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 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.     

A PDE representation of the density of the minimal entropy martingale measure in stochastic volatility markets

Under general conditions stated in Rheinländer [An entropy approach to the stein/stein model with correlation. Preprint, 2003, ETH Zürich.], we prove that in a stochastic volatility market the Radon–Nikodym density of the minimal entropy martingale measure (MEMM) can be expressed in terms of the solution of a semilinear PDE. The semilinear PDE is suggested by the dynamic programming approach to the utility indifference pricing problem of contingent claims. One of our main results is the existence and uniqueness of a classical solution of the semilinear PDE in the case of a general stochastic volatility model with additive noise correlated with the asset price. Our results are applied to the Stein–Stein and Heston stochastic volatility models.     

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.     

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.     

Market Viability and Martingale Measures under Partial Information

We consider a financial market model with a single risky asset whose price process evolves according to a general jump-diffusion with locally bounded coefficients and where market participants have only access to a partial information flow. For any utility function, we prove that the partial information financial market is locally viable, in the sense that the optimal portfolio problem has a solution up to a stopping time, if and only if the (normalised) marginal utility of the terminal wealth generates a partial information equivalent martingale measure (PIEMM). This equivalence result is proved in a constructive way by relying on maximum principles for stochastic control problems under partial information. We then characterize a global notion of market viability in terms of partial information local martingale deflators (PILMDs). We illustrate our results by means of a simple example.     

Pricing currency options under two-factor Markov-modulated stochastic volatility models

This article investigates the valuation of currency options when the dynamic of the spot Foreign Exchange (FX) rate is governed by a two-factor Markov-modulated stochastic volatility model, with the first stochastic volatility component driven by a lognormal diffusion process and the second independent stochastic volatility component driven by a continuous-time finite-state Markov chain model. The states of the Markov chain can be interpreted as the states of an economy. We employ the regime-switching Esscher transform to determine a martingale pricing measure for valuing currency options under the incomplete market setting. We consider the valuation of the European-style and American-style currency options. In the case of American options, we provide a decomposition result for the American option price into the sum of its European counterpart and the early exercise premium. Numerical results are included.     

Option Pricing for Time-Change Exponential Levy Model Under Memm

The purpose of this article is to study the rational evaluation of European options price when the underlying price process is described by a time-change Levy process. European option pricing formula is obtained under the minimal entropy martingale measure （MEMM） and applied to several examples of particular time-change Levy processes. It can be seen that the framework in this paper encompasses the Black-Scholes model and almost all of the models proposed in the subordinated market.     

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.     

General dynamic term structures under default risk

We consider the problem of modelling the term structure of defaultable bonds, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term in the forward rate approach by Heath et al. (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability. In this framework, we derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes.     

Minimal Martingale Measures for Discrete-time Incomplete Financial Markets

Abstract In this note, we give a characterization of the minimal martingale measure for a general discrete-time incomplete financial market.Then we concretely work out the minimal martingale messure for a specificdiscrete-time market model in which the assets'returns in different times are independent.     

Discrete-time local risk minimization of payment processes and applications to equity-linked life-insurance contracts

We develop a theory of local risk minimization for payment processes in discrete time, and apply this theory to the pricing and hedging of equity-linked life-insurance contracts. Thus, we extend the work of Møller (2001a) in several directions: from risk minimization (which is done under a martingale measure) to local risk minimization (which is done under an arbitrary measure), from single claims to payment processes, from complete financial markets to possibly incomplete financial markets, from a single risky asset to several risky assets, and from finite state spaces to general state spaces.Moreover, we show that, when tradable financial assets are independent of mortality, a locally risk-minimizing hedging strategy for most claims in the combined financial and mortality market (such as those arising from equity-indexed annuities) may be expressed as the product of two simpler locally risk-minimizing hedging strategies: one for a purely financial claim, the other for a traditional (i.e. non-equity-linked) life-insurance claim.Finally, we also show, under general assumptions, that the minimal measure for the combined market is the product of the minimal measure for the financial market and the physical measure for the mortality.     

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.     

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.     

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.     

A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a Lévy process financial market

In [Riesner, M., 2006. Hedging life insurance contracts in a Lévy process financial market. Insurance Math. Econom. 38, 599–608] the (locally) risk-minimizing hedging strategy for unit-linked life insurance contracts is determined in an incomplete financial market driven by a Lévy process. The considered risky asset is not a martingale under the original measure and therefore, a change of measure to the minimal martingale measure is performed.The goal of this paper is to show that the risk-minimizing hedging strategy under the new martingale measure which is found in the paper cited above is not the locally risk-minimizing strategy under the original measure. Finally, the real locally risk-minimizing strategy is derived and a relationship between the number of risky assets held in the proposed portfolio cited in the above-mentioned paper and the one proposed here is given.     

On hedging in finite security markets

A market is considered where trading can take place only at discrete time points, the trading frequency cannot grow without bound, and the number of states of nature is finite. The main objectives of the paper are to show that the market can be completed also with highly correlated risky assets, and to describe an efficient algorithm to compute a self-financing hedging strategy. The algorithm consists off-line of a backwards recursion and on-line of the solution, in each period, of a system of linear equations; it is a consequence of a proof where, using a well-known mathematical property, it is shown that uniqueness of the martingale measure implies completeness also in our setting. The significance of ‘multistate’ models versus the familiar binomial model is discussed and it is shown how the evolution of prices of the (correlated) risky assets can be chosen so that a given probability measure is already the unique equivalent martingale measure.     

Coupling and option price comparisons in a jump-diffusion model

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures.

Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.     

Pricing Forward Starting Call Options under a Markov-Modulated Jump Diffusion Process

The pricing problem of forward starting call options under a Markov-modulated jump diffusion process is studied. Under the assumption that the dynamics of risky asset follows a Markov-modulated jump diffusion process, the explicit analytical formula of forward starting call options is obtained by the change of measure and no arbitrage pricing theory. Moreover, the numerical results of option value are provided by the Monte Carlo method, and the value of forward starting call options is compared when the risky asset satisfies different financial models.     

Unit-linked life insurance policies: Optimal hedging in partially observable market models

In this paper we investigate the hedging problem of a unit-linked life insurance contract via the local risk-minimization approach, when the insurer has a restricted information on the market. In particular, we consider an endowment insurance contract, that is a combination of a term insurance policy and a pure endowment, whose final value depends on the trend of a stock market where the premia the policyholder pays are invested. To allow for mutual dependence between the financial and the insurance markets, we use the progressive enlargement of filtration approach. We assume that the stock price process dynamics depends on an exogenous unobservable stochastic factor that also influences the mortality rate of the policyholder. We characterize the optimal hedging strategy in terms of the integrand in the Galtchouk–Kunita–Watanabe decomposition of the insurance claim with respect to the minimal martingale measure and the available information flow. We provide an explicit formula by means of predictable projection of the corresponding hedging strategy under full information with respect to the natural filtration of the risky asset price and the minimal martingale measure. Finally, we discuss applications in a Markovian setting via filtering.