Pricing power options with a generalized jump diffusion

In this article, the valuation of power option is investigated when the dynamic of the stock price is governed by a generalized jump-diffusion Markov-modulated model. The systematic risk is characterized by the diffusion part, and the non systematic risk is characterized by the pure jump process. The jumps are described by a generalized renewal process with generalized jump amplitude. By introducing NASDAQ Index Model, their risk premium is identified respectively. A risk-neutral measure is identified by employing Esscher transform with two families of parameters, which represent the two parts risk premium. In this article, the non systematic risk premium is considered, based on which the price of power option is studied under the generalized jump-diffusion Markov-modulated model. In the case of a special renewal process with log double exponential jump amplitude, the accurate expressions for the Esscher parameters and the pricing formula are provided. By numerical simulation, the influence of the non systematic risk’s price and the index of the power options on the price of the option is depicted.     

Risk-minimizing pricing and hedging foreign currency options under regime-switching jump-diffusion models

This article mainly investigates risk-minimizing European currency option pricing and hedging strategy when the spot foreign exchange rate is driven by a Markov-modulated jump-diffusion model. We suppose the domestic and foreign money market floating interest rates, the drift, and the volatility of the exchange rate dynamics all depend on the state of the economy, which is modeled by a continuous-time hidden Markov chain. The model considered in this article will provide market practitioners with flexibility in characterizing the dynamics of the spot foreign exchange rate. Using the minimal martingale measure, we obtain a system of coupled partial-differential-integral equations satisfied by the currency option price and find the corresponding hedging strategies and the residual risk. According to simulation of currency option prices in the special case of double exponential jump-diffusion regime-switching model, we further discuss and show the effects of the parameters on the prices.     

Analytical and numerical studies on the second-order asymptotic expansion method for European option pricing under two-factor stochastic volatilities

The celebrated Black–Scholes model made the assumption of constant volatility but empirical studies on implied volatility and asset dynamics motivated the use of stochastic volatilities. Christoffersen in 2009 showed that multi-factor stochastic volatilities models capture the asset dynamics more realistically. Fouque in 2012 used it to price European options. In 2013, Chiarella and Ziveyi considered Christoffersen’s ideas and introduced an asset dynamics where the two volatilities of the Heston type act separately and independently on the asset price, and using Fourier transform for the asset price process and double Laplace transform for the two volatilities processes, solved a pricing problem for American options. This paper considers the Chiarella and Ziveyi model and parameterizes it so that the volatilities revert to the long-run-mean with reversion rates that mimic fast (for example daily) and slow (for example seasonal) random effects. Applying asymptotic expansion method presented by Fouque in 2012, we make an extensive and detailed derivation of the approximation prices for European options. We also present numerical studies on the behavior and accuracy of our first- and second-order asymptotic expansion formulas.     

Empirical Performance and Asset Pricing in Hidden Markov Models

Abstract

To improve the empirical performance of the Black-Scholes model, many alternative models have been proposed to address leptokurtic feature, volatility smile, and volatility clustering effects of the asset return distributions. However, analytical tractability remains a problem for most alternative models. In this article, we study a class of hidden Markov models including Markov switching models and stochastic volatility models, that can incorporate leptokurtic feature, volatility clustering effects, as well as provide analytical solutions to option pricing. We show that these models can generate long memory phenomena when the transition probabilities depend on the time scale. We also provide an explicit analytic formula for the arbitrage-free price of the European options under these models. The issues of statistical estimation and errors in option pricing are also discussed in the Markov switching models.     

A memory reduction method in pricing American options

This paper is concerned with the pricing of American options by simulation methods. In the traditional methods, in order to determine when to exercise, we have to store the simulated asset prices at all time steps on all paths. If N time steps and M paths are used, then the storage requirement is O(MN). In this paper, we present a simulation method for pricing American options where the number of storage required only grows like O(M). The only additional computational cost is that we have to generate each random number twice instead of once. For machines with limited memory, we can now use a larger N to improve the accuracy in pricing the options.     

Efficient Monte Carlo option pricing under CEV model

One of the financial model with nonconstant volatiltiy is the constant elasticity of varinace model, or CEV model for short. The CEV model is an altrnative to the Black–Scholes model of stock price movements. In this diffusion process, unlike the Black–Scholes model, the volatility is a function of the stock price and involves two parameters. In this article, we propose an efficient Monte-Carlo algorithm for pricing arithmetic Asian option under CEV model. In an earlier work by Mehrdoust, an efficient Monte Carlo simulation algorithm for pricing arithmetic Asian options under Black–Scholes model is proposed. The proposed algorithm has proved extremely successful in decreasing the standard deviation and the error of simulation in pricing of the arithmetic Asian options. In this article, we find that the proposed algorithm under the geometric Brownian motion assumption in the Black–Scholes model can effectively apply for pricing arithmetic Asian options when the stock price process follows the CEV model. Numerical experiments show that our algorithm gives very accurate results.     

Valuing equity-indexed annuities with icicled barrier options

Inspired by the recent popularity of autocallable structured products, this paper intends to enhance equity-indexed annuities (EIAs) by introducing a new class of barrier options, termed icicled barrier options. The new class of options has a vertical (icicled) barrier along with the horizontal one of the ordinary barrier options, which may act as an additional knock-in or knock-out trigger. To improve the crediting method of EIAs, we propose a new EIA design, termed autocallable EIA, with payoff structure similar to the autocallable products except for the minimum guarantee, and further investigate the possibility of embedding various icicled barrier options into the plain point-to-point or the ratchet EIAs. Explicit pricing formulas for the proposed EIAs and the icicled barrier options are obtained under the Black–Scholes model. To the purpose, we derive the joint distribution of the logarithmic returns at the icicled time and the maturity, and their running maximum. As an application of the well-known reflection principle, the derivation itself is an interesting probability problem and the joint distribution plays a key role in the subsequent pricing stage. Our option pricing result can be easily transferred to EIAs or other equity-linked products. The pricing formulas for the EIAs and the options are illustrated through numerical examples.     

Hedging of contingent claims written on non traded assets under Markov-modulated models

ABSTRACT

This paper studies the hedging problem of European contingent claims when the underlying asset is non traded. We assume that the share prices of the assets are governed by Markov-modulated processes; that is, the market parameters switch over the time according to a finite-state continuous time Markov chain. Due to the presence of Markov chain the non traded asset, the market which we consider is incomplete, we shall use the local risk minimization method to obtain an optimal hedging strategy in a closed-form for an investor. Finally, numerical illustrations of an optimal hedging strategy are given by the Monte Carlo simulation.     

A new hybrid Monte Carlo simulation for Asian options pricing

The aim of this paper is to present a new hybrid algorithm for pricing financial derivatives in the arithmetic Asian options. In this paper, two variance reduction techniques are combined, the multiple control variates (MCV) and the antithetic variates (AV). We propose an efficient algorithm for pricing arithmetic Asian options based on the AV and the MCV procedures. A detailed numerical study illustrates the efficiency of the proposed algorithm.     

A Stochastic Volatility Model With Realized Measures for Option Pricing

Abstract

Based on the fact that realized measures of volatility are affected by measurement errors, we introduce a new family of discrete-time stochastic volatility models having two measurement equations relating both observed returns and realized measures to the latent conditional variance. A semi-analytical option pricing framework is developed for this class of models. In addition, we provide analytical filtering and smoothing recursions for the basic specification of the model, and an effective MCMC algorithm for its richer variants. The empirical analysis shows the effectiveness of filtering and smoothing realized measures in inflating the latent volatility persistence—the crucial parameter in pricing Standard and Poor’s 500 Index options.     

Stability in distribution of Markov-modulated stochastic differential delay equations with reflection

In this paper, we discuss Markov-modulated stochastic differential delay equations with reflection. The aim of this paper is to extend the stability criterion in distribution as in [Systems and Control Letters Vol 50 (2003) 195–207] to the equations with reflection. Interesting examples are provided to demonstrate not only our theory, but also the importance of Markov-modulating.     

An analytical approximation method for pricing barrier options under the double Heston model

Abstract

The purpose of the paper is to provide an efficient pricing method for single barrier options under the double Heston model. By rewriting the model as a singular and regular perturbed BS model, the double Heston model can separately mimic a fast time-scale and a slow time-scale. With the singular and regular perturbation techniques, we analytically derive the first-order asymptotic expansion of the solution to a barrier option pricing partial differential equation. The convergence and efficiency of the approximate method is verified by Monte Carlo simulation. Numerical results show that the presented asymptotic pricing method is fast and accurate.     

Pricing arithmetic Asian option under a two-factor stochastic volatility model with jumps

This paper presents an efficient Monte Carlo simulation scheme based on the variance reduction methods to evaluate arithmetic average Asian options in the context of the double Heston's stochastic volatility model with jumps. This paper consists of two essential parts. The first part presents a new flexible stochastic volatility model, namely, the double Heston model with jumps. In the second part, by combining two variance reduction procedures via Monte Carlo simulation, we propose an efficient Monte Carlo simulation scheme for pricing arithmetic average Asian options under the double Heston model with jumps. Numerical results illustrate the efficiency of our method.     

Fourier-cosine method for pricing forward starting options with stochastic volatility and jumps

This article provides an efficient method for pricing forward starting options under stochastic volatility model with double exponential jumps. The forward characteristic function of the log asset price is derived and thereby forward starting options are well evaluated by Fourier-cosine technique. Based on adaptive simulated annealing algorithm, the model is calibrated to obtain the estimated parameters. Numerical results show that the pricing method is accurate and fast. Double exponential jumps have pronounced impacts on long-term forward starting options prices. Stochastic volatility model with double exponential jumps fits forward implied volatility smile pretty well in contrast to stochastic volatility model.     

A quadratically convergent algorithm for first passage time distributions in the Markov-modulated Brownian motion

A Markov-modulated Brownian motion (MMBM) is a substantial generalization of the classical Brownian Motion and is obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As with Brownian Motion, the time-dependent analysis of the MMBM becomes easy once the first passage times between levels are determined. However, in the MMBM those distributions cannot be obtained explicitly, and we need efficient algorithms to compute them. In this article, we provide a powerful approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows without Brownian components that weakly converge to the MMBM. Our main result is a Riccati equation for an associated matrix of transforms that satisfies conditions for the Newton scheme to have quadratic convergence and thus yields a very practical tool. The solution of that Riccati equation determines needed first passage times in the MMBM without much additional work. The success of our approach, which is based essentially on first-order fluid flows and a stochastic limit process, is argued to be due to the way we have isolated certain terms involving the quadratic variation effects of the Brownian. As an illustration of our algorithm, we present a numerical example of time-dependent results for a MMBM considered by Asmussen for which he determined (only) the eventual first return probabilities which we use here as an accuracy check.     

The homotopy perturbation method for the Black–Scholes equation

The homotopy perturbation method is designed to obtain a quick and accurate solution to the Black–Scholes equation and boundary conditions for a European option pricing problem. The problem of pricing a European option can be cast a partial differential equation. The analytical solution of the equation is calculated in the form of a convergent power series with easily computable components.     

On the arbitrage price of European call options

We show that in a discrete price and discrete time model for option pricing, specifically that given by the Cox–Ross–Rubinstein model, the arbitrage price of a European call option can depend on parameters other than volatility (the standard deviation of the log asset price). We provide two theorems to illustrate this phenomenon. Our first theorem considers two securities with the same volatility so that at a specified time n₀, with probability near 1, the two securities are equal. If their call options differ, both the discounted securities will be martingales. Our second theorem considers two securities with the same volatility so that at times n = 0, ..., N ? 1 the securities are equal with probability near 1. If their call options differ, one of the discounted securities will be a martingale and the other discounted security will be a supermartingale.     

Hedging processes for catastrophe options

This paper considers the problem of pricing and hedging of certain catastrophe (CAT) options. In particular, a self-financing hedging strategy that minimizes the risk in the loss period and replicates the option in the development period is proposed.     

Brownian–Laplace Motion and Its Use in Financial Modelling

Brownian-Laplace motion is a Lévy process which has both continuous (Brownian) and discontinuous (Laplace motion) components. The increments of the process follow a generalized normal Laplace (GNL) distribution which exhibits positive kurtosis and can be either symmetrical or exhibit skewness. The degree of kurtosis in the increments increases as the time between observations decreases. This and other properties render Brownian-Laplace motion a good candidate model for the motion of logarithmic stock prices. An option pricing formula for European call options is derived and it is used to calculate numerically the value of such an option both using nominal parameter values (to explore its dependence upon them) and those obtained as estimates from real stock price data.     

Saddlepoint approximations to tail expectations under non-Gaussian base distributions: option pricing applications

The saddlepoint approximation formulas provide versatile tools for analytic approximation of the tail expectation of a random variable by approximating the complex Laplace integral of the tail expectation expressed in terms of the cumulant generating function of the random variable. We generalize the saddlepoint approximation formulas for calculating tail expectations from the usual Gaussian base distribution to an arbitrary base distribution. Specific discussion is presented on the criteria of choosing the base distribution that fits better the underlying distribution. Numerical performance and comparison of accuracy are made among different saddlepoint approximation formulas. Improved accuracy of the saddlepoint approximations to tail expectations is revealed when proper base distributions are chosen. We also demonstrate enhanced accuracy of the generalized saddlepoint approximation formulas under non-Gaussian base distributions in pricing European options on continuous integrated variance under the Heston stochastic volatility model.