well as for increasing domestic investments, as it offers the possibility of reducing or eliminating risks that deter investors. In the Republic of Serbia, there is a regulatory and institutional framework for trade in all types of financial derivatives, including exotic options as well. Bearing in mind that the financial market in the Republic of Serbia is becoming a part of the global financial market, exotic options are one of the instruments for integrating our economy into global economic flows. However, the domestic financial market lacks financial derivative instruments that would give an additional impetus to its development and trade.

Exotic Options With Non-Constant Interest Rate & Local Volatility

A barrier option is a type of option that is path-dependent, and the value is determined by the share history from inception until maturity. In our case, we are evaluating a European up-and-out call option if the value of the option is greater that L throughout its lifetime, the option value becomes zero. An inherent difficulty in calculating this type of option, is an analytical or closed-form solution is not tractable.

Path independence of exotic options and convergence of binomial approximations

The analysis of the convergence of tree methods for pricing barrier and lookback options has been the subject of numerous publications aimed at describing, quantifying and improving the slow and oscillatory convergence in such methods. For barrier and lookback options, we find path-independent options whose price is exactly that of the original path-dependent option. The usual binomial models converge at a speed of order 1 / √n to the Black–Scholes price. Our new path-independent approach yields a convergence of order 1 / n. Further, we derive a closed-form formula for the coefficient of 1 / n in the expansion of the error of our path-independent pricing when the underlying is approximated by the Cox, Ross and Rubinstein (CRR) model. Using this, we obtain a corrected model with a convergence of order n-3/2 to the price of barrier and lookback options in the Black–Scholes model. Our results are supported and illustrated by numerical examples.

Integral equation formulation for shout options

We use an integral equation formulation approach to value shout options, which are exotic options giving an investor the ability to ``shout” and lock in profits while retaining the right to benefit from potentially favourable movements in the underlying asset price. Mathematically, the valuation is a free boundary problem involving an optimal exercise boundary which marks the region between shouting and not shouting. We also find the behaviour of the optimal exercise boundary for one- and two-shout options close to expiry. doi:10.1017/S1446181118000160

Proactive Hedging European Call Option Pricing with Linear Position Strategy

Proactive hedging option is an exotic European stock option designed for hedgers. Such option requires option holders to buy in (or sell out) the underlying asset (stock) and allows them to adjust the holdings of the underlying asset per its price changes within an option period. The proactive hedging option is an attractive choice for hedgers because its price is lower than that of classical options and because it completely hedges the risk of exposure for option holders. In this study, the underlying asset price movement is assumed to follow geometric fractional Brownian motion. The pricing formula for proactive hedging call options is derived with a linear position strategy by applying the risk-neutral evaluation principle. We use simulations to confirm that the price of this exotic option is always no more than that of the classical European option under the same parameters.

Collocating Local Volatility: A Competitive Alternative to Stochastic Local Volatility Models

We discuss a competitive alternative to stochastic local volatility models, namely the Collocating Volatility (CV) model, introduced in Grzelak (2016). The CV model consists of two elements, a 'kernel process' that can be efficiently evaluated and a local volatility function. The latter, based on stochastic collocation – e.g. Babuska et al. (2007), Witteveen et al. (2012) – connects the kernel process to the market and allows the CV model to be perfectly calibrated to European-type options. In this article we consider three different kernel process choices: the Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes and the Heston model. The kernel process controls the forward smile and allows for an accurate and efficient calibration to exotic options, while the perfect calibration to liquid market quotes is preserved. We confirm this by numerical experiments, in which we calibrate the OU-CV, CIR-CV and Heston-CV models to FX barrier options.

Laplace Transform Method for Pricing American CEV Strangles Option with Two Free Boundaries

Laplace transform method (LTM) has a lot of applications in the evaluation of European-style options and exotic options without early exercise features. However the Laplace transform methods for pricing American options have unsatisfactory accuracy and suffer from the instability. The aim of this paper is to develop a Laplace transform method for pricing American Strangles options with the underlying asset price following the constant elasticity volatility (CEV) models. By approximating the free boundaries, the Laplace transform is taken on a fixed space region to replace the moving boundaries space. After solving the linear system in Laplace space, Gaver-Stehfest formula (GSF) and hyperbola contour integral method (HCIM) are applied to compute the Laplace inversion. Numerical results show that the LTM-HCIM outperform the LTM-GSF in regard to the accuracy and stability for the option values.

Efficient control variate methods with applications to exotic options pricing under subordinated Brownian motion models

In this paper we apply both biased and multivariate control variate methods to some exotic option pricing problems with exponential subordinated Brownian motion models for the underlying asset prices. For both arithmetic Asian and basket options, control variates conditional on geometric means of asset prices are constructed. To reduce biases, we derive expressions of expectations in terms of expectations of subordinators for relevant random variables. Numerical results show that the constructed control variates are more efficient than the classical control variates in reducing variances when pricing Asian and basket options under the normal inverse Gaussian and variance gamma models. We believe that the great variance reduction gains are due to the high correlations between the option playoffs and the constructed control variates. The efficiencies realized by the control variates are even much more significant when they are combined with quasi-Monte Carlo methods. Details are given in Section 6.

Static hedging and pricing of exotic options with payoff frames

AbstractWe develop a general framework for statically hedging and pricing European‐style options with nonstandard terminal payoffs, which can be applied to mixed static–dynamic and semistatic hedges for many path‐dependent exotic options including variance swaps and barrier options. The goal is achieved by separating the hedging and pricing problems to obtain replicating strategies. Once prices have been obtained for a set of basis payoffs, the pricing and hedging of financial securities with arbitrary payoff functions is accomplished by computing a set of “hedge coefficients” for that security. This method is particularly well suited for pricing baskets of options simultaneously, and is robust to discontinuities of payoffs. In addition, the method enables a systematic comparison of the value of a payoff (or portfolio) across a set of competing model specifications with implications for security design.

INTEGRAL EQUATION FORMULATION FOR SHOUT OPTIONS

We use an integral equation formulation approach to value shout options, which are exotic options giving an investor the ability to “shout” and lock in profits while retaining the right to benefit from potentially favourable movements in the underlying asset price. Mathematically, the valuation is a free boundary problem involving an optimal exercise boundary which marks the region between shouting and not shouting. We also find the behaviour of the optimal exercise boundary for one- and two-shout options close to expiry.