Risk Minimizing Option Pricing in a Regime Switching Market

Abstract

We study option pricing in a regime switching market where the risk free interest rate, growth rate and the volatility of a stock depends on a finite state Markov chain. Using a minimal martingale measure we show that the risk minimizing option price satisfies a system of Black–Scholes partial differential equations with weak coupling.     

Valuation and hedging strategy of currency options under regime-switching jump-diffusion model

The main purpose of this thesis is in analyzing and empirically simulating risk minimizing European foreign exchange option pricing and hedging strategy when the spot foreign exchange rate is governed by a Markov-modulated jump-diffusion model. The domestic and foreign money market interest rates, the drift and the volatility of the exchange rate dynamics all depend on a continuous-time hidden Markov chain which can be interpreted as the states of a macro-economy. In this paper, we will provide a practical lognormal diffusion dynamic of the spot foreign exchange rate for market practitioners. We employing the minimal martingale measure to demonstrate a system of coupled partial-differential-integral equations satisfied by the currency option price and attain the corresponding hedging schemes and the residual risk. Numerical simulations of the double exponential jump diffusion regime-switching model are used to illustrate the different effects of the various parameters on currency option prices.     

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.     

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.     

Pricing Defaultable Bonds in a Markov Modulated Market

We address the problem of pricing defaultable bonds in a Markov modulated market. Using Merton's structural approach we show that various types of defaultable bonds are combination of European type contingent claims. Thus pricing a defaultable bond is tantamount to pricing a contingent claim in a Markov modulated market. Since the market is incomplete, we use the method of quadratic hedging and minimal martingale measure to derive locally risk minimizing derivative prices, hedging strategies and the corresponding residual risks. The price of defaultable bonds are obtained as solutions to a system of PDEs with weak coupling subject to appropriate terminal and boundary conditions. We solve the system of PDEs numerically and carry out a numerical investigation for the defaultable bond prices. We compare their credit spreads with some of the existing models. We observe higher spreads in the Markov modulated market. We show how business cycles can be easily incorporated in the proposed framework. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low risk-free interest rate, high payout rate and high volatility.     

面对成比例交易成本的情况下外汇期权的定价

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor's expected utility. Option prices are determined from the investor's basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility.Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

FOREIGN CURRENCY OPTION PRICING WITH PROPORTIONAL TRANSACTION COSTS

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor＇s expected utility. Option prices are determined from the investor＇s basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility. Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

Option pricing when the regime-switching risk is priced

We study the pricing of an option when the price dynamic of the underlying risky asset is governed by a Markov-modulated geometric Brownian motion. We suppose that the drift and volatility of the underlying risky asset are modulated by an observable continuous-time, finite-state Markov chain. We develop a two- stage pricing model which can price both the diffusion risk and the regime-switching risk based on the Esscher transform and the minimization of the maximum entropy between an equivalent martingale measure and the real-world probability measure over different states. Numerical experiments are conducted and their results reveal that the impact of pricing regime-switching risk on the option prices is significant.     

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.     

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 Jump-Diffusion Processes with Regime Switching

We study an incomplete market model, based on jump-diffusion processes with parameters that are switched at random times. The set of equivalent martingale measures is determined. An analogue of the fundamental equation for the option price is derived. In the case of the two-state hidden Markov process we obtain explicit formulae for the option prices. Furthermore, we numerically compare the results corresponding to different equivalent martingale measures.     

Regime-switching stochastic volatility model: estimation and calibration to VIX options

We develop and implement a method for maximum likelihood estimation of a regime-switching stochastic volatility model. Our model uses a continuous time stochastic process for the stock dynamics with the instantaneous variance driven by a Cox–Ingersoll–Ross process and each parameter modulated by a hidden Markov chain. We propose an extension of the EM algorithm through the Baum–Welch implementation to estimate our model and filter the hidden state of the Markov chain while using the VIX index to invert the latent volatility state. Using Monte Carlo simulations, we test the convergence of our algorithm and compare it with an approximate likelihood procedure where the volatility state is replaced by the VIX index. We found that our method is more accurate than the approximate procedure. Then, we apply Fourier methods to derive a semi-analytical expression of S&P500 and VIX option prices, which we calibrate to market data. We show that the model is sufficiently rich to encapsulate important features of the joint dynamics of the stock and the volatility and to consistently fit option market prices.     

Rare Shock,Two-Factor Stochastic Volatility and Currency Option Pricing

Abstract

In this paper, we develop an option valuation model where the dynamics of the spot foreign exchange rate is governed by a two-factor Markov-modulated jump-diffusion process. The short-term fluctuation of stochastic volatility is driven by a Cox–Ingersoll–Ross (CIR) process and the long-term variation of stochastic volatility is driven by a continuous-time Markov chain which can be interpreted as economy states. Rare events are governed by a compound Poisson process with log-normal jump amplitude and stochastic jump intensity is modulated by a common continuous-time Markov chain. Since the market is incomplete under regime-switching assumptions, we determine a risk-neutral martingale measure via the Esscher transform and then give a pricing formula of currency options. Numerical results are presented for investigating the impact of the long-term volatility and the annual jump intensity on option prices.     

Multiscale analysis of a perpetual American option with the stochastic elasticity of variance

A perpetual American option is considered under a generalized model of the constant elasticity of variance model where the constant elasticity is perturbed by a small fast mean-reverting Ornstein–Uhlenbeck process. By using a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on option prices as well as optimal exercise prices. Our results improve the existing option price structure in view of flexibility and applicability through the market price of risk. The revealed results may provide useful information on real option problems.     

A numerical algorithm for pricing electricity derivatives for jump-diffusion processes based on continuous time lattices

We present a numerical algorithm for pricing derivatives on electricity prices. The algorithm is based on approximating the generator of the underlying price process on a lattice of prices, resulting in an approximation of the stochastic process by a continuous time Markov chain. We numerically study the rate of convergence of the algorithm for the case of the Merton jump-diffusion model and apply the algorithm to calculate prices and sensitivities of both European and Bermudan electricity derivatives when the underlying price follows a stochastic process which exhibits both fast mean-reversion and jumps of large magnitude.     

Pricing electricity derivatives within a Markov regime-switching model: a risk premium approach

In this paper we derive analytic formulas for electricity derivatives under assumption that electricity spot prices follow a 3-regime Markov regime-switching model with independent spikes and drops and periodic transition matrix. Since the classical derivatives pricing methodology cannot be used in the case of non-storable commodities, we employ the concept of the risk premium. The obtained theoretical results are then used for the European Energy Exchange data analysis. We calculate the risk premium in the case of the calibrated 3-regime MRS model. We find a time varying structure of the risk premium and an evidence for a negative risk premium (or positive forward premium), especially at short times before delivery. Finally, we use the obtained risk premium to calculate prices of European options written on spot, as well as, forward prices.     

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.     

无套利期权定价模型在一般均衡框架下的一致性研究

期权定价有无套利方法和一般均衡方法两种.本文在一般均衡框架下构造了一个允许连续消费的简单经济模型,并将基于无套利方法的期权定价模型中所假定的标的证券的价格变化动态过程内生化于理性预期均衡中.在常数相对风险厌恶(CRRA)的效用函数的条件下,我们推导出Merton(1973)期权定价公式,从而证明无套利方法与均衡方法的内在一致性,而CRRA这种类型的效用函数是无套利定价模型在一般均衡框架中成立的充分条件.本文进一步将此模型在一个简单经济中扩展到m种证券的情况,也得到相似的结论.     

Explicit solutions to European options in a regime-switching economy

We provide closed-form solutions for European option values when the dynamics of both the short rate and volatility of the underlying price process are modulated by a continuous-time Markov chain with a finite number of “economic states”. Extensions involving dividends, currencies and cost of carry are further explored.     

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.