Option Valuation with a Discrete-Time Double Markovian Regime-Switching Model

Abstract

This article develops an option valuation model in the context of a discrete-time double Markovian regime-switching (DMRS) model with innovations having a generic distribution. The DMRS model is more flexible than the traditional Markovian regime-switching model in the sense that the drift and the volatility of the price dynamics of the underlying risky asset are modulated by two observable, discrete-time and finite-state Markov chains, so that they are not perfectly correlated. The states of each of the chains represent states of proxies of (macro)economic factors. Here we consider the situation that one (macro)economic factor is caused by the other (macro)economic factor. The market model is incomplete, and so there is more than one equivalent martingale measure. We employ a discrete-time version of the regime-switching Esscher transform to determine an equivalent martingale measure for valuation. Different parametric distributions for the innovations of the price dynamics of the underlying risky asset are considered. Simulation experiments are conducted to illustrate the implementation of the model and to document the impacts of the macroeconomic factors described by the chains on the option prices under various different parametric models for the innovations.     

Pricing Options Under a Generalized Markov-Modulated Jump-Diffusion Model

Abstract

We consider the pricing of options when the dynamics of the risky underlying asset are driven by a Markov-modulated jump-diffusion model. We suppose that the market interest rate, the drift and the volatility of the underlying risky asset switch over time according to the state of an economy, which is modelled by a continuous-time Markov chain. The measure process is defined to be a generalized mixture of Poisson random measure and encompasses a general class of processes, for example, a generalized gamma process, which includes the weighted gamma process and the inverse Gaussian process. Another interesting feature of the measure process is that jump times and jump sizes can be correlated in general. The model considered here can provide market practitioners with flexibility in modelling the dynamics of the underlying risky asset. We employ the generalized regime-switching Esscher transform to determine an equivalent martingale measure in the incomplete market setting. A system of coupled partial-differential-integral equations satisfied by the European option prices is derived. We also derive a decomposition result for an American put option into its European counterpart and early exercise premium. Simulation results of the model have been presented and discussed.     

Option pricing under regime-switching models: Novel approaches removing path-dependence

A well-known approach for the pricing of options under regime-switching models is to use the regime-switching Esscher transform (also called regime-switching mean-correcting martingale measure) to obtain risk-neutrality. One way to handle regime unobservability consists in using regime probabilities that are filtered under this risk-neutral measure to compute risk-neutral expected payoffs. The current paper shows that this natural approach creates path-dependence issues within option price dynamics. Indeed, since the underlying asset price can be embedded in a Markov process under the physical measure even when regimes are unobservable, such path-dependence behavior of vanilla option prices is puzzling and may entail non-trivial theoretical features (e.g., time non-separable preferences) in a way that is difficult to characterize. This work develops novel and intuitive risk-neutral measures that can incorporate regime risk-aversion in a simple fashion and which do not lead to such path-dependence side effects. Numerical schemes either based on dynamic programming or Monte-Carlo simulations to compute option prices under the novel risk-neutral dynamics are presented.     

An explicit analytic formula for pricing barrier options with regime switching

This paper investigates the valuation of a European-style barrier option in a Markovian, regime-switching, Black–Scholes–Merton economy, where the price process of an underlying risky asset is assumed to follow a Markov-modulated geometric Brownian motion. An explicit analytic solution in infinite series form for the price of a European-style barrier option in a two-state regime is presented.     

A Simple Novel Approach to Valuing Risky Zero Coupon Bond in a Markov Regime Switching Economy

We have addressed the problem of pricing risky zero coupon bond in the framework of Longstaff and Schwartz structural type model by pricing it as a Down-and-Out European Barrier Call option on the company’s asset-debt ratio assuming Markov regime switching economy. The growth rate and the volatility of the stochastic asset debt ratio is driven by a continuous time Markov chain which signifies state of the economy. Regime Switching renders market incomplete and selection of a Equivalent martingale measure (EMM) becomes a subtle issue. We price the zero coupon risky bond utilizing the powerful technique of Risk Minimizing hedging of the underlying Barrier option under the so called “Risk Minimal” martingale measure via computing the bond default probability.     

广义Black-Scholes模型期权定价新方法--保险精算方法

利用公平保费原则和价格过程的实际概率测度推广了Mogens Bladt和Tina Hviid Rydberg的结果．在无中间红利和有中间红利两种情况下，把Black-Scholes模型推广到无风险资产(债券或银行存款)具有时间相依的利率和风险资产(股票)也具有时间相依的连续复利预期收益率和波动率的情况，在此情况下获得了欧式期权的精确定价公式以及买权与卖权之间的平价关系．给出了风险资产(股票)具有随机连续复利预期收益率和随机波动率的广义Black-Scholes模型的期权定价的一般方法．利用保险精算方法给出了股票价格遵循广义Ornstein-Uhlenback过程模型的欧式期权的精确定价公式和买权和卖权之间的平价关系．     

Equilibrium pricing bounds on option prices

We consider the problem of valuing European options in a complete market but with incomplete data. Typically, when the underlying asset dynamics is not specified, the martingale probability measure is unknown. Given a consensus on the actual distribution of the underlying price at maturity, we derive an upper bound on the call option price by putting two kinds of restrictions on the pricing probability measure. First, we put a restriction on the second risk-neutral moment of the underlying asset terminal value. Second, from equilibrium pricing arguments one can put a monotonicity restriction on the Radon-Nikodym density of the pricing probability with respect to the true probability measure. This density is restricted to be a nonincreasing function of the underlying price at maturity. The bound appears then as the solution of a constrained optimization problem and we adopt a duality approach to solve it. Explicit bounds are provided for the call option. Finally, we provide a numerical example.     

双跳-扩散过程下的脆弱期权定价

本文考虑含有交易对手违约风险的衍生产品的定价,以公司价值信用风险模型为基础,在标的资产价格和公司价值均服从跳-扩散过程的情况下,运用结构化的方法对脆弱期权定价进行建模,建立了双跳-扩散过程下的脆弱期权定价模型,分别在公司负债固定和随机的情况下推导出了脆弱期权的定价公式.     

Option Pricing and Hedging under a Markov Switching Lévy Process Model

In this paper,we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging.In this model,the market interest rate,the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process.We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure.The option price using this model is obtained by the Fourier transform method.We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

A game theoretic approach to option valuation under Markovian regime-switching models

In this paper, we consider a game theoretic approach to option valuation under Markovian regime-switching models, namely, a Markovian regime-switching geometric Brownian motion (GBM) and a Markovian regime-switching jump-diffusion model. In particular, we consider a stochastic differential game with two players, namely, the representative agent and the market. The representative agent has a power utility function and the market is a “fictitious” player of the game. We also explore and strengthen the connection between an equivalent martingale measure for option valuation selected by an equilibrium state of the stochastic differential game and that arising from a regime switching version of the Esscher transform. When the stock price process is governed by a Markovian regime-switching GBM, the pricing measures chosen by the two approaches coincide. When the stock price process is governed by a Markovian regime-switching jump-diffusion model, we identify the condition under which the pricing measures selected by the two approaches are identical.     

Option Pricing in Illiquid Markets with Jumps

ABSTRACT

The classical linear Black–Scholes model for pricing derivative securities is a popular model in the financial industry. It relies on several restrictive assumptions such as completeness, and frictionless of the market as well as the assumption on the underlying asset price dynamics following a geometric Brownian motion. The main purpose of this paper is to generalize the classical Black–Scholes model for pricing derivative securities by taking into account feedback effects due to an influence of a large trader on the underlying asset price dynamics exhibiting random jumps. The assumption that an investor can trade large amounts of assets without affecting the underlying asset price itself is usually not satisfied, especially in illiquid markets. We generalize the Frey–Stremme nonlinear option pricing model for the case the underlying asset follows a Lévy stochastic process with jumps. We derive and analyze a fully nonlinear parabolic partial-integro differential equation for the price of the option contract. We propose a semi-implicit numerical discretization scheme and perform various numerical experiments showing the influence of a large trader and intensity of jumps on the option price.     

固定执行价格下回望看涨期权保险精算定价

利用保险精算方法,将期权定价问题转化为纯保费确定问题,根据股票价格过程的实际概率测度推导出了无风险利率为常数时,固定执行价格下回望看涨期权定价公式,验证了当标的资产的期望收益率等于无风险利率时,保险精算定价和风险中性定价的一致性.最后通过实例分析了保险精算价格和风险中性价格的差异,并利用Matlab编程得到了保险精算价格与标的资产期望收益率之间的关系.     

SV模型下的期权定价和风险计量

本文利用SV（Stochastic Variance）模型对期权基础资产的收益过程进行统计描述,在同时给出期权定价和市场风险计量之后,又给出定价置信区间和风险置信区间的估计。文中对SV模型作了分析和比较,利用自适应滤波方法对模型的建立和参数的估计给出了简单的方法,最后还对SV模型作了模拟分析并计算了期权定价和风险计量的一个例子。     

Asian option as a fixed-point

We characterize the price of an Asian option, a financial contract, as a fixed-point of a non-linear operator. In recent years, there has been interest in incorporating changes of regime into the parameters describing the evolution of the underlying asset price, namely the interest rate and the volatility, to model sudden exogenous events in the economy. Asian options are particularly interesting because the payoff depends on the integrated asset price. We study the case of both floating- and fixed-strike Asian call options with arithmetic averaging when the asset follows a regime-switching geometric Brownian motion with coefficients that depend on a Markov chain. The typical approach to finding the value of a financial option is to solve an associated system of coupled partial differential equations. Alternatively, we propose an iterative procedure that converges to the value of this contract with geometric rate using a classical fixed-point theorem.     

Pricing swing options with regime switching

Gasoline price is highly volatile and exhibits Markov regime-switching process. In the electricity and the natural gas markets, “swing” options, which can provide some protection against day-to-day price fluctuations, are used to incorporate flexibility in delivering acquired energy. We propose a framework for pricing swing options for an underlying variable that follows a regime-switching process. We study the proposed framework in the gasoline industry for pricing swing options under price uncertainty by extracting the gasoline market information, estimating the parameters of the regime-switching process, and then presenting different numerical examples.     

The MEMMs for Markov-Modulated GBMs

In this paper, we consider the option pricing problem when the risky underlying assets are driven by Markov-modulated geometric Brownian motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the risky asset, depend on unobservable states of the economy which are modeled by a continuous-time hidden Markov chain. The market described by the Markov-modulated GBM model is incomplete in general, and, hence, the martingale measure is not unique. We adopt the minimal relative entropy martingale measure (MEMM) for the Markov-modulated GBM model as the suitable martingale measure and we obtain the MEMM for the market in general sense.     

Asymptotic analysis of option pricing in a Markov modulated market

We address asymptotic analysis of option pricing in a regime switching market where the risk free interest rate, growth rate and the volatility of the stocks depend on a finite state Markov chain. We study two variations of the chain namely, when the chain is moving very fast compared to the underlying asset price and when it is moving very slow. Using quadratic hedging and asymptotic expansion, we derive corrections on the locally risk minimizing option price.     

Pricing annuity guarantees under a double regime-switching model

This paper is concerned with the valuation of equity-linked annuities with mortality risk under a double regime-switching model, which provides a way to endogenously determine the regime-switching risk. The model parameters and the reference investment fund price level are modulated by a continuous-time, finite-time, observable Markov chain. In particular, the risk-free interest rate, the appreciation rate, the volatility and the martingale describing the jump component of the reference investment fund are related to the modulating Markov chain. Two approaches, namely, the regime-switching Esscher transform and the minimal martingale measure, are used to select pricing kernels for the fair valuation. Analytical pricing formulas for the embedded options underlying these products are derived using the inverse Fourier transform. The fast Fourier transform approach is then used to numerically evaluate the embedded options. Numerical examples are provided to illustrate our approach.     

A hidden Markov regime-switching model for option valuation

We investigate two approaches, namely, the Esscher transform and the extended Girsanov’s principle, for option valuation in a discrete-time hidden Markov regime-switching Gaussian model. The model’s parameters including the interest rate, the appreciation rate and the volatility of a risky asset are governed by a discrete-time, finite-state, hidden Markov chain whose states represent the hidden states of an economy. We give a recursive filter for the hidden Markov chain and estimates of model parameters using a filter-based EM algorithm. We also derive predictors for the hidden Markov chain and some related quantities. These quantities are used to estimate the price of a standard European call option. Numerical examples based on real financial data are provided to illustrate the implementation of the proposed method.     

Third-order extensions of Lo’s semiparametric bound for European call options

Computing semiparametric bounds for option prices is a widely studied pricing technique. In contrast to parametric pricing techniques, such as Monte-Carlo simulations, semiparametric pricing techniques do not require strong assumptions about the underlying asset price distribution. We extend classical results in this area. Specifically, we derive closed-form semiparametric bounds for the payoff of a European call option, given up to third-order moment (i.e., mean, variance, and skewness) information on the underlying asset price. We analyze how these bounds tighten the corresponding bounds, when only second-order moment (i.e., mean and variance) information is provided. We describe applications of these results in the context of option pricing; as well as in other areas such as inventory management, and actuarial science.